Vector Algebra Questions

We provide vector algebra practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on vector algebra skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.

Vector Algebra Questions

List of vector algebra Questions

Question NoQuestionsClass
1ff ( bar{a}, bar{b}, bar{c}, bar{d} ) are the position vectors of four points ( mathbf{A}, mathbf{B}, mathbf{C}, mathbf{D} ) in a plane such that ( |overline{boldsymbol{a}}-overline{boldsymbol{d}}|=|overline{boldsymbol{b}}-overline{boldsymbol{d}}|=|overline{boldsymbol{c}}-overline{boldsymbol{d}}| ) then the
point ( mathbf{D} ) is the ( cdots ) of ( Delta boldsymbol{A} boldsymbol{B} boldsymbol{C} )
A. centroid
B. circumcentre
c. orthocentre
D. incentre
12
2Given that ( A+B+C=0 . ) Out of three
vectors, two are equal in magnitude and the magnitude of third vector is ( sqrt{2} ) times that of either of the two having equal magnitude. Then, the angles between the vectors are given by.
A . ( 30^{circ}, 60^{circ}, 90^{circ} )
в. ( 45^{circ}, 45^{circ} ), ( 90^{circ} )
c. ( 45^{circ}, 60^{circ}, 90^{circ} )
D. ( 90^{circ}, 135^{circ}, 135^{circ} )
12
3The area of parallegram represented by the vectors. ( overrightarrow{boldsymbol{A}}=boldsymbol{2} hat{boldsymbol{i}}+boldsymbol{3} hat{boldsymbol{j}} ) and ( overrightarrow{boldsymbol{B}}=hat{boldsymbol{i}}+ )
( 4 hat{j} ) is :-
A. 14 units
B. 7.5 units
c. 10 units
D. 5 units
12
4The dot product of a vector with vectors ( hat{mathbf{i}}+hat{mathbf{j}}-mathbf{3} hat{mathbf{k}}, hat{mathbf{i}}+mathbf{3} hat{mathbf{j}}-mathbf{2} hat{mathbf{k}}, mathbf{2} hat{mathbf{i}}+hat{mathbf{j}}+mathbf{4} k ) are
0,5 and 8 respectively. Find the vector
12
5Motion of the aeroplane is an example
of motion.
A. zero dimensional
B. one dimensional
c. two dimensional
D. three dimensional
12
6A vector ( vec{a} ) has components 2 p and 1 with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect
to the new system, ( vec{a} ) components ( boldsymbol{p}+mathbf{1} ) and ( 1, ) then?
A. ( p=0 )
B. ( p=1 ) or ( p=-frac{1}{3} )
c. ( _{p}=-1 ) or ( p=frac{1}{3} )
D. ( p=1 ) or ( p=-1 )
12
7Let ( A B C D ) be a parallelogram and let ( L ) and ( mathrm{M} ) be the midpoints of the sides ( B C ) and ( C D ) respectively. Then ( overrightarrow{A L}+ ) ( boldsymbol{A} overrightarrow{boldsymbol{M}}= )
( A cdot overrightarrow{A C} )
В ( cdot frac{2}{3} overrightarrow{A C} )
c. ( frac{3}{2} overrightarrow{A C} )
D. ( 2 A C )
12
8Find the vector joining the points ( P(2,3,0) ) and ( Q(-1,-2,-4) ) directed
from ( boldsymbol{P} ) to ( boldsymbol{Q} )
12
989. Let a=i-,=i+i+and ċ be a vector such that
āxc+b=7 and a.c = 4, then |ēl is equal to:
[JEE M 2019-9 Jan (M)]
(a) (b) 9 (c) 8 (0) 17
12
10f ( vec{a}=vec{b}+vec{c}, ) then is true that ( |vec{a}|=|vec{b}|+ )
( |vec{c}| ) ? Justify your answer.
12
11[2005]
39. Let a, b and c be distinct non- negative numbers. If th
vectors ai + aj +ck, î+k and ci + cj +bk lie in a plan
then cis
(a) the Geometric Mean of a and b
(b) the Arithmetic Mean of a and b
(c) equal to zero
(d) the Harmonic Mean of a and b
12
12If ( I ) is the centre of a circle inscribed in
a triangle ( A B C, ) then ( |overline{B C}| overline{I A}+ ) ( |overline{C A}| overline{I B}+|overline{A B}| overline{I C} ) is
A. ( overline{0} )
в. ( overline{I A}+overline{I B}+overline{I C} )
c. ( frac{overline{I A}+overline{I B}+overline{I C}}{3} )
D. ( frac{overline{I A}+overline{I B}+overline{I C}}{2} )
12
13Assertion
( operatorname{Let} boldsymbol{A}(overline{boldsymbol{a}}) )
( & B(bar{b}) ) be two points in
space. Let ( boldsymbol{P}(bar{r}) ) be a variable point
which moves in space such that ( overline{boldsymbol{P} boldsymbol{A}} ) ( P B leq 0 ) such a variable points traces a
three-dimensional figure whose volume is given by ( frac{pi}{6}left{bar{a}^{2}+bar{b}^{2}-2 bar{a} cdot bar{b}right} )
( |bar{a}-bar{b}| )
Reason
Diameter of sphere subtends acute angle at any point inside the sphere ( & ) its volume is given by ( frac{4}{3} pi r^{3} )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
14If ( bar{a}, bar{b}, bar{c} ) are the position vectors of the points ( A, B, C ) respectively and ( 3 bar{a}+ ) ( 4 bar{b}-7 bar{c}=overline{0}, ) find the ratio in which
point ( B ) divides the segment ( A C )
12
15Find ( |vec{x}|, ) if for a unit vector ( vec{a},(vec{x}-vec{a}) )
( (vec{x}+vec{a})=12 )
Answer required
12
16If ( vec{b} ) and ( vec{c} ) are the position vectors of the points ( mathrm{B} ) and ( mathrm{C} ) respectively, then the position vector of the point D such that ( overrightarrow{B D}=4 overrightarrow{B C} ) is
A ( cdot 4(vec{c}-vec{b}) )
в. ( -4(vec{c}-vec{b}) )
c. ( 4 vec{c}-3 vec{b} )
D ( cdot 4 vec{c}+3 vec{b} )
12
17The vector ( (cos alpha cos beta) vec{i}+ )
( (cos alpha sin beta) vec{j}+sin alpha vec{k} ) is
A. Null vector
B. unit vector
C . parallel to ( (vec{i}+vec{j}+vec{k}) )
D. a vector parallel to ( (2 vec{i}+vec{j}-vec{k}) )
12
18Given that the vectors ( bar{a} ) and ( bar{b} ) are non-
collinear, the value of ( x ) and ( y ) for which
the vector quantity ( 2 bar{u}-bar{v}=bar{w} ) holds true if ( bar{u}=x bar{a}+2 y bar{b}, bar{v}=-2 y bar{a}+ )
( mathbf{3} boldsymbol{x} overline{boldsymbol{b}}, overline{boldsymbol{w}}=mathbf{4} overline{boldsymbol{a}}-mathbf{2} overline{boldsymbol{b}} ) are
A ( cdot x=frac{4}{7}, y=frac{6}{7} )
в. ( x=frac{10}{7}, y=frac{4}{7} )
c. ( x=frac{8}{7}, y=frac{2}{7} )
D. ( x=2, y=3 )
12
19The shortest distance between the lines
( frac{x-3}{3}=frac{y-8}{-1}=frac{z-3}{1} ) and
( frac{boldsymbol{x}+mathbf{3}}{mathbf{- 3}}=frac{boldsymbol{y}+mathbf{7}}{mathbf{2}}=frac{boldsymbol{z}-mathbf{6}}{mathbf{4}} ) is:
A ( cdot 2 sqrt{30} )
B. ( frac{7}{2} sqrt{30} )
( c .3 )
D. ( 3 sqrt{30} )
12
20If the vector ( -hat{mathbf{i}}+hat{boldsymbol{j}}-hat{boldsymbol{k}} ) bisects the
angles between the vector ( hat{c} ) and the vector ( 3 hat{i}+4 hat{j}, ) then the unit vector in
the direction of ( hat{boldsymbol{c}} )
( mathbf{A} cdot 11 hat{i}+10 hat{j}+2 hat{k} )
B . ( -(11 hat{i}+10 hat{j}+2 hat{k}) )
c. ( -frac{1}{15}(11 hat{i}+10 hat{j}+2 hat{k}) )
D. ( frac{1}{15}(11 hat{i}+10 hat{j}+2 hat{k}) )
12
2131. Let ā, ū and ē be non-zero vectors S
non-zero vectors such that
(āxb)
NA
ā. If O is the acute angle between
the vectors 7 and 7, then sin
equals
[2004]
(d)
12
22The position vectors of the points ( A, B, C ) are respectively (1,1,1)( ;(1,-1,2) ;(0,2,-1) )
Find a unit vector parallel to the plane determined by ABC & perpendicular to the vector (1,0,1)
A ( cdot pm frac{1}{3 sqrt{3}}(i+5 j-k) )
B . ( pm(i+5 j-k) )
( mathbf{c} cdot pm(-i-5 j-k) )
D ( : pm frac{1}{3 sqrt{3}}(-i-5 j-k) )
12
23Force ( 3 mathrm{N}, 4 mathrm{N} ) and ( 12 mathrm{N} ) act at a point in mutually perpendicular directions. The magnitutde of the resultant force is :-
( A cdot 19 N )
B. 13 N
c. ( 11 mathrm{N} )
D. 5 N
12
2418. Given: Ā = A cos 0 î + A sin oĝ. A vector B, which is
perpendicular to Ā, is given by
a. B cos o î – B sin eſ b. Bsin o î – Bcose ì
c. Bcos o î + Bsin d. Bsin o î + Bcose
12
25Let ( boldsymbol{u}=(mathbf{1},-mathbf{2}, mathbf{3}) ) and ( boldsymbol{v}=(mathbf{4}, mathbf{5},-mathbf{1}) )
find ( boldsymbol{u} cdot boldsymbol{v} )
12
26Define negative of a vector.12
27Given that ( 2 a-3 b=c ;-3 a+5 b=d )
expressing ( a ) and ( b ) as linear
combinations of ( c ) and ( d ) we get ( a= )
( k c+m d ) and ( b=h c+r d . ) Find ( k+ )
( boldsymbol{m}+boldsymbol{h}+boldsymbol{r} ? )
12
28Two different vectors having same
magnitude:
( mathbf{A} cdot hat{i}+2 hat{j}+3 widehat{k}, 3 hat{i}+2 hat{j}+widehat{k} )
в. ( hat{i}+widehat{j}-widehat{k}, 2 hat{i}-widehat{j}-bar{k} )
c. ( hat{i}-2 hat{j}+2 widehat{k}, hat{i}+widehat{j}+widehat{k} )
D. None
12
29If ( vec{a}, vec{b}, vec{c} ) are unit vectors, then ( |vec{a}-vec{b}|^{2}+ )
( |vec{b}-vec{c}|^{2}+|vec{c}-vec{a}|^{2} ) does not exceed :
A . 4
B. 9
( c cdot 8 )
D.
12
30If the position vectors of the points ( boldsymbol{A}, boldsymbol{B}, boldsymbol{C}, boldsymbol{D} operatorname{are}(boldsymbol{0}, boldsymbol{2}, boldsymbol{1}),(boldsymbol{3}, boldsymbol{1}, boldsymbol{1}) )
(-5,3,2),(2,4,1) respectively and if ( boldsymbol{P A}+boldsymbol{P B}+boldsymbol{P C}+boldsymbol{P D}=mathbf{0} ) then the
position vector of ( mathrm{P} ) is
A ( cdotleft(0, frac{5}{2}, frac{5}{4}right) )
в. ( left(frac{5}{2}, frac{5}{2}, frac{5}{4}right) )
c. ( left(frac{5}{2}, 0, frac{5}{4}right) )
D ( left(frac{5}{2}, frac{5}{4}, 0right) )
12
31Consider the set of eight vectors ( V= ) ( {a hat{i}+b hat{j}+c hat{k} ; a, b, c in{-1,1}} . ) Three
non-coplanar vectors can be chosen
from ( V ) in ( 2^{p} ) ways. Then ( p ) is
( A cdot 2 )
B. 3
( c cdot 4 )
D. 5
12
32Find the unit vector perpendicular to the plane of given vectors. ( vec{P}=i-2 j+hat{k} )
( vec{Q}=2 i+j-hat{k} )
12
33Show that the points whose position vectors are as given below are collinear:
( mathbf{3} hat{mathbf{i}}-mathbf{2} hat{mathbf{j}}+mathbf{4} hat{boldsymbol{k}}, hat{mathbf{i}}+hat{boldsymbol{j}}+hat{boldsymbol{k}} ) and ( -hat{mathbf{i}}+mathbf{4} hat{mathbf{j}}- )
( mathbf{2} hat{boldsymbol{k}} )
12
34A force ( overrightarrow{boldsymbol{F}}=mathbf{3} hat{mathbf{i}}+mathbf{4} hat{mathbf{j}}-mathbf{3} hat{boldsymbol{k}} ) is applied at
the point ( P, ) whose position vector is ( vec{r}=2 hat{i}-2 hat{j}-3 hat{k} . ) What is the
magnitude of the moment of the force about the origin?
A . 23 units
B. 19 units
c. 18 units
D. 21 units
12
35If ( vec{a} ) and ( vec{b} ) are not perpendicular and ( vec{c} ) and ( vec{d} ) are such that ( vec{b} times vec{c}=vec{b} times vec{a} ) and ( vec{a} cdot vec{d}=0 ) then ( bar{d} )
is equal to
A ( cdot vec{c}-left(frac{vec{a} cdot vec{c}}{vec{a} cdot vec{b}}right) cdot vec{b} )
B ( cdot frac{vec{a}}{p}-frac{(vec{c} vec{c} vec{a})}{p^{2}} vec{b} )
C ( cdot frac{vec{b}}{p}-frac{(vec{a} cdot vec{alpha}) bar{c}}{bar{p}^{2}} bar{c} )
D. ( frac{vec{p}}{p^{2}}-frac{(vec{b} cdot vec{c})}{vec{p}} vec{a} )
12
36If ( hat{i}+p hat{j}+hat{k}, 2 hat{i}+3 hat{j}+q hat{k} ) are parallel
vectors then ( (p, q)=? )
A ( cdotleft(2, frac{3}{2}right) )
в. (2,2)
c. ( left(frac{3}{2}, frac{3}{2}right) )
D. ( left(frac{3}{2}, 2right) )
12
37A force ( overrightarrow{boldsymbol{F}}=mathbf{3} hat{boldsymbol{i}}+mathbf{2} widehat{boldsymbol{j}}-boldsymbol{4} widehat{boldsymbol{k}} ) is applied at
the point ( (1,-1,2) . ) What is the moment of the force about the point (2,-1,3)( ? )
( mathbf{A} cdot hat{i}+4 widehat{j}+4 widehat{k} )
B. ( 2 hat{i}+widehat{j}+2 widehat{k} )
c. ( 2 hat{i}-7 hat{j}-2 widehat{k} )
D . ( 2 hat{i}+4 widehat{j}-widehat{k} )
12
38The vector component of the vector ( hat{mathbf{i}}+ ) ( hat{boldsymbol{j}}+hat{boldsymbol{k}} ) perpendicular to the vector ( boldsymbol{2} hat{boldsymbol{i}}- )
( hat{boldsymbol{j}}+boldsymbol{2} hat{boldsymbol{k}} ) is
A ( cdot frac{1}{3}(hat{i}+4 hat{j}+hat{k}) )
B ( cdot frac{1}{4}(2 hat{i}-2 hat{j}-3 hat{k}) )
c ( cdot frac{1}{2}(2 hat{i}+2 hat{j}-hat{k}) )
D ( cdot frac{1}{4}(2 hat{i}+2 hat{j}-hat{k}) )
12
39For given vectors, ( vec{a}=2 hat{i}-widehat{j}+2 widehat{k} ) and ( vec{b}=-hat{i}+widehat{j}-widehat{k}, ) find a unit vector in
the direction of the vector ( vec{a}+vec{b} )
12
40If ( vec{r}=3 vec{i}+2 vec{j}-5 vec{k}, vec{a}=2 vec{i}-vec{j}+vec{k} )
( vec{b}=vec{i}+3 vec{j}-2 vec{k} ) and ( vec{c}=-2 vec{i}+vec{j}-3 vec{k} )
such that ( vec{r}=lambda vec{a}+mu vec{b}+nu vec{c} ) then
A ( cdot mu, frac{lambda}{2}, nu ) are in ( A P )
B. ( lambda, mu, nu ) are in ( A P )
c. ( lambda, mu, nu ) are in ( H P )
D. ( mu, lambda, nu ) are in ( G P )
12
41If ( hat{boldsymbol{i}}-boldsymbol{3} hat{boldsymbol{j}}+boldsymbol{5} hat{boldsymbol{k}} ) bisects the angle between
( hat{a} ) and ( -hat{i}+2 hat{j}+2 hat{k}, ) where ( hat{a} ) is a unit
vector, then
A ( cdot hat{a}=frac{1}{105}(41 hat{i}+88 hat{j}-40 hat{k}) )
B・ ( hat{a}=frac{1}{105}(41 hat{i}+88 hat{j}+40 hat{k}) )
C ( quad hat{a}=frac{1}{105}(-41 hat{i}+88 hat{j}-40 hat{k}) )
D・ ( hat{a}=frac{1}{105}(41 hat{i}-88 hat{j}-40 hat{k}) )
12
42The vector ( overrightarrow{O A} ) where ( O ) is origin is ( operatorname{given} ) by ( overrightarrow{O A}=2 hat{i}+2 hat{j} . ) Now it is
rotated by ( 45^{circ} ) anticlockwise about ( O )
What will be the new vector?
A ( cdot 2 sqrt{2} hat{j} )
is ( sqrt{2} ) is ( sqrt{2} )
в. 2 ई
( c cdot 2 hat{imath} )
D. ( 2 sqrt{2} hat{imath} )
12
43In a triangle ( A B C, ) right angled at the
vertex ( A, ) if the position vectors ( A, B ) and ( C ) are respectively ( 3 tilde{i}+tilde{j}-tilde{k},-tilde{i}+ ) ( mathbf{3} tilde{j}+p tilde{k} ) and ( 5 tilde{i}+q tilde{j}-4 tilde{k}, ) then the
point ( (p, q) ) lies on a line
A. makeing an obtuse angle with the positive direction of ( x- ) axis
B. Parallel to ( x- ) axis
c. Parallel to ( y- ) axis
D. making an acute angle with the positive direction of ( x-a x i s )
12
44Let ( mathrm{ABC} ) be a triangle and let ( mathrm{S} ) be its circumcentre and ( mathrm{O} ) be its orthocentre.
The ( overline{mathbf{S A}}+overline{mathbf{S B}}+overline{mathbf{S C}}= )
A. 450
в. 3ज्ठ
c. 2 So
D. So
12
45Let ( overrightarrow{boldsymbol{a}}=hat{boldsymbol{i}}+2 hat{boldsymbol{j}} ) and ( overrightarrow{boldsymbol{b}}=boldsymbol{2} hat{boldsymbol{i}}+hat{boldsymbol{j}} )
( |vec{a}|=|vec{b}| ) ? Are the vectors ( vec{a} ) and ( vec{b} )
equal?
12
46Three vectors ( vec{A}, vec{B} ) and ( vec{C} ) are as shown in figure. If magnitude of ( vec{A} ) is ( 4 mathrm{m} ) and ( vec{A}+vec{B}+vec{C}=0, ) then the magnitude of ( vec{B} ) and ( vec{C} ) are respectively
A. ( 4 m, 4 m )
в. ( 4 sqrt{3} m, 4 sqrt{3} m )
c. ( 4 m, 4 sqrt{3} m )
D. ( 4 sqrt{3} m, 4 m )
12
47The minimum number of unequal
forces in a plane that can keep a particle in equilibrium is
( A cdot 4 )
B. 2
( c cdot 3 )
D.
12
4866.
Statement-1: The point A(1,0, 7)) is the mirror image of the
y-1
Z-2
point B(1,6, 3) in the line :
=
[2011]
N
Statement-2: The line
= v=1=252 bisects the line
1 2 3
segment joining A(1,0,7) and B(1,6, 3).
(a) Statement-1 is true, Statement-2 is true; Statement-2 is
not a correct explanation for Statement-1.
Statement-1 is true, Statement-2 is false.
) Statement-1 is false, Statement-2 is true.
Statement-1 is true, Statement-2 is true; Statement-2 is
a correct explanation for Statement-1.
12
49Express the vector ( vec{a}=5 hat{i}+5 hat{k} ) as sum
of two vector such that one is parallel to the ( vec{b}=3 hat{i}+hat{k} ) and other is
perpendicular to ( vec{b} )
12
50If ( a=2 hat{i}-hat{j}-m hat{k} ) and ( b=frac{4}{7} hat{i}-frac{2}{7} hat{j}+ )
( 2 hat{k} ) are collinear, then the value of ( m ) is
equal to
A . -7
в. – –
( c cdot 2 )
D.
E. -2
12
51A certain vector in the ( x y ) plane has an
( x ) component of ( 4 m ) and a ( y ) component
of ( 10 m . ) It is then rotated in the ( x y )
plane so that its ( x ) -component is doubled. Then its new ( y ) component is (approximately):
A. ( 20 m )
B. ( 7.2 m )
c. ( 5.0 m )
D. ( 4.5 m )
12
5210
15.
114, Bauu Cara voctors such that|Bl= |C. Prove that
(ATB) X (A + C] (
B C) (B+C) =0. (1997 – 5 Marks)
12
53The angle between the vectors ( vec{a} ) and ( vec{b} ) such that ( |vec{a}|=|vec{b}|=sqrt{2} ) and ( vec{a} cdot vec{b}=1 )
is
A ( cdot frac{pi}{2} )
в. ( frac{pi}{3} )
( c cdot frac{pi}{4} )
D.
12
54Point (4,0) lies on
A . ( overrightarrow{X O} )
в. ( overrightarrow{Y O} )
( c cdot O vec{X} )
D. ( overrightarrow{O Y} )
12
55Find the angle between the pair of lines
given by ( vec{r}=3 hat{i}+2 hat{j}-4 hat{k}+lambda(hat{i}+2 hat{j}+2 hat{k}) )
and ( vec{r}=5 hat{i}-2 hat{j}+mu(3 hat{i}+2 hat{j}+6 hat{k}) )
12
56If ( a, b, c ) are non-coplanar vectors, then
which of the following points are collinear whose position vectors are given by :
This question has multiple correct options
A. ( a-2 b+3 c, 2 a+3 b-4 c,-7 b+10 c )
в. ( 3 a-4 b+3 c,-4 a+5 b-6 c, 4 a-7 b+6 c )
c. ( 2 a+5 b-4 c, a+4 b-3 c, 4 a+7 b-6 c )
D. ( 6 a-b-2 c, 2 a+3 b+2 c,-a-9 b+7 c )
12
57Three forces of magnitude ( 6 N, 6 N ) and
( sqrt{72} N ) act at a corner of a cube along
three sides as shown in the figures.The
resultant of these forces is:
A . 12 N along ob
B. 18 N along of
c. ( 18 N ) along ( 0 c )
D. 12 N along of
12
5826. If A is perpendicular to B, then
a. Äx B=0 b. A LA + B] = 1²
C. Ã. B = AB d. Ä.[A + B]= A² + AB
12
59Three forces ( overrightarrow{boldsymbol{A}}={hat{boldsymbol{i}}+hat{boldsymbol{j}}+hat{boldsymbol{k}}}, overrightarrow{boldsymbol{B}}= )
( {2 hat{i}-hat{j}-3 hat{k}} ) and ( vec{C} ) acting on a body
To keep it in the equilibrium, the value of ( vec{C} ) is:
( mathbf{A} cdot-{3 hat{i}-2 hat{k}} )
B . ( -{2 hat{i}-3 hat{k}} )
c. ( 3 hat{i}+4 hat{j} )
D. ( 2 hat{i}-3 hat{k} )
12
6065.
The vectors ã and ū are not perpendicular and c and d
are two vectors satisfying Ō xo = 5 xà and ā.d = 0. Then
the vector à is equal to
[2011]
(b)
b +
12
61The linear relation between the vectors
( boldsymbol{a}+boldsymbol{3} overline{boldsymbol{b}}+boldsymbol{4} boldsymbol{c}, overline{boldsymbol{a}}-boldsymbol{2} overline{boldsymbol{b}}+boldsymbol{3} overline{boldsymbol{c}}, overline{boldsymbol{a}}+mathbf{5} overline{boldsymbol{b}}-boldsymbol{2} overline{boldsymbol{c}} )
( 6 bar{a}+14 bar{b}+4 bar{c} ) is
( mathbf{A} cdot(bar{a}+3 bar{b}+4 bar{c})+2(bar{a}-2 bar{b}+3 bar{c})+ )
[
text { 2 }(bar{a}+5 bar{b}-2 bar{c})-1(6 bar{a}+14 bar{b}+4 bar{c})=overline{0}
]
B . ( 1(bar{a}+3 bar{b}+4 bar{c})+2(bar{a}-2 bar{b}-3 bar{c})+ )
[
text { 3 }(bar{a}+5 bar{b}-2 bar{c})-2(7 bar{a}+14 bar{b}+4 bar{c})=overline{0}
]
C ( .1(bar{a}+3 bar{b}+4 bar{c})+2(bar{a}-2 bar{b}+3 bar{c})+ )
[
text { 3 }(bar{a}+5 bar{b}-2 bar{c})-1(6 bar{a}+14 bar{b}+4 bar{c})=overline{0}
]
D. ( (bar{a}+3 bar{b}+4 bar{c})+(bar{a}-2 bar{b}+3 bar{c})+ )
[
(bar{a}+5 bar{b}-2 bar{c})-(6 bar{a}+14 bar{b}+4 bar{c})=overline{0}
]
12
62The incentre of the triangle formed by the points ( hat{boldsymbol{i}}+hat{boldsymbol{j}}+hat{boldsymbol{k}}, boldsymbol{4} hat{boldsymbol{i}}+hat{boldsymbol{j}}+hat{boldsymbol{k}}, ) and
( 4 hat{i}+5 hat{j}+hat{k} ) is
A ( cdot frac{hat{i}+hat{j}+hat{k}}{3} )
B. ( hat{i}+2 hat{j}+3 hat{k} )
c. ( 3 hat{i}+2 hat{j}+hat{k} )
D. ( hat{i}+hat{j}+hat{k} )
12
63ung body is given by v=mXr,
11. The linear velocity of a rotating body is given by
where @ is the angular velocity and r is the radius vector.
The angular velocity of a body is ū=i-2j+2k and the
radius vector r=4j-3k, then lvl is
(a) 729 units (b) 731 units
(c) 37 units (d) 41 units
12
64( A B C D E ) is a pentagon. the resultant of the forces ( overrightarrow{A B}+overrightarrow{B C}+overrightarrow{D C}+overrightarrow{E D}+ )
( overrightarrow{A E}=k overrightarrow{A C}, ) then find the value of ( k )
12
65The ( x ) -y plane divides the line joining the points (-1,3,4) and (2,-5,6)
A. internally in the ratio 2: 3
B. externally in the ratio 2: 3
c. internally in the ratio 3: 2
D. externally in the ratio 3: 2
12
66Let ( vec{a}=hat{j}-hat{k} ) and ( vec{c}=hat{i}-hat{j}-hat{k}, ) then
the vector ( overrightarrow{boldsymbol{b}} ) satisfying ( overrightarrow{boldsymbol{a}} times overrightarrow{boldsymbol{b}}+overrightarrow{boldsymbol{c}}=mathbf{0} )
and ( vec{a} cdot vec{b}=3 ) is
( mathbf{A} cdot-hat{i}+hat{j}-2 hat{k} )
B . ( 2 hat{i}-hat{j}+2 hat{k} )
c. ( hat{i}-hat{j}-2 hat{k} )
D. ( hat{i}+hat{j}-2 hat{k} )
12
67If ( bar{e}=bar{l} bar{i}+m bar{j}+n bar{k} ) is a unit vector, the
maximum value of ( l m+m n+n l ) is
A ( cdot-frac{1}{2} )
B. 0
c. 1
D. ( frac{3}{2} )
12
68Unit vectors ( hat{P} ) and ( hat{Q} ) are inclined at an angle ( theta ), Prove that ( |hat{P}-hat{Q}|=2 sin (theta / 2) )
is?
12
69Assertion STATEMENT-1: If ( vec{a}, vec{b}, vec{c} ) are non coplanar unit vectors equally inclined to one
another at an angle ( theta, ) then there will
exist three scalar ( p, q, r ) such that ( vec{a} times ) ( overrightarrow{boldsymbol{b}}+overrightarrow{boldsymbol{b}} times overrightarrow{boldsymbol{c}}=boldsymbol{p} overrightarrow{boldsymbol{a}}+boldsymbol{q} overrightarrow{boldsymbol{b}}+boldsymbol{r} overrightarrow{boldsymbol{c}} )
Reason STATEMENT-2 : ( overrightarrow{boldsymbol{a}} times overrightarrow{boldsymbol{b}} ) is a vector
perpendicular to plane containing ( vec{a} ) and ( vec{b} ) and ( vec{b} times vec{c} ) is a vector perpendicular to plane containing ( overrightarrow{boldsymbol{b}} ) and ( overrightarrow{boldsymbol{c}} ) so ( overrightarrow{boldsymbol{a}} times overrightarrow{boldsymbol{b}}+overrightarrow{boldsymbol{b}} times )
( vec{c} ) will be non-zero only if ( vec{a}, vec{b}, vec{c} ) are
coplanar.
A. Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-
B. Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
c. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True
12
70Find the angle between two vectors ( a ) and ( b ) if ( |a+b|=|a-b| )12
71( |vec{a}+vec{b}|^{2}-|vec{a}-vec{b}|^{2} ) is equal to
A . ( 4 vec{a} . vec{b} )
B.
c. ( 4|vec{a} cdot vec{b}| )
D. None of these
12
722. You are given vectors Ā = 5î – 6.5j and B = 10î – 7j.
A third vector © lies in the x-y plane. Vector C is
perpendicular to vector , and the scalar product of Ĉ with
B is 15. From this information, find the components of C.
Tu
nontornī and R houe moonitude
200 and
12
73Find a unit vector perpendicular to each of the vectors ( vec{a}+vec{b} ) and ( vec{a}-vec{b} ) where ( vec{a}=3 hat{i}+2 hat{j}+2 hat{k} ) and ( vec{b}=hat{i}+2 hat{j}-2 hat{k} )12
74Let a, b, c be distinct non-negative numbers. If the vectors
ai + aj +ck, i +and cî +ci+bk lie in a plane, then cis
(1993 – 1 Marks)
(a) the Arithmetic Mean of a and b
(b) the Geometric Mean of a and b
(c) the harmonic Mean of a and b
(d) equal to zero
12
75If the points ( A ) and ( B ) are (1,2,-1) and
(2,1,-1) respectively, then ( overrightarrow{A B} ) is
A. ( hat{i}+hat{j} )
в. ( hat{i}-hat{j} )
c. ( 2 hat{i}+hat{j}-hat{k} )
D. ( hat{i}+hat{j}+hat{k} )
12
7635. Let P (3, 2, 6) be a point in space and Q be a point on the line
= (i – j + 2k) + ul-3î + j +5k)
Then the value of u for which the vector PQ is parallel to
the plane x – 4y + 3z=1 is
(2009)
12
77If ( vec{u}, vec{v} ) and ( vec{w} ) are three non-coplanar vectors, then ( (overrightarrow{boldsymbol{u}}+overrightarrow{boldsymbol{v}}-overrightarrow{boldsymbol{w}}) cdot(overrightarrow{boldsymbol{u}}-overrightarrow{boldsymbol{v}}) times )
( (vec{v}-vec{w}) ) equals
A ( . vec{u} .(vec{v} times vec{w}) )
B.
c. ( 2 vec{u} cdot(vec{v} times vec{w}) )
D. ( vec{u} .(vec{w} times vec{v}) )
12
78A girl walks ( 4 k m ) towards West, then
she walks ( 3 k m ) in a direction ( 30^{circ} ) East
of North and stops. Then, the girl’s displacement from her initial point of departure is
A ( -frac{5}{2} hat{i}+frac{3 sqrt{3}}{2} )
B. ( frac{1}{2} hat{i}+frac{sqrt{3}}{2} hat{j} )
c. ( -frac{1}{2} hat{i}+frac{3 sqrt{3}}{2} )
D. None of these
12
79Suppose ( vec{a} ) is a vector of magnitude 4.5 unit due north. What is the vector ( -4 vec{a} ) ?12
80A man starts from ( O ) moves 500 m turns
by ( 60^{circ} ) and moves ( 500 m ) again turns by
( 60^{circ} ) and moves ( 500 m ) and so on. Find
the displacement after
( (i) 5 ) th turn ( ,(i i) ) 3rd turn.
A . ( -500 m, 1000 m )
в. ( 500 m, 500 sqrt{3} m )
c. ( 1000 m, 500 sqrt{3} m )
D. none of these
12
8143. A vector A when added to the vector B = 3i + 4j yields
a resultant vector that is in the positive y-direction and has
a magnitude equal to that of B. Find the magnitude of A.
a. V10 b. 10 c. 5 d. 115
12
82Assertion
( (A): overline{G A}+overline{G B}+overline{G C}=overline{0} ) where ( G ) is
the centroid of triangle ( A B C ).
Reason
( (R): overline{A B}= ) P.V of ( B- ) P.V of ( A )
A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
B. Both Assertion & Reason are individually true but Reason is not the correct explanation of Assertion
c. Assertion is true but Reason is false
D. Assertion is false but Reason is true
12
83If ( vec{a}, vec{b}, vec{c}, vec{d}, vec{e}, vec{f} ) are position vectors of 6 points ( A, B, C, D, E & F ) respectively such that ( mathbf{3} overrightarrow{boldsymbol{a}}+mathbf{4} overrightarrow{boldsymbol{b}}=mathbf{6} overrightarrow{boldsymbol{c}}+overrightarrow{boldsymbol{d}}=mathbf{4} overrightarrow{boldsymbol{e}}+mathbf{3} overrightarrow{boldsymbol{f}}=overrightarrow{boldsymbol{x}} )
then
This question has multiple correct options
( mathbf{A} cdot overline{A B} ) is parallel to ( overline{C D} )
B. line AB, CD and EF are concurrent
c. ( vec{x} ) ( overline{7}^{text {is position vector of the point dividing CD in ratio } 1} )
6
D. A, B, C, D, E & F are coplanar
12
84If a quadrilateral ( A B C D ) is such that
( A B=b, A D=d ) and ( A C=m b+p d )
( (m+p geq 1), ) then the area of the
quadrilateral is ( k(p+m)|b times d|, ) where
( k ) is equal to
A ( cdot frac{1}{4} )
B. ( frac{1}{8} )
( c cdot frac{1}{2} )
D. None of these
12
85Vector ( vec{A} ) is ( 2 c m ) long and is ( 60^{circ} ) above
the ( x ) -axis in the first quadrant. Vector
( vec{B} ) is ( 2 c m ) long and is ( 60^{circ} ) below the ( x ) axis in the fourth quadrant. The sum ( vec{A}+vec{B} ) is a vector of magnitude
A. ( 2 c m ) along positive y-axis
B. 2cm along positive x-axis
c. ( 2 c m ) along negative y-axis
D. ( 2 c m ) along negative x-axis
12
86Let ( bar{alpha}, bar{beta}, bar{gamma} ) be three vectors such that
( overline{boldsymbol{alpha}} cdot(overline{boldsymbol{beta}}+bar{gamma})+overline{boldsymbol{beta}} cdot(bar{gamma}+overline{boldsymbol{alpha}})+bar{gamma} )
( (bar{alpha}+bar{beta})=mathbf{0} ) and ( |overline{boldsymbol{alpha}}|=sqrt{mathbf{3}},|bar{beta}|=mathbf{2} ) and
( |bar{gamma}|=mathbf{3} ) then ( |overline{boldsymbol{alpha}}+overline{boldsymbol{beta}}+bar{gamma}| ) is
A ( . sqrt{17} )
B. 4
( c .5 )
D. ( sqrt{15} )
12
87Find the sum of the following vectors:
( overrightarrow{boldsymbol{a}}=hat{boldsymbol{i}}-mathbf{2} hat{boldsymbol{j}}, overrightarrow{boldsymbol{b}}=mathbf{2} hat{boldsymbol{i}}-boldsymbol{3} hat{boldsymbol{j}}, overrightarrow{boldsymbol{c}}=mathbf{2} hat{boldsymbol{i}}+boldsymbol{3} hat{boldsymbol{k}} )
12
88If ( |bar{a}+bar{b}|<|bar{a}+bar{b}| ) then the angle
between the vectors ( bar{a}, bar{b} ) is
A. Acute angle
B. obtuse angle
c. Right angle
D. ( 45^{circ} )
12
89Assertion
Set of any four vectors is always linearly dependent
Reason
If a subset of ( n ) vectors is ( L . D, ) then set
of those ( n ) vectors is linearly dependent
( boldsymbol{L} cdot boldsymbol{D} )
A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
B. Both Assertion & Reason are individually true but Reason is not the , correct (proper) explanation of Assertion
c. Assertion is true but Reason is false
D. Assertion is false but Reason is true
12
9019.
Show, by vector methods, that the angular bisectors of a
triangle are concurrent and find an expression for the
position vector of the point of concurrency in terms of the
position vectors of the vertices. (2001 – 5 Marks)
12
91( A B C ) is an equilateral triangle of side ( a ) The value of ( overrightarrow{boldsymbol{A B}} cdot overrightarrow{boldsymbol{B C}}+overrightarrow{boldsymbol{B C}} cdot overrightarrow{boldsymbol{C A}}+ )
( overrightarrow{C A} cdot overrightarrow{A B} ) is equal to
A ( cdot frac{3 a^{2}}{2} )
В. ( 3 a^{2} )
c. ( -frac{3 a^{2}}{2} )
D. None of these
12
92Derive Poynting vector expression for a straight wire of resistance R carrying a current I having length12
93Prove that ( |overrightarrow{boldsymbol{a}} cdot overrightarrow{boldsymbol{b}}| leq|overrightarrow{boldsymbol{a}}||overrightarrow{boldsymbol{b}}| )12
944.
The points with position vectors 60i + 3j, 40 i-8j, ai – 52
j are collinear if
(1983 – 1 Mark)
(a) a=-40
(b) a=40
(c) a=20
(d) none of these
12
9510. If + -]=PQ, then angle between and is
(a) 0° (b) 30° (c) 45° (d) 60°
12
96( A B C D ) is a quadrilateral, ( E ) is the point of intersection of the line joining the midpoints of the opposite sides. If ( O ) is any point and ( overrightarrow{O A}+overrightarrow{O B}+overrightarrow{O C}+ ) ( overrightarrow{O D}=x vec{O} E, ) then ( x ) is equal to
( A cdot 3 )
B. 9
( c cdot 7 )
D.
12
97Illustration 3.16 Given A = 0.3 +0.4
value of c if A is a unit vector.
+ ck. Calculate the
12
98Let ( overrightarrow{O A}=vec{a}, overrightarrow{O B}=10 vec{a}+2 vec{b} ) and
( overrightarrow{O C}=vec{b}, ) where ( O, A ) and ( C ) are non
collinear points. Let ( p ) denote the area of
quadrilateral ( O A B C, ) and ( q ) denote the
area of the parallelogram with ( O A ) and ( O C ) as adjecent sides, then ( frac{p}{q} ) is equal
to
A. 4
в. 6
( ^{mathrm{c}} frac{1}{2} frac{|vec{a}-vec{b}|}{vec{a}} )
D. None of these
12
99If ( vec{V}=3 hat{i}+4 hat{j} ) then, with what scalar ‘C must it be multiplied so that ( C|overrightarrow{boldsymbol{V}}|= )
( mathbf{7 . 5}: )
A . 0.5
B . 2.
c. 1.5
D. 3.5
12
100The vector ( vec{a}+vec{b} ) bisects the angles between the vectors ( vec{a} ) and ( vec{b} ) if
A. ( |vec{a}|=1 )
1 ( 1==1=1 )
в. ( |vec{b}|=1 )
c. ( |vec{a}|=mid vec{b} )
D. ( vec{a} vec{b}=0 )
12
101Find the distance between the points whose position vectors are given as follows ( mathbf{4} hat{mathbf{i}}+mathbf{3} hat{mathbf{j}}-mathbf{6} hat{boldsymbol{k}},-mathbf{2} hat{mathbf{i}}+hat{boldsymbol{j}}-hat{boldsymbol{k}} )
A . ( sqrt{65} )
B. ( sqrt{69} )
c. 13
D. none of these
12
102Find the position vector of the midpoint of the vector joining the points ( P(2,2,4) ) and ( Q(4,1,-2) )12
10357.
The projections of a vector on the three coordinate axis are
6,-3,2 respectively. The direction cosines of the vector are:
[2009]
2
6
3 2
6 -32
(b) 777
(d) 6,-3,2
12
104If vector ( vec{a}=a_{1} hat{i}+a_{2} hat{j}+a_{3} hat{k} ) then, find
unit vector.
12
105Find the angle between the vectors ( vec{r}_{1}=(4 hat{i}-3 hat{j}+5 hat{k}) ) and ( vec{r}_{2}=(3 hat{i}+ )
( 4 hat{j}+5 hat{k}) )
A ( cdot frac{pi}{3} )
B.
( c cdot frac{pi}{7} )
D. ( frac{pi}{6} )
12
106Let A be vector parallel to line of intersection of planes
Pand P2. Plane P, is parallel to the vectors
2j+3k and 4j – 3k and that P2 is parallel to
j-k and 3* +3ị, then the angle between vector A and a
given vector 2î + j – 2k is
(2006 – 5M, -1)
12
107The projection of the vector ( hat{mathbf{i}}-mathbf{2} hat{mathbf{j}}+hat{boldsymbol{k}} )
on the vector ( 4 hat{i}-4 hat{j}+7 hat{k} ) is
A ( cdot frac{5}{19} sqrt{5} )
в. ( frac{19}{9} )
c. ( frac{9}{19} )
D. ( frac{1}{19} sqrt{6} )
12
108Check whether the vector ( 4 hat{mathbf{i}}+mathbf{1 3} hat{mathbf{j}}- )
( 18 hat{k} ) is a linear combination of the vectors ( hat{i}-2 hat{j}+3 hat{k} ) and ( 2 hat{i}+3 hat{j}-4 hat{k} )
12
109When ( overrightarrow{boldsymbol{A}} cdot overrightarrow{boldsymbol{B}}=-|boldsymbol{A}||boldsymbol{B}|, ) then
( mathbf{A} cdot vec{A} ) and ( vec{B} ) are perpendicular to each other
B. ( vec{A} ) and ( vec{B} ) Act in the same direction
C . ( vec{A} ) and ( vec{B} ) Act in the opposite direction
D. ( vec{A} ) and ( vec{B} ) can act in any direction
12
110Three forces of magnitudes 1,2 and 3 dynes meet in a point and act along diagonals of three adjacent faces of a cube. The resultant force is?
A. 114 dynes
B. 6 dynes
c. 5 dynes
D. None of the above
12
111If the vectors ( vec{A}=2 hat{i}+4 hat{j} ) and ( vec{B}= ) ( mathbf{5} hat{mathbf{i}}-boldsymbol{P} hat{boldsymbol{j}} ) are parallel to each other, the
magnitude of ( vec{B} ) is:
A ( 5 sqrt{5} )
5
B. 10
c. 15
D. ( 2 sqrt{5} )
12
112( mid f hat{i}+hat{j}+hat{k}, 2 hat{i}+5 hat{j}, 3 hat{i}+2 hat{j}-3 hat{k} ) and
( hat{mathbf{i}}-mathbf{6} hat{mathbf{j}}-hat{boldsymbol{k}} ) are the position vectors of
points ( A, B, C ) and ( D ) respectively, then find the angle between ( overrightarrow{A B} ) and ( overrightarrow{C D} )
12
113If ( vec{a}+vec{b}+vec{c}=overrightarrow{0} ) show that ( vec{a} times vec{b}=vec{b} times )
( overrightarrow{boldsymbol{c}}=overrightarrow{boldsymbol{c}} times overrightarrow{boldsymbol{a}} )
12
1143. If two vectors 2î +39 – k and 4ỉ – 6.9 – ak are parallel
to each other, then value of 2 will be
(a) 0
(b) 2
(d) 4
(c) 3
12
115Illustration 3.28 Find the dot product of two vectors
A = 3i +29-4k and B = 2i – 39 – ok.
12
116What should be added in vector ( 3 i+ )
( 4 j-2 k ) to get its resultant a unit vector
( i )
A . ( -2 i-4 j+2 k )
B. ( 2 i+4 j+2 k )
( c .-2 i+4 j+2 k )
D. None of these
12
117Find sum of the vectors ( 3 hat{i}+7 hat{j}-4 hat{k} ) and ( hat{i}-5 hat{j}-8 hat{k} hat{i}-5 hat{j}-8 hat{k} ) and hence
find the unit vector along the sum of these vectors.
12
11819. The angle which the vector à = 2î +3j makes with
the y-axis, where i and i are unit vectors along x- and
y-axes, respectively, is
a. cos (3/5)
b. cos (2/3)
c. tan (2/3)
d. sin (2/3)
12
119Let ( bar{a} ) and ( bar{b} ) be two unit vectors and let ( theta )
be angle between them then ( (bar{a}+bar{b})^{2}- ) ( (bar{a}-bar{b})^{2}= )
A ( cdot cos theta )
B . ( 2 cos theta )
( c cdot 3 cos theta )
D. ( 4 cos theta )
12
120If the position vectors of three consecutive vertex of any parallelogram ( operatorname{are} ) respectively ( hat{i}+hat{j}+hat{k}, hat{i}+3 hat{j}+ )
( mathbf{5} hat{boldsymbol{k}}, mathbf{7} hat{boldsymbol{i}}+mathbf{9} hat{boldsymbol{j}}+mathbf{1 1} hat{boldsymbol{k}} ) then the position
vector of its fourth vertex is:
A ( cdot 6(hat{i}+hat{j}+hat{k}) )
в. ( 7(hat{i}+hat{j}+hat{k}) )
c. ( 2 hat{j}-4 hat{k} )
D. ( 6 hat{i}+8 hat{j}+1 hat{0} k )
12
121set the following vectors in the increasing order of their magnitudes.
a) ( 3 i+4 j )
b) ( 2 i+4 j+6 k )
c)2i+2j+2k
A. b,a c
B. ( c, a, b )
( c cdot a, c, b )
( D cdot a, b, c )
12
122ABCD is a parallelogram. The position vectors of ( A ) and ( C ) are respectively, ( 3 hat{i}+ ) ( 3 hat{j}+5 hat{k} ) and ( hat{i}-5 hat{j}-5 hat{k} . ) If ( mathrm{M} ) is the
mid-point of the diagonal DB, then the magnitude of the projection of ( boldsymbol{O} boldsymbol{M} ) on ( overrightarrow{O C}, ) where 0 is the origin is?
A ( cdot frac{7}{sqrt{50}} )
B . ( 7 sqrt{50} )
c. ( frac{7}{sqrt{51}} )
D. ( 7 sqrt{51} )
12
123f ( boldsymbol{A}=(-mathbf{3}, mathbf{2}, mathbf{5}), boldsymbol{B}=(-mathbf{3}, mathbf{4}, mathbf{5}) ) and
( C=(-3,4,7) ) are the position vectors
of vertices of ( Delta A B C ) then its
circumcentre is
A. (-3,3,5)
(年 ( cdot(-3,3,5) )
В. (-3,3,6)
c. (-3,4,6)
a
D. (-3,4,7)
12
1243. Given that A+B=C and that Ĉ is perpendicular to
A. Further if A = C), then what is the angle between
A and B?
radian (b) radiand o
El+
radian
(d) a radian
+
12
125Let a and b be two non-collinear unit vectors. Ifu=a-(a.b)
b and v=ax b, then vis
(1999 – 3 Marks)
(a) Jul
(b) | ul+u.a
(c) |u|+|u.b!
(d) |u +u.(a+b)
11 11.
12
126Find ( |vec{a} times vec{b}|, ) if ( vec{a}=hat{i}-7 hat{j}+7 hat{k} ) and ( vec{b}= )
( mathbf{3} hat{mathbf{i}}-mathbf{2} hat{mathbf{j}}+mathbf{2} hat{boldsymbol{k}} )
12
127What is the maximum number of
rectangular components into which a vector can be split in it’s own plane?
( A cdot 2 )
B. 3
( c cdot 4 )
D. Infinite
12
128( |vec{a}+vec{b}|^{2}-|vec{a}-vec{b}|^{2} ) is equal to
( mathbf{A} cdot 4|vec{a}||vec{b}| sin (theta) )
B . ( 4|vec{a}||vec{a}| sin (theta) )
( mathbf{c} cdot mathbf{4}|vec{a}||vec{b}| cos (theta) )
( mathbf{D} cdot mathbf{4}|vec{a}||vec{b}| tan (theta) )
12
129For non zero vectors ( a, b ) and ( c, ) if ( a+ ) ( b+c=0 ) then which relation true:
A ( a=b=c=0 )
0
В. ( a . b=b . c=c . a )
( mathbf{c} cdot a times b=b times c=c times a )
D. None
12
130Vector is represented by directed line segment where length of line gives
of vector.
A. Direction
B. Magnitude
c. Both (A) and (B)
D. None of the above
12
131( boldsymbol{A}=(mathbf{0}, mathbf{1}, mathbf{2}), boldsymbol{B}=(mathbf{3}, mathbf{0}, mathbf{1}), boldsymbol{C}= )
( (4,3,6), D=(2,3,2) ) are the
rectangular cartesian co-ordinates. Find the area of the triangle ( A B C ).
A. ( sqrt{90} )
B. ( sqrt{frac{45}{2}} )
c. 45
D. 18
12
132The position vectors of ( boldsymbol{A}, boldsymbol{B}, boldsymbol{C}, boldsymbol{D} ) are
( bar{a}, bar{b}, bar{c}, bar{d} ) respectively and ( |bar{a}-overline{boldsymbol{d}}|= )
( |bar{b}-bar{d}|=|bar{c}-bar{d}|, ) then for the triangle
( A B C, D ) is
A. Ortho centre
B. Centroid
c. circumcentre
D. Incentre
12
13314. If |AX B = 13Ā. B , then the value of A+ Blis
(c) (A? + B2 + 13 AB)1/2 (d) (A2 + B2 + AB)1/2
12
134Let ( vec{u} ) be a vector on rectangular system
with sloping angle ( 60^{circ} . ) Suppose that ( |vec{u}-hat{i}| ) is geometric mean of ( |vec{u}| ) and ( |vec{u}-2 hat{i}|, ) where ( vec{i} ) is the unit vector along
( x ) -axis, then ( |vec{u}| ) has the value equal to ( sqrt{a}-sqrt{b}, ) where ( a, b in N . ) Find the value ( frac{(boldsymbol{a}+boldsymbol{b})^{3}+(boldsymbol{a}-boldsymbol{b})^{3}}{mathbf{7}} )
12
13512. The position vectors of points A, B, C and D are
A = 3ỉ +4ġ + 5k, B = 4 +5j +6k, C = 7î +9j+3k, and
D=4i +6j, then the displacement vectors AB and CD are
(a) Perpendicular (b) Parallel
(c) Antiparallel (d) Inclined at an angle of 60°
12
136Let two vectors ( vec{x} ) and ( vec{y} ) satisfy the following conditions ( : vec{x}+vec{y}=vec{a} ; vec{x} times )
( vec{y}=vec{b} times vec{a}, vec{x} cdot vec{a}=1 ) where ( vec{a} ) and ( vec{b} )
are non-zero perpendicular vectors,
then
( ^{mathrm{A}} cdot_{vec{x}}=frac{|vec{a}|^{2} vec{b}+vec{a}}{|vec{a}|^{2}} )
( ^{mathrm{B}} cdot_{vec{x}}=frac{|vec{a}|^{2} vec{b}-vec{a}}{|vec{a}|^{2}} )
( stackrel{mathrm{C}}{*} vec{y}=frac{|vec{a}|^{2}(vec{a}-vec{b})-vec{a}}{|vec{a}|^{2}} )
( vec{y}=frac{vec{a}(vec{a} cdot vec{y})-|vec{a}|^{2} vec{b}}{|vec{a}|^{2}} )
12
137The vectors ( vec{a}(x)=cos x hat{i}+sin x hat{j} ) and
( vec{b}(x)=x hat{i}+sin (x) hat{j} ) are collinear for
A cdot Unique value of ( x, 0<x<frac{pi}{6} )
B. Unique value of ( x, frac{pi}{6}<x<frac{pi}{3} )
c. No value of ( x )
D cdot Infinite values of ( x, 0<x<frac{pi}{2} )
12
138If ( 2 vec{a} cdot vec{b}=|vec{a}||vec{b}| ) then the angle between ( vec{a} ) and ( vec{b} ) is
A ( .30^{circ} )
в. ( 0^{circ} )
( c cdot 90^{circ} )
D. ( 60^{circ} )
12
139The position vectors of the points ( A, B, C ) and ( mathrm{D} ) are ( 3 hat{i}-2 hat{j}-hat{k}, 2 hat{i}-3 hat{j}+ )
( mathbf{2} hat{boldsymbol{k}}, mathbf{5} hat{boldsymbol{i}}-hat{boldsymbol{j}}+mathbf{2} hat{boldsymbol{k}} ) and ( mathbf{4} hat{boldsymbol{i}}-hat{boldsymbol{j}}+boldsymbol{lambda} hat{boldsymbol{k}} )
respectively. If the points ( A, B, C ) and ( D ) lie on a plane, the value of ( lambda ) is?
( A cdot 0 )
B.
( c cdot 2 )
D. -4
12
140Find the dot product of the following vectors:
( vec{a}=2 hat{i}+hat{j}+3 hat{k} ) and ( vec{b}=3 hat{i}+5 hat{j}-2 hat{k} )
12
141cosine of an angle between the vectors ( vec{a}+vec{b} ) and ( vec{a}-vec{b} ) if ( |vec{a}|=2,|vec{b}|=1 ) and ( vec{a} )
( wedge vec{b}=60^{circ} ) is?
A ( cdot sqrt{frac{3}{7}} )
B. ( frac{9}{sqrt{21}} )
c. ( frac{3}{sqrt{7}} )
D. None
12
14230. Letā =i+2j+k, b =î – i+k and c =î+j-k. A
vector in the plane of a and ū whose projection on c is
(2006 – 3M, -1)
(a) 41 – it 4 (6) 3i +9 – 3
(C) 2î + j-2 (d) ti ti – 4
Tois
1:
12
143A bird moves from point ( (1 m,-2 m, 3 m) ) to ( (4 m, 2 m, 3 m) . ) If
the speed of the bird is ( 10 m / s ), then the
velocity vector of the bird in ( boldsymbol{m} / boldsymbol{s} ) is:
( mathbf{A} cdot 5(hat{i}-2 hat{j}+3 hat{k}) )
B . ( 5(4 hat{i}+2 hat{j}+3 hat{k}) )
c. ( 0.6 hat{i}+0.8 hat{j} )
D. ( 6 hat{i}+8 hat{j} )
12
144Let ( vec{a} ) be an unit vector and ( m ) being a
scalar then find the value of ( m ) from the
equation ( |boldsymbol{m} overrightarrow{boldsymbol{a}}|=mathbf{2} )
12
145The unit vector perpendicular to both ( L_{1} )
and ( boldsymbol{L}_{2} ) is
( ^{mathbf{A}} cdot frac{-hat{i}+7 hat{j}+7 hat{k}}{sqrt{99}} )
( ^{text {B. }} frac{-hat{i}-7 hat{j}+5 hat{k}}{5 sqrt{3}} )
( frac{-hat{i}+7 hat{j}+5 hat{k} hat{k}}{5 sqrt{3}} )
D. ( frac{7 hat{i}-7 hat{j}-hat{k}}{sqrt{99}} )
12
146( vec{A}=3 hat{i}+4 hat{j}+2 hat{k} )
( vec{B}=6 hat{i}-hat{j}+3 hat{k} )
Find a vector parallel to ( vec{A} ) whose magnitude is equal to that of ( vec{B} ).
A ( cdot sqrt{frac{46}{29}}(3 hat{i}+4 hat{j}+2 hat{k}) )
В. ( sqrt{frac{46}{29}}(6 hat{i}-hat{j}+3 hat{k}) )
c. ( sqrt{frac{29}{46}}(3 hat{i}+4 hat{j}+2 hat{k}) )
D. None
12
147The vectors ( 2 hat{i}-3 hat{j}+4 widehat{k} ) and ( -4 hat{i}+ ) ( 6 widehat{j}-8 widehat{k} ) are collinear.
A. True
B. False
12
148Assertion
The resultant of the three vector
( overrightarrow{O A}, overrightarrow{O B} ) and ( overrightarrow{O C} ) as shown in the figure is ( boldsymbol{R}(1+sqrt{2}) . ) Here ( boldsymbol{R} ) is the radius of the
circle.(Here ( left.angle A O B=angle B O C=45^{circ}right) )
Reason
( overrightarrow{O A}+overrightarrow{O C} ) is along ( overrightarrow{O B} ) and ( (overrightarrow{O A}+ ) ( overrightarrow{O C})+overrightarrow{O B} ) is along ( overrightarrow{O B} )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
149A unit vector perpendicular to the lines
( frac{boldsymbol{x}+mathbf{1}}{mathbf{3}}=frac{boldsymbol{y}+mathbf{2}}{mathbf{1}}=frac{boldsymbol{z}+mathbf{1}}{mathbf{2}} ) and ( frac{boldsymbol{x}-mathbf{2}}{mathbf{1}}= )
( frac{boldsymbol{y}+boldsymbol{2}}{boldsymbol{2}}=frac{boldsymbol{z}-boldsymbol{3}}{boldsymbol{3}} ) is
( ^{mathbf{A}} cdot frac{-hat{i}+7 hat{j}+7 hat{k}}{sqrt{99}} )
( ^{text {B. }} frac{-hat{i}-7 hat{j}+5 hat{k}}{5 sqrt{3}} )
( frac{-hat{i}+7 hat{j}+5 hat{k}}{5 sqrt{3}} )
D. ( frac{7 hat{i}-7 hat{j}-hat{k}}{sqrt{99}} )
12
150A girl walks ( 4 k m ) towards west, then
she walk ( 3 k m ) in a direction ( 30^{circ} ) east of
north and stops. Determine the girl’s displacement from her initial point of
departure.
12
151The two adjacent sides of a
parallelogram, ( vec{a}=2 i+4 j-5 k ) and ( vec{b}=i+2 j+3 k, ) then find the unit
vector parallel to its diagonal.
12
152e.,
Illustration 3.38 Calculate the area of the parallelogram
when adjacent sides are given by the vectors Ari +2j+3k
and B = 2-3j+ k.
12
153Find ( overrightarrow{A B}, ) if ( A(4,3,4) ) and ( B(2,5,0) )12
154If ( bar{A}=3 hat{i}+5 hat{j}-2 hat{k}, ) and vector ( bar{B}= )
( -3 hat{j}+6 hat{k} . ) Find a vector ( bar{C} ) such that 2 ( bar{A}+7 bar{B}+4 bar{C}=0 )
( mathbf{A} cdot-6 hat{i}+11 hat{j}-38 hat{k} )
B . ( 11 hat{j}-4 hat{k} )
c. ( -1.5 hat{i}+2.75 hat{j}-9.5 hat{k} )
D. ( 4 hat{j}-6 hat{k} )
12
155Let ( vec{a}, vec{b} ) and ( vec{c} ) be non-coplanar unit vectors equally inclined to one another at an acute angle ( theta ) then ( [vec{a} vec{b} vec{c}] ) in terms of ( theta ) is equal to :
A ( cdot(1+cos theta) sqrt{cos 2 theta} )
B . ( (1+cos theta) sqrt{1-2 cos 2 theta} )
C ( cdot(1-cos theta) sqrt{1+2 cos theta} )
D cdot ( (1-sin theta) sqrt{1+2 cos 2 theta} )
12
156If ( |vec{a}+vec{b}|<|vec{b}-vec{a}| ) then angle between ( vec{a} ) and ( vec{b} ) is
A ( cdot frac{pi}{2} )
в.
c. acute
D. obtuse
12
157If three consecuting vertices of a
parallelogram are ( boldsymbol{A}(mathbf{4}, mathbf{3}, mathbf{5}), boldsymbol{B}(mathbf{0}, mathbf{6}, mathbf{0}), boldsymbol{C}(-mathbf{8}, mathbf{1}, mathbf{4}) ) and ( boldsymbol{D} )
is the fourth vertex then the anlge between ( overline{A C} ) and ( overline{B D} ) is
( ^{mathbf{A}} cdot cos ^{-1}left[frac{55}{sqrt{149} sqrt{161}}right] )
( ^{mathbf{B}} cdot cos ^{-1}left[frac{65}{sqrt{149} sqrt{161}}right] )
( ^{mathbf{c}} cdot cos ^{-1}left[frac{15}{sqrt{149} sqrt{161}}right] )
D. ( cos ^{-1}left[frac{3}{sqrt{149} sqrt{161}}right] )
12
158Find a vector in the direction of a vector
( vec{a}=hat{i}-widehat{j}+widehat{k}, ) which has magnitude 8
units.
12
159Unit vector perpendicular to the plane of the triangle ( A B C ) with position vectors of the vertices ( A, B, C, ) is (where ( Delta )
is the area of the triangle ( A B C ) ).
( frac{(vec{a} times vec{b}+vec{b} times vec{c}+vec{c} times vec{a})}{Delta} )
B. ( frac{(vec{a} times vec{b}+bar{b} times vec{c}+vec{c} times vec{a})}{2 Delta} )
( overbrace{frac{(vec{a} times vec{b}+vec{b} times vec{c}+vec{c} times vec{a})}{3 Delta}}^{text {c. }} )
( frac{(vec{a} times vec{b}+vec{b} times vec{c}+vec{c} times vec{a})}{4 Delta} )
12
16077. Let a, b and c be three non-zero vectors such that no two
of them are collinear and (axb)
fo is the
angle between vectors band c , then a value of sin 0 is :
[JEE M 2015]
12
161Two vectors ( a ) and ( b ) are said to be equal
if
।. ( |boldsymbol{a}|=|boldsymbol{b}| )
Il. they have same or parallel support.
III. the same sense.
Which of the following is true?
A. Onlyı
c. Both I and III
D. All three
12
162Assertion
Vectors ( vec{a}, vec{b}, vec{c} ) are coplanar.
Reason
( (x vec{a}+y vec{b})=overrightarrow{0} Rightarrow x=0 ) and ( y=0 )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
163( A B C ) is a triangle and ( P ) is any point on
BC. If ( P Q ) is the resultant of the vectors ( overrightarrow{A P}, overrightarrow{P B} ) and ( overrightarrow{P C} ) then ( A C Q B ) is
A. Rectangle
B. Square
c. Rhombus
D. Parallelogram
12
164A certain vector in the xy-plane has an
x-component of ( 4 mathrm{m} ) and a y-component of ( 10 mathrm{m} . ) It is then rotated in the xy-plane so that its ( x ) -component is doubled Then, its new y-component is (approximately)
A. ( 20 mathrm{m} )
B. 7.2 ( m )
c. ( 5.0 mathrm{m} )
D. ( 4.5 mathrm{m} )
12
165Direction angle of a vector is ( 30^{circ}, ) then
find the vector.
B. ( sqrt{3} a hat{i}+b hat{j} )
c. ( a hat{i}+b hat{j} )
D. ( sqrt{3} a hat{i}+sqrt{3} 3 hat{j} )
12
166Find a vector in the direction of vector ( vec{a}=hat{i}-2 hat{j} ) that has magnitude 7 units.
A ( cdot frac{7}{sqrt{5}}(hat{i}-2 hat{j}) )
B. ( frac{7}{sqrt{5}}(hat{i}+2 hat{j}) )
c. ( frac{7}{sqrt{5}}(-hat{i}-2 hat{j}) )
D. None of these
12
16727. If the angle between the vectors ä and b is an acute
angle, then the difference a – b is
a. The major diagonal of the parallelogram
b. The minor diagonal of the parallelogram
c. Any of the above
d. None of the above
– – –
11
12
168Let ( vec{a}, vec{b} ) and ( vec{c} ) br non-coplanar unit vectors equally inclined to one another at an acute angle ( theta . ) Then ( [vec{a} vec{b} vec{c}] ) in terms
of ( theta ) is equal to
A ( cdot(1+cos theta) sqrt{cos 2 theta} )
B. ( (1+cos theta) sqrt{1-2 cos 2 theta} )
c. ( (1-cos theta) sqrt{1+2 cos 2 theta} )
D. none of these
12
169UUD.
Vector á has components 2p and 1 with respect to a
tangular cartesian system. This system is rotated through
a certain angle about the origin in the counter clockwise
sense. If, with respect to the new system, à has components
p+1 and 1, then
(1986 – 2 Marks)
(a) p=0
(b) p=1 or p= –
| © p=-1 or p=
(e) none of these
(d) p=1 or p=-1
12
170Show that the position vector of the point ( P, ) which divides the line joining the points ( A ) and ( B ) having position vector ( vec{a} ) and ( vec{b} ) internally in the ratio ( m: n ) is ( frac{m vec{b}+n vec{a}}{m+n} )12
171What vector when added to ( (2 hat{i}-2 hat{j}+ )
( hat{boldsymbol{k}}) ) and ( (mathbf{2} hat{boldsymbol{i}}-hat{boldsymbol{k}}) ) will give a unit vector
along negative ( y- ) axis?
12
172The vectors ( bar{a}, bar{b}, bar{c} ) are equal in length and taken pairwise they make equal angles. If ( bar{a}=bar{i}+bar{j}, bar{b}=bar{j}+bar{k} ) and ( bar{c} )
makes an obtuse angle with ( x ) -axis
then ( bar{c}= )
A. ( bar{i}+bar{k} )
the
B . ( -bar{i}+4 bar{j}-bar{k} )
c. ( frac{1}{3}(-bar{i}+4 bar{j}-bar{k}) )
D ( frac{1}{3}(bar{i}-4 bar{j}+bar{k}) )
12
173If ( vec{a}, vec{b}, vec{c} ) are unit vectors such that ( vec{a} ) ( overrightarrow{boldsymbol{b}}=mathbf{0},(overrightarrow{boldsymbol{a}}-overrightarrow{boldsymbol{c}}) cdot(overrightarrow{boldsymbol{b}}+overrightarrow{boldsymbol{c}})=mathbf{0} ) and ( overrightarrow{boldsymbol{c}}= )
( lambda vec{a}+mu vec{b}+omega(vec{a} times vec{b}), ) where ( lambda, mu, omega ) are
scalars then?
A ( cdot mu^{2}+omega^{2}=1 )
в. ( lambda-mu=1 )
C ( cdot(mu+1)^{2}+mu^{2}+omega^{2}=1 )
D. ( lambda^{2}+mu^{2}=1 )
12
174If ( |overline{boldsymbol{a}}|=mathbf{1},|overline{boldsymbol{b}}|=mathbf{2},|overline{boldsymbol{a}}-overline{boldsymbol{b}}|^{2}+|overline{boldsymbol{a}}+mathbf{2} overline{mathbf{b}}|^{2}= )
( mathbf{2 0}, ) then ( (overline{boldsymbol{a}}, overline{boldsymbol{b}})= )
A ( cdot frac{pi}{3} )
в.
c.
D. ( frac{2 pi}{3} )
12
17520. Consider points A, B, C and D with position
vectors Ti – 49+7k,i – 6j +10k, -i -3j+4k and
si – j+5k respectively. Then ABCD is a [2003]
(a) parallelogram but not a rhombus
(b) square
rhombus
(d) rectangle.
(c)
!
12
176Find the area of the parallelogram whose two sides are ( 2 bar{i}-3 bar{j} ) and ( 3 bar{i}-bar{j} )12
177If ( vec{a}=hat{i}+hat{j}-hat{k}, vec{b}=-hat{i}+2 hat{j}+2 hat{k} & )
( vec{c}=-hat{i}+2 hat{j}-hat{k}, ) find a unit vectors normal to the vectors ( vec{a}+vec{b} ) and ( vec{b}+vec{c} ).
( ^{mathbf{A}} cdot frac{-hat{i}-2 hat{j}+6 hat{k}}{sqrt{40}} )
B. ( frac{-hat{i}-2 hat{j}+6 hat{k}}{sqrt{41}} )
c. ( frac{hat{i}-2 hat{j}+6 hat{k}}{sqrt{41}} )
D. ( frac{-hat{i}+2 hat{j}+6 hat{k}}{sqrt{41}} )
12
178If ( A((2,1,3), B(5,3,9), C(1,-1,3) text { and } D(2, )
3, 11 ), find the angle between the lines ( A B ) and ( C D )
12
179( P ) and ( Q ) are two points with position vectors ( 3 vec{a}-2 vec{b} ) and ( vec{a}+vec{b} )
respectively. Write the position vector of a point R which divides the line
segment ( boldsymbol{P Q} ) in the ratio ( mathbf{2}: mathbf{1} ) externally
12
180State whether the given statement is true or false if A triangle OAB where ( overrightarrow{O A}=a, overrightarrow{O B}=b ) Find the value of ( a times )
( b+b times c+c times a ) will be perpendicular
to plane ABC.
12
181If the position vector of a point ( boldsymbol{A} ) is ( vec{a}+overrightarrow{2 b} ) and ( vec{a} ) divides ( overrightarrow{A B} ) in the ratio
( 2: 3, ) then the position vector of ( B ) is
A ( cdot overrightarrow{2 a}-vec{b} )
B . ( vec{b}-overrightarrow{2 a} )
c. ( vec{a}-overrightarrow{3 b} )
D. ( vec{b} )
12
182If ( bar{a}, bar{b}, bar{c} ) are non-coplanar vectors and if ( bar{d} ) is such that ( bar{d}=frac{1}{x}(bar{a}+bar{b}+bar{c}) ) and
( overline{boldsymbol{d}}=frac{mathbf{1}}{boldsymbol{y}}(overline{boldsymbol{b}}+overline{boldsymbol{c}}+overline{boldsymbol{d}}) ) where ( mathbf{x} ) and ( mathbf{y} ) are non
zero real numbers, then ( frac{1}{x y}(bar{a}+bar{b}+ ) ( overline{boldsymbol{c}}+overline{boldsymbol{d}})= )
B. ( -bar{a} )
( c . overline{0} )
D . ( 2 bar{a} bar{a} bar{a} )
12
183Find the components along the coordinate axes of the position vector of each of the following point:
( boldsymbol{R}(-mathbf{1 1},-mathbf{9}) )
12
184( (vec{a} cdot vec{i})^{2}+(vec{a} cdot vec{jmath})^{2}+(vec{a} cdot vec{k})^{2} ) is equal to
A ( left.left.cdot|vec{a}|^{2}right|^{2}right|^{vec{l}} )
B. 3
c ( cdot|vec{a} cdot(vec{i}+vec{j}+vec{k})|^{2} )
D. None of these
12
185If ( bar{z}_{1}=a hat{i}+b hat{j} ) and ( bar{z}_{2}=c hat{i}+d hat{j} ) are two
vectors in ( hat{i} ) and ( hat{j} ) system where ( left|bar{z}_{1}right|= )
( left|bar{z}_{2}right|=r ) and ( bar{z}_{1} cdot bar{z}_{2}=0, ) then ( bar{omega}_{1}=a hat{i}+ )
( c hat{j} ) and ( bar{omega}_{2}=b hat{i}+d hat{j} ) satisfy
( mathbf{A} cdotleft|bar{omega}_{1}right|=r )
В ( cdotleft|bar{omega}_{2}right| neq r )
begin{tabular}{l}
c. ( bar{omega}_{1}, bar{omega}_{2}=0 ) \
hline
end{tabular}
D. None of these
12
186If ( overline{O A}=i+j+k, overline{A B}=3 i-2 j+ )
( k, overline{B C}=i+2 j-2 k ) and ( overline{C D}=2 i+ )
( j+3 k ) then find the vector ( overline{O D} )
A. ( 7 i-2 j-6 k )
B. ( 7 i+2 j+3 k )
c. ( 7 i+2 j+5 k )
D. None of these
12
187Six vectors, a through f have the magnitudes and directions indicated in
the figure. Which of the following
statements is true?
A. ( b+e=f )
B. ( b+c=f )
c. ( d+c=f )
D. ( d+e=f )
12
188If ( vec{a}=hat{i}+hat{j}+2 hat{k} ) and ( vec{b}=3 hat{i}+2 hat{j}-hat{k} )
find the value of ( (vec{a}+3 vec{b}) cdot(2 vec{a}-vec{b}) )
12
189If ( a, b, c ) are unit vectors, then the
maximum value of ( |boldsymbol{a}+mathbf{2 b}|^{2}+mid boldsymbol{b}+ )
( left.3 cright|^{2}+|c+4 a|^{2} ) is
A . 50
B . 21
c. 48
D. 58
12
190If ( overrightarrow{boldsymbol{a}} cdot hat{boldsymbol{i}}=overrightarrow{boldsymbol{a}} cdot(hat{boldsymbol{i}}+hat{boldsymbol{j}})=overrightarrow{boldsymbol{a}} cdot(hat{boldsymbol{i}}+hat{boldsymbol{j}}+hat{boldsymbol{k}}) )
then find the unit vector ( vec{a} ).
12
191( operatorname{Let} overrightarrow{A B}=3 hat{i}+hat{j}-hat{k} ) and ( overrightarrow{A C}=hat{i}- )
( hat{j}+3 hat{k} . ) If the point ( P ) on the line
segment ( B C ) is equidistant from ( A B ) and ( A C ) then ( overrightarrow{A P} ) is
A ( cdot 2 hat{i}-hat{k} )
в. ( hat{i}-2 hat{k} )
( c cdot 2 hat{i}+hat{k} )
D. None of these
12
192If ( bar{a} ) is unit vector, then ( |overline{boldsymbol{a}} times hat{mathbf{i}}|^{2}+mid overline{boldsymbol{a}} times )
( left.hat{boldsymbol{j}}right|^{2}+|overline{boldsymbol{a}} times hat{boldsymbol{k}}|^{2}= )
( A cdot 2 )
B.
( c cdot 0 )
D. 3
12
193Find the values of ( x ) and ( y ) such that the vectors ( 2 hat{i}+3 hat{j} ) and ( x hat{i}+y hat{j} ) are equal.12
194Let ( vec{a}, vec{b} ) be the position vectors of points ( A ) and ( B ) with respect to ( O ) and ( |vec{a}|=a,|vec{b}|=b ) the points ( C ) and ( D )
divides ( A ) internally and externally in the ratio 2: 3 If ( overrightarrow{O C} ) and ( overrightarrow{O D} ) are
perpendicular then
A ( cdot 9 a^{2}=4 b^{2} )
В. ( 4 a^{2}=9 b^{2} )
c. ( 9 a=4 b )
D. ( 4 a=9 b )
12
19512 coplanar non collinear forces (all of equal magnitude) maintain a body in equilibrium, then the angle between any two adjacent forces is:
A . 15
в. 30
c. 45
D. ( 60^{circ} )
12
196Equation of the plane passing through a point with position vector ( 3 hat{i}-3 hat{j}+hat{k} )
& normal to the line joining the points with position vectors ( 3 hat{i}+4 hat{j}-hat{k} & ) ( mathbf{2} hat{mathbf{i}}-hat{boldsymbol{j}}=mathbf{5} hat{boldsymbol{k}} ) is
( mathbf{A} cdot bar{r} cdot(-hat{i}-5 hat{j}+6 hat{k})+18=0 )
B ( cdot bar{r} cdot(hat{i}-5 hat{j}+6 hat{k})=22 )
c. ( bar{r} .(hat{i}+5 hat{j}-6 hat{k})+18=0 )
D ( cdot bar{r} .(-hat{i}+5 hat{j}+6 hat{k})+12=0 )
12
19728. If Ā=2î +j and B = { – ġ, sketch vectors graphically
and find the component of Ā along B and perpendicular
to B.
12
1981. The magnitudes of vectors A, B and are 3, 4 and 5
units respectively. If A+B=C, the angle between A
and B is
(b) cos- (0.6)
4.
(C)
(c) tan’
tan-1
(d) *
nl forces of 10 N each are annlied at one point
12
1991. The sum and difference of two perpendicular vectors of
equal length are
a. Perpendicular to each other and of equal length
b. Perpendicular to each other and of different lengths
c. Of equal length and have an obtuse angle between
them
d. Of equal length and have an acute angle between
them
12
200The vector ( overline{O P}=hat{i}+2 hat{j}+2 hat{k} ) turns
through a right angle passing through the positive ( x ) -axis on the way. Find the vector in its new position.
A ( cdot frac{4}{sqrt{2}} hat{i}-frac{1}{sqrt{2}} hat{j}-frac{1}{sqrt{2}} hat{k} )
B. ( -frac{4}{sqrt{2}} hat{i}+frac{1}{sqrt{2}} hat{j}-frac{1}{sqrt{2}} hat{k} )
( ^{mathrm{C}} cdot-frac{4}{sqrt{2}} hat{i}-frac{1}{sqrt{2}} hat{j}+frac{1}{sqrt{2}} hat{k} )
D. ( frac{4}{sqrt{2}} hat{i}+frac{1}{sqrt{2}} hat{j}-frac{1}{sqrt{2}} hat{k} )
12
201Find the unit vectors orthogonal to both
(2,3,5) and (2,-1,4)
12
202If ( vec{a}=x vec{i}-y vec{j}, vec{b}=y vec{i}+x vec{j} ) and ( mid vec{a} times )
( vec{b} mid=5, ) then locus of ( (x, y) ) is
A. Hyperbola
B. Parabola
c. Ellipse
D. Circle
12
203paragraph, find the resultant vectors ( boldsymbol{A} )
and ( B ) in figure.
( |vec{R}|=13.54 m s^{-1}, tan beta=frac{3}{13.196} ) with respect to ( 8 m s^{-1} )
vector
3 ( |vec{R}|=15.54 m s^{-1}, tan beta=frac{3}{13.196} ) with respect to ( 8 m s^{-1} )
vector
( ^{mathrm{C}}|vec{R}|=15.54 m s^{-1}, tan beta=frac{3}{13.196} ) with respect to ( 9 m s^{-1} )
vector
( |vec{R}|=13.54 m s^{-1}, tan beta=frac{3}{13.196} ) with respect to ( 9 m s^{-} )
vector
12
204ulen
2. The minimum number of vectors having different planes
which can be added to give zero resultant is
a. 2 b. 3 . c. 4 d. 5
12
205Find the angle between the vectors:
( boldsymbol{a}={-1,2,-2} ) and ( boldsymbol{b}={mathbf{6}, mathbf{3},-mathbf{6}} )
12
206Given ( overrightarrow{boldsymbol{a}}=hat{boldsymbol{i}}+hat{boldsymbol{j}}-hat{boldsymbol{k}}, overrightarrow{boldsymbol{b}}=-hat{boldsymbol{i}}+boldsymbol{2} hat{boldsymbol{j}}+hat{boldsymbol{k}} )
and ( vec{c}=-hat{mathbf{i}}+mathbf{2} hat{mathbf{i}}-hat{boldsymbol{k}} cdot boldsymbol{A} ) unit vector
perpendicular to both ( vec{a}+vec{b} & vec{b}+vec{c} ) is
A ( cdot frac{2 hat{i}+hat{j}+hat{k}}{sqrt{6}} )
B. ( j )
( c cdot hat{k} )
D. ( frac{hat{i}+hat{j}+hat{k}}{sqrt{3}} )
12
207If ( theta ) is the angle between the lines
( A B, A C ) where ( A, B, C ) are the three points with coordinates (1,2,-1),(2,0,3),(3,-1,2)
respectively, then ( sqrt{462} cos theta ) is equal to
A . 20
в. 2
c. 22
D. 26
12
20832. Mark the correct statement.
a. ā + b 2 lā+
5 b . + b slā + 5
d. All of the above
12
209A line passes through the points whose position vectors ( hat{mathbf{i}}+hat{mathbf{j}}-mathbf{2} hat{boldsymbol{k}} ) and ( hat{mathbf{i}}- )
( 3 hat{j}+hat{k} . ) Then the position vector of a
point on it at a unit distance from the first point is
A ( cdot frac{1}{5}(5 hat{i}+hat{j}-7 hat{k}) )
B ( cdot frac{1}{5}(5 hat{i}+9 hat{j}-13 hat{k}) )
c. ( (hat{i}-4 hat{j}+3 hat{k}) )
D. ( (hat{i}+4 hat{j}+3 hat{k}) )
12
2105.
Let a, b, c, be three non-coplana
cy be three non-coplanar vectors and
Бхс
p, q, r, are vectors defined by the relations p = Tab c]’

cx a

axb
2 ſabore then the value of the expression
Tā +). + (7 +c).9+(c +a), r is equal to
(1988 – 2 Marks)
(a) 0 (6) 1 (c) 2 (d) 3.
12
211Given ( left|overrightarrow{boldsymbol{A}}_{1}right|=2,left|overrightarrow{boldsymbol{A}}_{2}right|=overrightarrow{mathbf{3}} ) and
( left|vec{A}_{1}+vec{A}_{2}right|=3 . ) Find the value of ( left(overrightarrow{boldsymbol{A}}_{1}+mathbf{2} overrightarrow{boldsymbol{A}}_{2}right) cdotleft(boldsymbol{3} overrightarrow{boldsymbol{A}}_{1}-boldsymbol{4} boldsymbol{vec { A }}_{2}right) )
A . -64
B. 60
c. -60
D. 64
12
212Let ( a=2 hat{i}+hat{j}+hat{k}, b=hat{i}+2 hat{j}-hat{j} ) an a
unit vector ( hat{c} ) be coplanar. If ( hat{c} ) is perpendicular to ( a ), then ( c ) is equal to
( mathbf{A} cdot(-hat{j}+hat{k}) )
B. ( pm frac{1}{sqrt{2}}(-hat{j}+hat{k}) )
( ^{mathrm{C}} pm frac{1}{sqrt{2}}(hat{j}+hat{k}) )
D. None of these
12
213If ( bar{a}=bar{i}+2 bar{j}, bar{b}=-2 bar{i}+bar{j}, bar{c}=4 bar{i}+3 bar{j} )
find ( x ) and ( y ) such that ( bar{c}=x bar{a}+y bar{b} )
12
214Show that the lines ( widehat{P Q} ) are ( overrightarrow{R Q} ) paralle
where ( P, Q, R, S ) are the points
(2,3,4),(4,7,8),(-1,-2,1) and
(1,2,5) respectively.
12
215Find the projection of the vector ( hat{mathbf{i}}+ ) ( mathbf{3} hat{boldsymbol{j}}+hat{boldsymbol{k}} ) on the vector ( mathbf{7} hat{boldsymbol{i}}-hat{boldsymbol{j}}+boldsymbol{8} hat{boldsymbol{k}} )12
216The expression ( frac{1}{sqrt{2}}(hat{i}+hat{j}) ) is a :
A. unit vector
B. null vector
C. vector of magnitude ( sqrt{2} )
D. vector of magnitude ( frac{1}{sqrt{2}} )
12
217Let ( boldsymbol{a}=overline{boldsymbol{i}}+overline{boldsymbol{j}}-overline{boldsymbol{k}}, overline{boldsymbol{b}}=overline{mathbf{5}} boldsymbol{i}-overline{mathbf{3}} boldsymbol{j}- )
( mathbf{3 k}, overline{boldsymbol{c}}=overline{mathbf{3}} mathbf{i}-boldsymbol{j}-mathbf{2} overline{mathbf{k}} . ) If collinear with ( overline{boldsymbol{c}} )
and ( |overline{boldsymbol{r}}|=|overline{boldsymbol{a}}+overline{boldsymbol{b}}| . ) Then ( overline{boldsymbol{r}} ) equals:
( mathbf{A} cdot pm 3 c )
B ( cdot pm frac{3}{2} sqrt[3]{c} )
( c cdot pm 2 c )
( mathbf{D} cdot pm frac{3}{2} overline{4 c} )
12
218If ( hat{a}, hat{b}, hat{c} ) are unit vectors such that ( hat{a} cdot hat{b}=0=hat{a} cdot hat{c} ) and the angle between ( hat{b} )
and ( hat{c} ) is ( pi / 3, ) then the value of ( |hat{boldsymbol{a}} times hat{boldsymbol{b}}-hat{boldsymbol{a}} times hat{boldsymbol{c}}| )
A . ( 1 / 2 )
B. 1
( c cdot 2 )
D. none of these
12
219Let ( vec{a} ) and ( vec{b} ) be two unit vectors if the vectors ( vec{c}=vec{a}+2 vec{b} ) and ( vec{d}=2 vec{a}- )
( 4 vec{b} ) are perpendicular to each other Then the angle between ( vec{a} ) and ( vec{b} ) is:
A ( cdot frac{pi}{6} )
в. ( frac{pi}{2} )
( c cdot frac{pi}{3} )
D.
12
220Find the cosine of the angle between two diagonals of a cube?
A ( cdot frac{1}{3} )
B. ( frac{2 sqrt{2}}{3} )
( c cdot frac{1}{2} )
D. None of thes
12
221If ( vec{a} ) be the position vector whose tip is ( (5,-3), ) find the coordinates of a point ( B ) such that ( overrightarrow{A B}=vec{a}, ) the coordinates of ( A )
being (4,-1)
A. (9,-4)
B. (-9,-4)
( c cdot(9,4) )
D. none of these
12
222f ( a, b, c ) and ( d ) are linearly independent
set of vectors and ( k_{1} a+k_{2} b+k_{3} c+ )
( k_{4} d=0, ) then
A. ( k_{1}+k_{2}+k_{3}+k_{4}=0 )
В. ( k_{1}+k_{3}=k_{2}+k_{4}=0 )
c. ( k_{1}+k_{2}=k_{3}+k_{4} )
D. None of the above
12
223If ( 2 bar{a}-4 hat{i}-2 widehat{j}+widehat{k}=0 ) then find ( bar{a} )
( mathbf{A} cdot 2 bar{j}+overline{mathbf{k}} )
B . ( 2 hat{i}+hat{j}-frac{1}{2} widehat{k} )
c. ( 2 bar{i}+overline{mathbf{j}} )
D. ( 2 overline{i-j} )
12
22423. Three lines L:r=ni,a ER
L:r=k+uġ, ue R and
Lir=i+ i + vk,VER
are given. For which point(s) Q on L, can we find a point P
on L, and a point R on L, so that P, Q and R are collinear?
(JEE Adv. 2019
(a) Â –
(b) Å
(d) +
12
225If ( vec{a} ) and ( vec{b} ) are unit vectors, then angle between ( vec{a} ) and ( vec{b} ) for ( sqrt{3} vec{a}-vec{b} ) to be unit
vector is
A ( .60^{circ} )
B. ( 90^{circ} )
( c cdot 45 )
D. 30
12
22644.
The values of a, for which points A, B, C with position
vectors 2 – j +Â, î -3j – 5k and ai – 3j+ k respectively are
the vertices of a right angled triangle with C = * are
(a) 2 and 1
(b) – 2 and -1 [2006]
(C) -2 and 1
(d) 2 and -1
12
227Vectors ( bar{a}, bar{b} ) are non-collinear find ( x, s ) o that the vectors ( (x-1) bar{a}+bar{b} ) and ( (2+ ) ( 3 x) bar{a}-2 bar{b} ) are collinear12
228If ( vec{a}=2 hat{m}+hat{n}, vec{b}=hat{m}-2 hat{n} ) and angle
between the unit vectors ( hat{m} ) and ( hat{n} ) is ( 60^{circ} )
( a, b ) are the sides of a
parallelogram,then the lengths of the diagonals are
A ( . sqrt{7}, sqrt{5} )
B. ( sqrt{13}, sqrt{5} )
C ( . sqrt{7}, sqrt{13} )
D. ( sqrt{11}, sqrt{13} )
12
229If ( vec{a}+vec{b} perp vec{a} ) and ( |vec{b}|=sqrt{2}|vec{a}| ), then?
( mathbf{A} cdot(2 vec{a}+vec{b}) | vec{b} )
B . ( (2 vec{a}+vec{b}) perp vec{b} )
c. ( (2 vec{a}-vec{b}) perp vec{b} )
D. ( (2 vec{a}+vec{b}) perp vec{a} )
12
23027. Let ā,b and
be three non-zero vectors such that no two
of these are collinear. If the vector a + 25 is collinear with
7 and 5 + 30 is collinear with a (a being some non-zero
scalar) then a + 25 + 60 equals
[2004]
(a) (b) no (c) no (d) na
12
231The vector ( bar{x} ) which is perpendicular to
(2,-3,1) and (1,-2,3) and which satisfies the condition
( bar{x}(bar{i}+overline{2} bar{j}-7 k)=10 ) is
12
2325.
Ifa=i+j+k, b = 4i +3j + 4k and c=i+aj +Bk are linearly
dependent vectors and|c1= 13, then (1998 – 2 Marks)
(a) a=1, B=-1 (b) a=1, B = +1
(c) a=-1, B=+1
(d) a= +1, B =1
12
233The vectors ( hat{i}-2 x hat{j}-3 y hat{k} ) and ( hat{i}+ ) ( 3 x hat{j}+2 y hat{k} ) are orthogonal to each other.
Then the locus of the point ( (x, y) ) is
A. Hyperbola
B. Ellipse
c. Parabola
D. circle
12
23420. If a,b,c andă are unit vectors such that
1
1
(axb).(ēxd) = 1 and àč = 5, then
(a) ā,b,c are non-coplanar
(b) b,c,d are non-coplanar
(c) b,d are non-parallel
(d) ă ă are parallel and b,c are parallel
12
235If ( 3 vec{a}+4 vec{b}-7 vec{c}=0 ) then the ratio in
which ( C(vec{c}) ) divides the join of ( boldsymbol{A}(overrightarrow{boldsymbol{a}}) ) and ( B(vec{b}) ) is
A .1: 2
B. 2: 3
( c cdot 3: 2 )
D. 4: 3
12
236If three vectors along coordinate axis represents the adjacent sides of a cube of length b, then the unit vector along its diagonal passing through the origin will be
( mathbf{A} cdot frac{hat{i}+hat{j}+hat{k}}{sqrt{2}} )
B. ( frac{i+3+hat{k}}{sqrt{36}} )
( mathbf{c} cdot hat{i}+hat{j}+hat{k} )
D. ( frac{i+vec{j}+hat{k}}{sqrt{3}} )
12
237If ( C ) is the mid point of ( A B ) and ( P ) is any
point outside ( A B, ) then
( mathbf{A} cdot overline{P A}+overline{P B}+overline{P C}=0 )
в. ( overline{P A}+overline{P B}+2 overline{P C}=overline{0} )
c. ( overline{P A}+overline{P B}=overline{P C} )
D. ( overline{P A}+overline{P B}=2 overline{P C} )
12
238The position vector of a point ( boldsymbol{P} ) such
that ( overline{O P} ) is inclined to ( O X ) at ( 45^{circ} ) and to
( O Y ) at ( 60^{circ} ) and ( O P=12 ) units, is
( mathbf{A} cdot 6(sqrt{2} hat{i}+hat{j} pm hat{k}) )
B . ( 3(sqrt{2} hat{imath}+hat{j} pm hat{k}) )
c. ( 2(sqrt{2} hat{i}+hat{j} pm hat{k}) )
D . ( 12(sqrt{2} hat{i}+hat{j} pm hat{k}) )
12
239The ratio ( frac{A X}{X D} ) is
( A cdot 5 )
2
B. 6
( c cdot 7 )
( overline{3} )
( D )
12
240The vector sum of (N) coplanar forces, each of magnitude ( F, ) when each force is making an angle of
( frac{2 pi}{N} ) with that preceding one, is :
( A cdot F )
B. ( frac{N F}{2} )
c. ( mathrm{NF} )
D. zero
12
241In a unit cube, find the angle between
the diagonal of the cube and a diagonal of a skew to it?
12
242Given ( A=hat{i}+2 hat{j}-3 hat{k} . ) When a vector
( B ) is added to ( A, ) we get a unit vector
along ( x ) -axis. Then ( B ) is
( mathbf{A} cdot-2 hat{j}+3 hat{k} )
В. ( -1 hat{i}-2 hat{j} )
c. ( -1 hat{i}+3 hat{k} )
D. 2 ( hat{j}-3 hat{k} )
12
243f ( vec{a} ) and bare unit vectors and ( theta ) is the
angle between them, show that ( sin frac{boldsymbol{theta}}{mathbf{2}}=frac{mathbf{1}}{mathbf{2}}|overrightarrow{boldsymbol{a}}-overrightarrow{boldsymbol{b}}| ? )
12
244( vec{A} ) and ( vec{B} ) are two vectors, find the angle
between them, if ( |overrightarrow{boldsymbol{A}} times overrightarrow{boldsymbol{B}}|=sqrt{mathbf{3}}(overrightarrow{boldsymbol{A}} cdot overrightarrow{boldsymbol{B}}) ) the value of is :-
A ( .90^{circ} )
В. ( 60^{circ} )
( c cdot 45 )
D. 30
12
245If the vertices of a ( Delta A B C ) are ( A= )
( (1,-1,-3), B=(2,1,-2) ) and ( C= )
(-5,2,-6) then the length of the
internal bisector of angle ( boldsymbol{A} ) is
A ( cdot frac{3 sqrt{10}}{2} )
в. ( frac{3 sqrt{10}}{5} )
c. ( frac{3 sqrt{10}}{7} )
D. ( frac{3 sqrt{10}}{4} )
12
246If ( overline{boldsymbol{a}}=hat{boldsymbol{i}}+hat{boldsymbol{j}}-mathbf{2} hat{boldsymbol{k}}, overline{boldsymbol{b}}=mathbf{2} hat{boldsymbol{i}}-mathbf{0} hat{boldsymbol{j}}+hat{boldsymbol{k}}, overline{boldsymbol{c}}= )
( mathbf{3} hat{mathbf{i}}-hat{boldsymbol{k}} ) and ( overline{boldsymbol{c}}=boldsymbol{m} overline{boldsymbol{a}}+boldsymbol{n} overline{boldsymbol{b}} ) then ( boldsymbol{m}+boldsymbol{n}= )
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
D. –
12
247If ( 4 i+7 j+8 k, 2 i+7 j+7 k ) and ( 3 i+ )
( 5 j+7 k ) are the position vectors of the
vertices ( A, B ) and ( C ) respectively of triangle ( A B C . ) The position vector of the point where the bisector of angle ( boldsymbol{A} )
meets ( B C ).
A ( cdot frac{1}{3}(5 j+12 k) )
B. ( frac{1}{3}(6 i+13 j+18 k) )
c. ( frac{2}{3}(6 i+8 j+6 k) )
D ( cdot frac{2}{3}(-6 i-8 j-6 k) )
12
24838. The resultant C of A and B is perpendicular to Ā.Also,
A = čl. The angle between A and B is
b. 31Trad
a. I rad
c. St rad
-rad
20
0
1
12
249The unit vector in the direction of ( overrightarrow{boldsymbol{a}} ) is
A ( cdot frac{vec{a}}{|vec{a}|} )
в. ( vec{a}|vec{a}| )
( c cdot a^{2} )
D.
12
250If ( [overrightarrow{boldsymbol{a}} overrightarrow{boldsymbol{b}} overrightarrow{boldsymbol{c}}]=1 ) then ( frac{overrightarrow{boldsymbol{a}} cdot overrightarrow{boldsymbol{b}} times overrightarrow{boldsymbol{c}}}{overrightarrow{boldsymbol{c}} times overrightarrow{boldsymbol{a}} cdot overrightarrow{boldsymbol{b}}}+frac{overrightarrow{boldsymbol{b}} cdot overrightarrow{boldsymbol{c}} times overrightarrow{boldsymbol{a}}}{overrightarrow{boldsymbol{a}} times overrightarrow{boldsymbol{b}} cdot overrightarrow{boldsymbol{c}}}+ )
( frac{vec{c} cdot vec{a} times vec{b}}{vec{b} times vec{c} cdot vec{a}} ) is equal to
( A cdot 3 )
B. 1
( c cdot-1 )
D. None of these
12
251If ( bar{a}=bar{i}+bar{j}+t bar{k} ) and ( bar{b}=bar{i}+2 bar{j}+3 bar{k} )
then the value of ( t ) for which ( bar{a}+bar{b} ) and
( bar{a}-bar{b} ) are perpendicular is
( A ldots pm 2 )
В. ( pm 2 sqrt{2} )
( mathbf{c} cdot pm 2 sqrt{3} )
( mathrm{D} cdot pm 3 )
12
2529. The angle between the vectors (î + ) and (ỉ + k) is
(a) 30° (b) 45° (c) 60° (d) 90°
12
253Evaluate ( int_{0}^{pi / 2} frac{cos ^{2} x}{1+3 sin ^{2} x} d x )
( A cdot frac{pi}{4} )
в.
c.
D.
12
254If ( vec{a}, vec{b}, vec{c} ) are three vectors such that ( vec{a} times ) ( overrightarrow{boldsymbol{b}}=overrightarrow{boldsymbol{c}}, overrightarrow{boldsymbol{b}} times overrightarrow{boldsymbol{c}}=overrightarrow{boldsymbol{a}}, overrightarrow{boldsymbol{c}} times overrightarrow{boldsymbol{a}}=overrightarrow{boldsymbol{b}} ) then prove
that ( |overrightarrow{boldsymbol{a}}|=|overrightarrow{boldsymbol{b}}|=|overrightarrow{boldsymbol{c}}| )
12
255If ( bar{x}, bar{y}, bar{z} ) are mutually perpendicular vectors of eqaul magnitude, then find the measure of an angle that ( bar{x}+bar{y}+bar{z} )
makes with any of the three vectors.
12
256If ( overrightarrow{boldsymbol{a}}=2 hat{boldsymbol{i}}+hat{boldsymbol{j}}+hat{boldsymbol{k}}, overrightarrow{boldsymbol{b}}=boldsymbol{3} hat{boldsymbol{i}}-boldsymbol{4} hat{boldsymbol{j}}+ )
( mathbf{2} hat{boldsymbol{k}}, overrightarrow{boldsymbol{c}}=hat{boldsymbol{i}}-mathbf{2} hat{boldsymbol{j}}+mathbf{2} hat{boldsymbol{k}} ) then the projection
of ( vec{a}+vec{b} ) on ( vec{c} ) is
A ( cdot frac{17}{3} )
в.
( c cdot frac{4}{3} )
D. ( frac{17}{sqrt{43}} )
12
257Let ( boldsymbol{O A B C} ) be a parallelogram and ( boldsymbol{D} ) the midpoint of ( boldsymbol{O} boldsymbol{A} ). The ratio in which
( O B ) divides ( C D ) in the ratio
A .1: 2
B. 1: 3
( c cdot 1: 4 )
D. 2: 1
12
258Let ā =î – j, b = j – Â , c = k – î . If d is a unit vector
such that a.d = 0 = [7 c d], then d equals (19958)
(a) + i + 9 – 2k a i + 9 – ĥ
+
16
c) i+jR
i +
it
(D) FR
12
259The position vectors of the points ( A, B, C ) and ( D ) are ( 3 hat{i}-2 hat{j}-hat{k}, 2 hat{i}+ )
( mathbf{3} hat{boldsymbol{j}}-mathbf{4} hat{boldsymbol{k}},-hat{mathbf{i}}+hat{boldsymbol{j}}+mathbf{2} hat{boldsymbol{k}} ) and ( mathbf{4} hat{boldsymbol{i}}+mathbf{5} hat{boldsymbol{j}}+boldsymbol{lambda} hat{boldsymbol{k}} )
respectively. If the points ( A, B, C ) and ( D )
lie on a plane, find the value of ( lambda )
12
2605.
The position vectors of the points A, B, C and D are
3î – 2j – k, 2î + 3j – 4k,-i + i + 2k and 4ỉ +5j +ak,
respectively. If the points A, B, C and D lie on a plane, find
the value of a.
(1986-2/2 Marks)
12
261If ( vec{b} ) is the vector whose initial point divides the joining ( 5 hat{i} ) and ( 5 hat{j} ) in the ratio ( lambda: 1 ) and terminal point is at origin. If ( |vec{b}| leq sqrt{37}, ) then ( lambda in )
( ^{mathbf{A}} cdot(-infty,-6] cupleft[-frac{1}{6}, inftyright) )
B ( cdot(-infty,-3) cupleft[-frac{1}{4}, inftyright) )
( ^{mathbf{c}} cdot(-infty, 0) cupleft(frac{1}{2}, inftyright) )
D. ( left[-6,-frac{1}{6}right] )
12
262If ( vec{n} ) is a unit vector in the direction of
the vector ( vec{A}, ) then:
( mathbf{A} cdot vec{n}=|vec{A}| / vec{A} )
В . ( vec{n}=vec{n} times vec{n} )
C ( cdot vec{n}=vec{A} /|vec{A}| )
D . ( vec{n}=vec{A}|vec{A}| )
12
263If ( vec{x} ) is a vector whose initial point divides the line joining ( 5 hat{i} ), and ( 5 hat{j} ) in the ratio ( lambda: 1 ) and the terminal point is the
origin. Also given ( |vec{x}| leq sqrt{37}, ) then ( lambda ) belongs to
( mathbf{A} cdotleft[-frac{1}{6}, frac{1}{6}right] )
B ( cdot(-infty,-6) cupleft(-frac{1}{6}, inftyright) )
c. ( (-infty,-8) )
D. ( (1, infty) )
12
264Using vectors, prove that angle in a semicircle is a right angle.12
265Find the area of a parallelogram having ( bar{a}=(4,-3,1), bar{b}=(2,-4,5) ) as
diagonals
12
266Given ( boldsymbol{A}=(mathbf{1}, mathbf{2}, mathbf{5}), boldsymbol{B}=(mathbf{5}, mathbf{7}, mathbf{9}) ) and
( C=(3,2,-1) . ) Find a until vector
normal to the plane of the triangle ABC.
A. ( frac{-13 i+9 j-7 k}{sqrt{(299)}} )
в. ( frac{13 i-9 j+7 k}{sqrt{(299)}} )
c. ( frac{-15 i+16 j-5 k}{sqrt{(506)}} )
D. ( frac{15 i-16 j+5 k}{sqrt{(506)}} )
12
267A unit vector is represented as ( 0.8 hat{i}+ ) ( b hat{j}+0.4 hat{k} . ) Hence the value of ( b ) must be
A . 0.4
B. ( sqrt{0.6} )
( c .0 .2 )
D. ( sqrt{0.2} )
12
268-22-1
12. A line I passing through the origin is perpendicular to the
lines
(JEE Adv. 2013)
4:03+t)i +(-1+2t)j+(4+2t)k, -oo«t<oo
12:(3+2s){ +(3+28) j+(2+s)k, -o0<s<oo
Then, the coordinate(s) of the point(s) only at a distance of
V17 from the point of int
12
269Show that the line of intersection of the
planes
( bar{r} cdot(hat{i}+3 hat{j}-2 hat{k})=0 ) and
( bar{r} cdot(2 hat{i}+4 hat{j}-3 hat{k})=0 ) is equally incline
to ( hat{i} ) and ( hat{j} ). Also find the angle it make with ( hat{k} ? )
12
270If ( vec{a} cdot hat{i}=4, ) then ( (vec{a} times hat{j}) cdot(2 hat{j}-3 hat{k}) ) is
equal to
A . 12
B . 2
c. 0
D. -12
12
271Find the direction angle of the vector ( boldsymbol{v}=mathbf{2}left(cos mathbf{3 0}^{o} hat{boldsymbol{i}}+sin mathbf{3 0}^{boldsymbol{o}} hat{boldsymbol{j}}right) )
A ( .90^{circ} )
B. ( 60^{circ} )
( c cdot 30^{circ} )
D. ( 0^{circ} )
12
272If ( boldsymbol{a}=(mathbf{0}, mathbf{1},-mathbf{1}) ) and ( boldsymbol{c}=(mathbf{1}, mathbf{1}, mathbf{1}) ) are
given vectors, then find ( |b|^{2}, ) where ( b )
satisfies ( a times b+c=0 ) and ( a cdot b=3 )
( A cdot 6 )
B. 12
( c cdot 18 )
D. 3
12
273The non-zero vectors ( bar{a}, bar{b} ) and ( bar{c} ) are related ( bar{a}=8 bar{b} ) and ( bar{c}=-7 bar{b} . ) Then the
angle between ( bar{a} ) and ( bar{c} )
( mathbf{A} cdot mathbf{0} )
в. ( pi / 4 )
c. ( pi / 2 )
D. ( pi )
12
274Let ( alpha, beta, gamma ) be the distinct real numbers. The vectors ( boldsymbol{alpha} hat{boldsymbol{i}}+boldsymbol{beta} hat{boldsymbol{j}}+boldsymbol{r} hat{boldsymbol{k}} )
( beta hat{i}+r hat{j}+alpha hat{k} ; gamma hat{i}+alpha hat{j}+beta hat{k} ) are
A. collinear
B. form an isosceles triangle
c. right angled triangle
D. equilatetral triangle
12
275VII LLIULPU
15 1.
7. Can you find at least one vector perpendicular to
3i – 4 + 7k?
1.
1.
:
:
12
276If ( S ) is the circumcenter, ( O ) is the orthocenter of ( triangle A B C, ) then ( overrightarrow{S A}+ ) ( overrightarrow{boldsymbol{S B}}+overrightarrow{boldsymbol{S C}}= )
A ( cdot 2 overrightarrow{O S} )
B . 2 SO
( c cdot partial s )
D. ( overrightarrow{S O} )
12
277If ( a, b ) and ( c ) are unit vectors then
( |a-b|^{2}+|b-c|^{2}+|c-a|^{2} ) does not
exceed
( mathbf{A} cdot mathbf{4} )
B. 9
( c cdot 8 )
D. 6
12
278Calculalte R,when ( boldsymbol{R}= ) ( sqrt{A^{2}+B^{2}+2 A B cos theta} ) given that, ( A=6 )
( mathrm{m}, mathrm{B}=8 mathrm{m} ) and angle is ( 60^{0} )
12
279The axes of co-ordinates are rotated
about z-axis through an angle of ( pi / 4 ) in anti-clockwise direction and the
components of a vector are ( 2 sqrt{2}, 3 sqrt{2}, 4 )
Then the components of the same vector in the original system are -1,5,4
A. ( a=5 i-j-4 k )
B. ( a=-5 i+j+4 k )
c. ( a=i-5 j-4 k )
D. ( a=-i+5 j+4 k )
12
280A point is on the ( x ) -axis. What are its ( y )
coordinate and ( z ) -coordinates?
12
281Prove using vectors : If the diagonals of a parallelogram are equal in length, then it is a rectangle.12
282( vec{a}=4 hat{i}-hat{j}+hat{k} quad ) and ( quad vec{b}=p hat{i}+2 hat{j}+ )
( 3 hat{k} ) are mutually perpendicular, then
find the value of ( mathrm{p} )
12
283Find the angle between the vectors ( 3 i+ )
( 2 j-6 k ) and ( 4 i-3 j+k )
12
284Let ( mathbf{A B C} ) be a triangle and let ( mathbf{D}, mathbf{E} ) be the midpoints of the sides ( mathbf{A B}, mathbf{A C} ) respectively,then ( hat{B} E+hat{D} C= )
( mathbf{A} cdot ) ВС
в. ( frac{1}{2} hat{B C} )
c. ( frac{3}{2} hat{B C} )
D. ( frac{3}{4} ) for
12
285A position-dependent force ( boldsymbol{F}=mathbf{4} boldsymbol{x}^{mathbf{3}}+ )
( 3 x^{2}+2 x-4 ) newton acts on a small
body of mass ( 2 k g ) and displaces it from ( x=0 ) to ( x=2 m . ) Calculate the work
done.
A . ( 20 J )
в. ( 24 J )
c. 16.5
D. ( 12 J )
12
286A terahedron has vertices ( boldsymbol{P}(mathbf{1}, mathbf{2}, mathbf{1}) )
( Q(2,1,3), R(-1,1,2) ) and ( O(0,0,0) )
The angle between the faces OPQ and PQR is :
A ( cdot cos ^{-1}left(frac{9}{35}right) )
B. ( cos ^{-1}left(frac{19}{35}right) )
c. ( cos ^{-1}left(frac{17}{31}right) )
D. ( cos ^{-1}left(frac{7}{31}right) )
12
287If ( vec{a}=x hat{i}+y hat{j}+z hat{k} ) makes equal angles with ( vec{b}=y hat{i}-2 z hat{j}+3 x hat{k} ) and ( vec{c}=2 z hat{i}+3 x hat{j}-y hat{k} ) and ( vec{a} perp vec{d}, ) where
( vec{d}=hat{i}-hat{j}+2 hat{k} . ) If ( |vec{a}|=2 sqrt{3}, ) then
( vec{a} cdot vec{b}= )
A . 12
B. -12
( c cdot 24 )
D. -24
12
288If ( vec{a}=x hat{i}+(x-1) hat{j}+hat{k} ) and ( vec{b}= )
( (x+1) hat{i}+hat{j}+a hat{k} ) always make an
acute angle with each other for every
value of ( x in R, ) then
A ( . a in(-infty, 2) )
B . ( a in(2, infty) )
c. ( a in(-infty, 1) )
D. ( a in(1, infty) )
12
289( overline{boldsymbol{x}}=(mathbf{1}, mathbf{1}, mathbf{1}) overline{boldsymbol{y}}=(mathbf{4}, mathbf{3}, mathbf{4}) boldsymbol{a} boldsymbol{n} boldsymbol{d} bar{z}= )
( (1, alpha, beta) ) are linearly dependent vectors
and ( |bar{z}|=sqrt{3}, ) then findaand ( beta )
12
29049
Ifa line makes an angle of 1/4 with the positive directions
of each of x- axis and y-axis, then the angle that the line
makes with the positive direction of the z-axis is [2007]
wa
12
29112.
If a, b and are three non coplanar vectors, then (1995
(ā + + c). [(a + b) x (a + c)] equals
(a) 0
(b) [ã b c]
(c) 2 [ã ē]
(d) – [a b c]
12
292The unit vector in the direction of ( vec{A}= )
( mathbf{3} hat{mathbf{i}}+mathbf{4} hat{mathbf{j}} ) is :
A ( cdot hat{A}=frac{hat{i}+hat{j}}{7} )
B. ( hat{A}=frac{3 hat{i}+4 hat{j}}{5} )
c. ( hat{A}=hat{i}+hat{j} )
D. ( _{hat{A}}=frac{4 hat{i}-3 hat{j}}{5} )
12
293If ( vec{r} times vec{b}=vec{c} times vec{b} & vec{r}, vec{a}=0 ) where
( overrightarrow{boldsymbol{a}}=boldsymbol{2} hat{boldsymbol{i}}+boldsymbol{3} hat{boldsymbol{j}}-hat{boldsymbol{k}}, overrightarrow{boldsymbol{b}}= )
( mathbf{3} hat{mathbf{i}}-hat{boldsymbol{j}}-boldsymbol{k}, overrightarrow{boldsymbol{c}}=hat{boldsymbol{i}}+hat{boldsymbol{j}}+hat{boldsymbol{k}} ) then ( overrightarrow{boldsymbol{r}} )
A ( cdot 2(hat{i}-hat{j}+hat{k}) )
B ( cdot 2(hat{i}+hat{j}-hat{k}) )
c. ( 2(-hat{i}+hat{j}+hat{k}) )
D. ( 2(hat{i}+hat{j}+hat{k}) )
12
294Show that the four points ( A, B, C, D ) with position vectors ( vec{a}, vec{b}, vec{c}, vec{d} ) respectively such that ( 3 vec{a}-2 vec{b}+5 vec{c}-6 vec{d}=overrightarrow{0}, ) are
coplanar. Also find the position vector of the point of intersection of the line
segments ( A C ) and ( B D )
12
295If ( bar{x} cdot bar{y}=bar{x} cdot bar{z} ) and ( bar{x} times bar{y}=bar{x} times bar{z} ) and
( bar{x} neq overline{0}, ) then prove that ( bar{y}=bar{z} )
12
29612. A vector B which has a magnitude 8.0 is added to a
vector A which lies along the x-axis. The sum of these
two vectors is a third vector which lie along the y-axis and
has a magnitude that is twice the magnitude of A. Find
the magnitude of A.
L
. Dia 2.21 bono mognitude
12
297If ( vec{A}=2 i+3 j ) and ( vec{B}=2 hat{j}+3 hat{k} ) the
component of ( bar{B} ) along ( bar{A} ) is
A. 6
B. 1/6
( c cdot 6 / 13 )
D. ( frac{6}{sqrt{13}} )
12
298The sine of the angle between the vectors ( hat{boldsymbol{i}}+boldsymbol{3} hat{boldsymbol{j}}+boldsymbol{2} hat{boldsymbol{k}} ) and ( boldsymbol{2} hat{boldsymbol{i}}-boldsymbol{4} hat{boldsymbol{j}}+hat{boldsymbol{k}} ) is
A ( cdot sqrt{frac{155}{156}} )
в. ( sqrt{frac{115}{116}} )
c. ( sqrt{frac{115}{147}} )
D. ( sqrt{frac{157}{158}} )
12
299If ( vec{a}, vec{b}, vec{c} ) are three vectors such that ( |vec{a}|=|vec{c}|=1,|vec{b}|=4,|vec{b} times vec{c}|=2 )
and ( 2 vec{b}=vec{c}+lambda vec{a} ) then ( lambda ) is
A ( cdot sqrt{65-8 sqrt{3}} )
B. ( sqrt{17} )
( c cdot sqrt{3} )
D. ( sqrt{frac{17}{2}(2+sqrt{3})} )
12
300If a, b,c are vectors such that [a b c]= 4 then
[ax b bxс сха] =
(a) 16 (6) 64
[2002]
(d) 8
(c) 4
12
301Expressing ( vec{a} ) and ( vec{b} ) as linear combinations of ( vec{c} ) and ( vec{d} ) we ( operatorname{get} vec{a}= ) ( boldsymbol{k} overrightarrow{boldsymbol{c}}+boldsymbol{m} overrightarrow{boldsymbol{d}} ) and ( overrightarrow{boldsymbol{b}}=boldsymbol{h} overrightarrow{boldsymbol{c}}+boldsymbol{r} overrightarrow{boldsymbol{d}} )
Find integer part of ( boldsymbol{k}+boldsymbol{m}+boldsymbol{h}+boldsymbol{r} ? )
( -boldsymbol{a}+boldsymbol{b}=-boldsymbol{c} ; mathbf{2} boldsymbol{a}-boldsymbol{b}=boldsymbol{c}+boldsymbol{d} )
12
302If vectors ( a ) and ( b ) be respectively equal ( operatorname{to} 3 hat{i}-4 hat{j}+5 hat{k} ) and ( 2 hat{i}+3 hat{j}-4 hat{k} . ) Find
the unit vector parallel to ( overrightarrow{boldsymbol{a}}+overrightarrow{boldsymbol{b}} )
12
303The non-zero vectors ( bar{a}, bar{b} ) and ( bar{c} ) are related ( bar{a}=8 bar{b} ) and ( bar{c}=-7 bar{b} . ) Then the
angle between ( bar{a} ) and ( bar{c} ) is –
A . 0
в.
c. ( frac{pi}{2} )
D. ( pi )
12
304The force ( F ) is exerted on a particle of mass 5 kg. If the particle starts from rest from origin and its position after 3
s is ( (4.5 hat{i}+13.5 hat{j}) m, ) then find ( vec{F} ).
A ( cdot(5 hat{i}+13.5 hat{j}) N )
B . ( (5 hat{i}+15 hat{j}) N )
c. ( (5 hat{i}+13 hat{j}) N )
D. ( (15 hat{i}+5 hat{j}) N )
12
305Four forces act on a point object. The
object will be in equilibrium, if:
A. all of them are in the same plane
B. they are opposite to each other in pairs
C. the sum of ( x, y ) and ( z ) – components of forces zero separately
D. they form a closed figure of 4 sides when added as Polygon law
12
306Assertion
Let ( bar{a}, bar{b}, bar{r} ) be the vectors such that ( bar{r}+ ) ( overline{boldsymbol{r}} times overline{boldsymbol{a}}=overline{boldsymbol{b}} operatorname{then}|overline{boldsymbol{r}}|^{2}=frac{(overline{boldsymbol{a}} cdot overline{boldsymbol{b}})^{2}+|overline{boldsymbol{b}}|^{2}}{mathbf{1}+|overline{boldsymbol{a}}|^{2}} )
Reason
( overline{boldsymbol{r}}=frac{(overline{boldsymbol{a}} cdot overline{boldsymbol{b}}) overline{boldsymbol{a}}+overline{boldsymbol{b}}+overline{boldsymbol{a}} times overline{boldsymbol{b}}}{mathbf{1}+|overline{boldsymbol{a}}|^{2}} )
A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
B. Both Assertion & Reason are individually true but Reason is not the correct (proper) explanation of Assertion
C. Assertion is true but Reason is false
D. Assertion is false but Reason is true
12
307Which of the following algebraic operations with scalar and vector physical quantities are meaningful.
(i) adding any two vectors
(ii) adding a scalar to a vector of the same dimensions
(iii) multiplying any vector by any scalar
(iv) multiplying any two scalars
(v) adding any two scalars
(vi) adding a component of a vector to the same vector.
A. Only (iii) & (iv) are permissible
B. Only (ii) & (iv) are permissible
c. only (iii) & (v) are permissible
D. Only (i) (iii) (iv) (v)& (vi) are permissible
12
308Vectors ( vec{a}, vec{b} ) and ( vec{c} ) are such that ( vec{a}+vec{b}+ ) ( vec{c}=overrightarrow{0} ) and ( |vec{a}|=3, mid vec{b}=5 ) and ( |vec{c}|=7 )
Find the angle between ( vec{a} ) and ( vec{b} )
12
309Illustration 3.14 Find the unit vector of A = 2î +3j + 2k.12
310Let ( vec{a}=hat{i}+hat{j}+hat{k}, vec{b}=hat{i}+3 hat{j}+5 hat{k} ) and
( vec{c}=7 hat{i}+9 hat{j}+11 hat{k} . ) Then the area of the
parallelogram with diagonals ( vec{a}+vec{b} ) and ( vec{b}+vec{c} ) is
A ( .4 sqrt{6} )
в. ( frac{1}{2} sqrt{21} )
c. ( frac{sqrt{6}}{2} )
D. ( sqrt{6} )
E ( cdot frac{1}{sqrt{6}} )
12
311Find the values of ( x ) for which the angle between the vectors ( vec{a}=x hat{i}-3 hat{j}-1 hat{k} ) and ( vec{b}=2 x hat{i}+x hat{j}-1 hat{k} ) is acute and
the angle between the vector b and ( y- ) axis is obtuse.
( mathbf{A} cdot x0 )
c. ( _{x}frac{1}{2} )
12
312( boldsymbol{A E} )
A ( cdot overrightarrow{A E}=frac{1}{2}(bar{a}+bar{b}) )
B ( cdot overrightarrow{A E}=frac{1}{2}(bar{a}-bar{b}) )
C ( cdot overrightarrow{A E}=frac{1}{2}(bar{b}-bar{a}) )
D. none of these
12
313Orthocentre of an equilateral triangle
( A B C ) is the origin O. If ( A=bar{a}, B= ) ( boldsymbol{b}, boldsymbol{C}=overline{boldsymbol{c}} ) then ( overline{boldsymbol{A} boldsymbol{B}}+mathbf{2} overline{boldsymbol{B} boldsymbol{C}}+boldsymbol{3} overline{boldsymbol{C} boldsymbol{A}}= )
A . ( 3 bar{c} bar{c} bar{c} )
B. ( 3 bar{a} bar{a} bar{a} ) a
( c cdot 0 )
( D .3 bar{b} bar{b} )
12
314||( vec{a}|=3,| vec{b}|=4,| vec{c} mid=5, vec{a} perp(vec{b}+ )
( vec{c}), vec{b} perp(vec{c}+vec{a}) ) and ( vec{c} perp(vec{a}+vec{b}) ) then
( sqrt{2}|vec{a}+vec{b}+vec{c}| ) is equal to
12
315resultant of the three forces
hown in the figure is
( (sqrt{2}+1) )
12
316If ( a ) and ( b ) are two non-zero non-collinear
vectors then ( a+3 b ) and ( a-3 b ) are:
A. Linearly independent.
B. Linearly dependent
c. May be both
D. None of these
12
317If ( hat{boldsymbol{i}}+widehat{boldsymbol{j}}, widehat{boldsymbol{j}}+widehat{boldsymbol{k}}, hat{boldsymbol{i}}+widehat{boldsymbol{k}} ) are the position
vectors of the vertices of a ( Delta A B C )
taken in order, then ( angle A ) is equal to
A ( cdot frac{pi}{2} )
в.
c.
D.
E ( cdot frac{pi}{3} )
12
318If ( vec{a}, vec{b}, vec{c} ) are three unit vector such that ( overrightarrow{boldsymbol{a}} times overrightarrow{boldsymbol{b}}=overrightarrow{boldsymbol{c}}, overrightarrow{boldsymbol{b}} times overrightarrow{boldsymbol{c}}=overrightarrow{boldsymbol{a}}, overrightarrow{boldsymbol{c}} times overrightarrow{boldsymbol{a}}=overrightarrow{boldsymbol{b}} . ) Show
that ( vec{a}, vec{b}, vec{c} ) from an orthogonal right handed tried of unit vectors.
12
319If a vector ( vec{A} ) makes angles ( alpha, beta ) and ( gamma )
with ( X, Y ) and ( Z ) axes respectively then ( sin ^{2} alpha+sin ^{2} beta+sin ^{2} gamma= )
A .
B.
( c cdot 2 )
D. 3
12
320If ( bar{a}, bar{b}, bar{c} ) be three unit vectors, such that ( bar{a}+bar{b}+bar{c} ) is also a unit vector and ( alpha_{1} )
( alpha_{2}, alpha_{3} ) be the angles between ( bar{a} ) and ( bar{b}, bar{b} )
and ( bar{c}, bar{c} ) and ( bar{a} ) respectively, then ( alpha_{1}, alpha_{2} )
( boldsymbol{alpha}_{3} )
A. All are acute angles
B. All are right angles
C. Has at least one among them obtuse
D. None of these
12
321The resultant of two forces acting at an
angle of ( 150^{circ} ) is ( 10 mathrm{N} ) and its
perpendicular to one of the forces.The
two other force is:
A ( cdot frac{20}{sqrt{3}} N )
в. ( frac{10}{sqrt{3}} N )
c. ( 20 N )
D. ( frac{20}{3} N )
12
32241. Two adjacent sides of a parallelogram ABCD are given by
AB = 2î+10î+11k and AD = i +2j+2Ã
The side AD is rotated by an acute angle a in the plane of
the parallelogram so that AD becomes AD’. If AD’ makes a
right angle with the side AB, then the cosine of the angle a
is given by
(2010)
8
(a) Ō (b)
(d) 4/5
12
323A particle moves from the point ( (2.0 hat{i}+ ) ( 40 hat{j}) m, ) at ( t=0, ) with an initial velocity ( (5.0 hat{i}+4.0 hat{j}) m s^{-1} . ) It is acted upon by a
constant force which produces a constant acceleration ( (4.0 hat{i}+ ) ( 4.0 hat{j}) m s^{-2} . ) What is the distance of the
particle from the origin at time ( 2 s ? )
A ( .20 sqrt{2} mathrm{m} )
B. ( 10 sqrt{2} mathrm{m} )
( c .5 mathrm{m} )
D. ( 15 mathrm{m} )
12
32410.
If a, b, c are non coplanar unit vectors such that
axbx) = (b + c), then the angle between a and 5
(1995)
ã
(e) 1/2
(d) i
12
325Show that the vectors ( 2 hat{i}-hat{j}+hat{k}, hat{i}- ) ( mathbf{3} hat{boldsymbol{j}}-mathbf{5} hat{boldsymbol{k}} quad ) and ( quad mathbf{3} hat{mathbf{i}}-mathbf{4} hat{boldsymbol{j}}-mathbf{4} hat{boldsymbol{k}} ) form the
vertices of a right angled triangle.
12
326If ( |overrightarrow{boldsymbol{a}}|=mathbf{5},|overrightarrow{boldsymbol{a}}-overrightarrow{boldsymbol{b}}|=mathbf{8} ) and ( |overrightarrow{boldsymbol{a}}+overrightarrow{boldsymbol{b}}|= )
( mathbf{1 0}, ) then ( |overrightarrow{boldsymbol{b}}| ) is equal to:
A .
в. ( sqrt{57} )
( c cdot 3 )
D. 57
12
327Let ( overrightarrow{boldsymbol{a}}=boldsymbol{a}_{1} hat{boldsymbol{i}}+boldsymbol{a}_{2} hat{boldsymbol{j}}+boldsymbol{a}_{3} hat{boldsymbol{k}}, overrightarrow{boldsymbol{b}}=boldsymbol{b}_{1} hat{boldsymbol{i}}+ )
( boldsymbol{b}_{2} hat{boldsymbol{j}}+boldsymbol{b}_{3} hat{boldsymbol{k}}, overrightarrow{boldsymbol{c}}=boldsymbol{c}_{1} hat{boldsymbol{i}}+boldsymbol{c}_{2} hat{boldsymbol{j}}+boldsymbol{c}_{3} hat{boldsymbol{k}} . ).If ( |overrightarrow{boldsymbol{c}}|= )
1 and ( (vec{a} times vec{b}) times vec{c}=0 ) then
( left|begin{array}{lll}boldsymbol{a}_{1} & boldsymbol{a}_{2} & boldsymbol{a}_{3} \ boldsymbol{b}_{1} & boldsymbol{b}_{2} & boldsymbol{b}_{3} \ boldsymbol{c}_{1} & boldsymbol{c}_{2} & boldsymbol{c}_{3}end{array}right|^{2}= )
A . 0
в.
c. ( |vec{a}|^{2}|vec{b}|^{2} )
D. ( |vec{a} times vec{b}|^{2} )
12
328The unit vector which is orthogonal to the vector ( vec{a}=3 vec{i}+2 vec{j}+6 vec{k} ) and is coplanar with the vectors ( overrightarrow{boldsymbol{b}}=mathbf{2} overrightarrow{mathbf{i}}+overrightarrow{boldsymbol{j}}+ )
( vec{k} ) and ( vec{c}=vec{i}-vec{j}+vec{k} ) is
A ( cdot frac{1}{sqrt{41}}(2 vec{i}-6 vec{j}+vec{k}) )
B. ( frac{2 vec{i}-3 vec{j}}{sqrt{13}} )
c. ( frac{3 vec{j}-vec{k}}{sqrt{10}} )
D. ( frac{4 vec{i}+3 vec{j}-3 vec{k}}{sqrt{14}} )
12
3294. If Ax B=C, then which of the following statements is
wrong
(a) CIA
(b) CLB
(c) ČLA+B) (d) ČI A x B)
12
330Which of the following is/are true?
This question has multiple correct options
A. Two unit vectors may not be equal unless they have the same direction.
B. A unit vector is self reciprocal.
C. Two unit vectors may be equal irrespective of their direction.
D. All of the above are correct.
12
33183. Let a = 2î+j-2k and 5=i+j. Let č be a vector such
that |-ā= 3, (axb)x 7 = 3 and the angle between a
and a xh be 30°. Then a.c is equal to: [JEE M 2017
-100
(c) 2
(d) 5
12
332A vector ( vec{B} ) which has a magnitude 8.0 is added to a vector ( vec{A} ) which lie along
the ( x ) -axis. The sum of these two vectors
is a third vector which lie along the ( y )
axis and has a magnitude that is twice the magnitude of ( vec{A} . ) The magnitude of
( vec{A} ) is:
A. ( frac{8}{sqrt{5}} )
B. ( frac{10}{sqrt{5}} )
c. ( frac{8}{sqrt{3}} )
D. ( frac{8}{sqrt{6}} )
12
333Given that ( vec{a} cdot vec{b}=0 ) and ( vec{a} times vec{b}=overrightarrow{0} )
What can you conclude about the vectors ( vec{a} ) and ( vec{b} ? )
12
334() LUI
2. If A=3î + + 2Â and B = 2ỉ – 2î + 4k, then value of
|AX BI will be
(a) 872 (b) 813 (c) 875 (d) 518
T
.
12
33513. When A · B = -AI IBI then
(a) A and B are perpendicular to each other
(b) A and B act in the same direction
(c) A and B act in the opposite direction
(d) A and B can act in any direction
12
336A vector ( vec{A} ) points vertically downward(south)and ( vec{B} ) points towards east, then the vector product ( vec{A} times vec{B} ) is:
A . Along west
B. Along east
c. zero
D. Outwards or inwards
12
337If pitqj is a unit vector perpendicular to ( 4 i+3 j, ) then
A. P= ( -3 / 5 ) and ( q=-4 / 5 )
B. P=4/5 and q=3/5
c. ( P=2 / 5 ) and ( q=1 / 5 )
D. None
12
338If ( |vec{A}+vec{B}|=|vec{A}|=|vec{B}|, ) then angle
between ( vec{A} ) and ( vec{B} ) is
A ( cdot 0^{circ} )
B. ( 60^{circ} )
( c .90^{circ} )
D. ( 120^{circ} )
12
339Illustration 3.32 Find the value of m so that the
vector 3i – 2j+k may be perpendicular to the vector
zi +67 + mk.
12
340If ( G ) is the centroid of the triangle ( A B C ) then ( overrightarrow{G A}+overrightarrow{G B}+overrightarrow{G C} ) is equal to
( mathbf{A} cdot overrightarrow{A B} )
в. ( overrightarrow{B C} )
( c cdot 4 overrightarrow{G A} )
D.
12
341The position vectors of ( boldsymbol{A}, boldsymbol{B}, boldsymbol{C}, boldsymbol{D} ) are
( vec{a}, vec{b}, 2 vec{a}+3 vec{b} ) and ( vec{a}-2 vec{b} ) respectively
Show that ( overrightarrow{D B}=3 vec{b}-vec{a} ) and ( overrightarrow{A C}= ) ( vec{a}+3 vec{b} )
12
342Find the angle between the vectors ( hat{i}- ) ( 2 widehat{j}+3 widehat{k} ) and ( 3 hat{i}-2 widehat{j}+widehat{k} )12
34314. If vectors Ā=î +2j + 4k and B = 5î represent the two
sides of a triangle, then the third side of the triangle can
have length equal to
a. 6
b. 56
c. Both of the above d. None of the above

1 – 1
12
344If ( vec{a}, vec{b}, vec{c} ) are unit vectors such that ( vec{a} cdot vec{b}=0=vec{a} cdot vec{c} ) and the angle between ( vec{b} )
and ( vec{c} ) is ( pi / 3, ) then the value to ( |vec{a} times vec{b}-vec{a} times vec{c}| ) is ( ? )
A. ( 1 / 2 )
B.
( c cdot 2 )
D. 3
12
345Evaluate the following:
( [mathbf{2} hat{mathbf{i}} hat{mathbf{j}} hat{mathbf{k}}]+[hat{mathbf{i}} hat{mathbf{k}} hat{mathbf{j}}]+[hat{boldsymbol{k}} hat{mathbf{j}} mathbf{2} hat{mathbf{i}} )
12
346Vectors ( vec{a}=hat{i}+2 hat{j}+3 hat{k}, vec{b}=2 hat{i}-hat{j}+hat{k} )
and ( vec{c}=3 hat{i}+hat{j}+4 hat{k} ) are so placed that
the end point of one vector is the starting point of the next vector, then
the vectors are
A. Not coplanar
B. Coplanar but cannot form a triangle
c. coplanar and form a triangle
D. coplanar and can form a right-angled triangle
12
347( operatorname{Let} vec{a}=vec{i}+2 vec{j}+vec{k}, vec{b}=vec{i}-vec{j}+vec{k} ) and
( vec{c}=vec{i}+vec{j}-vec{k} . A ) vector in the plane of ( vec{a} )
and ( vec{b} ) has projection ( frac{1}{sqrt{3}} ) on ( vec{c} ). Then, one
such vector is
A ( cdot 4 vec{i}+vec{j}-4 vec{k} )
B. ( 3 vec{i}+vec{j}-3 vec{k} )
c. ( 4 vec{i}-vec{j}+4 vec{k} )
D. ( 2 vec{i}+vec{j}-2 vec{k} )
12
348If ( alpha, beta ) are roots of the equation ( x^{2}+ ) ( mathbf{2} boldsymbol{x}+mathbf{5}=mathbf{0} ) and ( overrightarrow{boldsymbol{a}}=(boldsymbol{alpha}+boldsymbol{beta}) hat{mathbf{i}}+ )
( boldsymbol{alpha} boldsymbol{beta} hat{boldsymbol{j}} overrightarrow{boldsymbol{b}}=boldsymbol{alpha} boldsymbol{beta} hat{boldsymbol{i}}+(boldsymbol{alpha}+boldsymbol{beta}) hat{boldsymbol{j}}+left(boldsymbol{alpha}^{2}+boldsymbol{beta}^{2}right) hat{boldsymbol{k}} )
( operatorname{then} vec{a} times vec{b}= )
A ( . hat{i}+12 hat{j}+12 hat{k} )
B . ( -30 hat{i}+12 hat{j}-4 hat{k} )
c. ( -30 hat{i}-12 hat{j}-21 hat{k} )
D. ( hat{i}-12 hat{j}+29 hat{k} )
12
349If a vector ( vec{A}=3 hat{i}-7 ) jbk ( a n d vec{B}=2 hat{j}- )
( 3 hat{k} ) are perpendicular find to each other find value ( =mathbf{b} )
12
350A motor car is going due north at a speed of ( 50 mathrm{km} / mathrm{hr} ). It makes a ( 90^{circ} ) left
turn without changing the speed. The change in the velocity of the car is about.
A. ( 50 sqrt{2} mathrm{km} / mathrm{hr} )
B. ( 70 mathrm{km} / mathrm{hr} )
c. ( 80 mathrm{km} / mathrm{hr} )
D. zero
12
35125. When two vectors of magnitudes P and Q are mm
an angle e, the magnitude of their resultant is 2P. When
the inclination is changed to 180° – 0, the magnitude of
the resultant is halved. Find the ratio of P to Q.
12
3524. Find the resultant of three vectors OA, OB and
OC shown in the following figure. Radius of the circle
is R.
450
(a) 2R
(c) RV2
(b) R(1+V2)
(d) R (V2 – 1)
12
353( overrightarrow{boldsymbol{a}}=hat{boldsymbol{i}}+hat{boldsymbol{j}}+hat{boldsymbol{k}}, overrightarrow{boldsymbol{b}}=2 hat{boldsymbol{i}}-boldsymbol{4} hat{boldsymbol{k}}, overrightarrow{boldsymbol{c}}=hat{boldsymbol{i}}+ )
( lambda hat{j}+3 hat{k} ) are coplanar, then the value of
( lambda ) is
( A cdot frac{5}{2} )
B. ( frac{3}{5} )
( c cdot frac{5}{3} )
D. None of these
12
354If the magnitudes of two vectors ( vec{a} ) and ( vec{b} ) are equal then which one of the following is correct?
A ( cdot(vec{a}+vec{b}) ) is parallel to ( (vec{a}-vec{b}) )
в. ( (vec{a}+vec{b}) bullet vec{a}-vec{b} )
C . ( (vec{a}+vec{b}) ) is perpendicular to ( (vec{a}-vec{b}) )
D. None of the above
12
355The position vectors of ( boldsymbol{A}, boldsymbol{B}, boldsymbol{C} ) are ( overline{boldsymbol{i}}+ )
( overline{boldsymbol{j}}+overline{boldsymbol{k}}, boldsymbol{4} overline{boldsymbol{i}}+overline{boldsymbol{j}}+overline{boldsymbol{k}}, boldsymbol{4} overline{boldsymbol{i}}+mathbf{5} overline{boldsymbol{j}}+overline{boldsymbol{k}} . ) Then
the position vector of the circumcentre
of the triangle ( A B C ) is
( mathbf{A} cdot 3 hat{i}+2 hat{j}+hat{k} )
B. ( frac{1}{2}(6 hat{i}+hat{j}+hat{k}) )
c ( cdot frac{1}{2}(5 hat{i}+6 hat{j}+2 hat{k}) )
D. ( frac{1}{2}(9 bar{i}+7 bar{j}+3 bar{k}) )
12
356Direction angle of a vector is ( 30^{circ}, ) then
find the vector.
B. ( sqrt{3} b hat{i}+b hat{j} )
c. ( a hat{i}+b hat{j} )
D. ( sqrt{3} a hat{i}+sqrt{3} 3 hat{j} )
12
35711. A particle acted on by constant forces 4ỉ +-3k and
3i+ – is displaced from the point i+2j-3ſ to the
point 5î+49+. The total work done by the forces is
(a) 50 units
(b) 20 units [2003]
(c) 30 units
1) 40 units.
12
358the last sum 10 UN
12. A vector A points vertically upward and B points
towards north. The vector product AxB istom
(a) Zero
(b) Along west
(c) Along east
(d) Vertically downward
12
359Prove that the triangle, whose position vectors of the vertices are ( 2 hat{i}+4 hat{j}- ) ( hat{boldsymbol{k}}, boldsymbol{4} hat{boldsymbol{i}}+mathbf{5} hat{boldsymbol{j}}+hat{boldsymbol{k}} ) and ( boldsymbol{3} hat{boldsymbol{i}}+boldsymbol{6} hat{boldsymbol{j}}-boldsymbol{3} hat{boldsymbol{k}} )
respectively, is an isosceles right angled triangle.
12
360The points ( O, A, B, C, D ) are such that ( boldsymbol{O A}=boldsymbol{a}, boldsymbol{O B}=boldsymbol{b}, boldsymbol{O C}=boldsymbol{2} boldsymbol{a}+boldsymbol{3 b} ) and
( boldsymbol{O} boldsymbol{D}=boldsymbol{a}-boldsymbol{2 b} . ) If ( |boldsymbol{a}|=boldsymbol{3}|boldsymbol{b}|, ) then the
angle between ( B D ) and ( A C ) is
A . ( pi )
в. ( frac{pi}{2} )
( c cdot frac{pi}{3} )
D. None of these
12
36181. Let a, b and c be three unit vectors such that
ax bxc
b+ c . If b is not parallel to c, then
the angle between a and b is:
[JEE M 2016]
12
362Prove that ( ,(vec{a} cdot vec{b})^{2} leqslant|vec{a}|^{2} cdot|vec{b}|^{2} )12
363Find a vector in which two of the three
direction angles are ( boldsymbol{alpha}=mathbf{7} mathbf{5}^{boldsymbol{o}} ) and ( boldsymbol{beta}= )
( mathbf{5 5}^{circ} )
( mathbf{A} cdot v=cos 55^{o} hat{i}+cos 75^{circ} hat{j}+cos 39^{o} hat{k} )
B . ( v=cos 75^{circ} hat{i}+cos 55^{circ} hat{j}+cos 39^{circ} hat{k} )
c. ( v=cos 55^{circ} hat{imath}+cos 39^{circ} hat{jmath}+cos 75^{circ} hat{k} )
D . ( v=cos 75^{circ} hat{i}+cos 39^{circ} hat{j}+cos 55^{circ} hat{k} )
12
364In a triangle ( A B C, ) right angled at the
vertex ( A, ) if the position vectors of ( A, B ) and ( C ) are respectively ( 3 hat{i}+hat{j}-hat{k},-hat{i}+ ) ( mathbf{3} hat{boldsymbol{j}}+boldsymbol{p} hat{boldsymbol{k}} ) and ( mathbf{5} hat{boldsymbol{i}}+boldsymbol{q} hat{boldsymbol{j}}-boldsymbol{4} hat{boldsymbol{k}}, ) then the
point ( (p, q) ) lies on a line:
A. Making an obtuse angle with the positive direction of x-axis
B. Making an acute angle with the positive direction of x axis
c. Parallel to x-axis
D. Parallel to y-axis
12
365( A B C ) is a triangle, the point ( P ) is on side ( B C ) such that ( 3 overrightarrow{B P}=2 overrightarrow{P C} ), the
point ( Q ) is on the line ( overrightarrow{C A} ) such that ( 4 overrightarrow{C Q}=overrightarrow{Q A} . ) If ( R ) is the common point of ( overrightarrow{A P} & overrightarrow{B Q}, ) then the ratio in which the line joining ( boldsymbol{C} boldsymbol{R} ) divides ( overrightarrow{boldsymbol{A}} boldsymbol{B} ) is
A .2: 5
B. 3: 8
c. 4: 1
D. 6: 1
12
366If ( vec{a} ) and ( vec{b} ) are non-collinear vectors and
( boldsymbol{A}=(boldsymbol{p}+mathbf{4} boldsymbol{q}) boldsymbol{a}+(boldsymbol{2} boldsymbol{p}+boldsymbol{q}+mathbf{1}) boldsymbol{b} )
( boldsymbol{B}=(-mathbf{2} boldsymbol{p}+boldsymbol{q}+mathbf{2}) boldsymbol{a}+(mathbf{2} boldsymbol{p}-boldsymbol{3} boldsymbol{q}-mathbf{1}) boldsymbol{b} )
then determine ( p ) and ( q, ) so that ( 3 A= )
( 2 B )
A. ( p=1 ) and ( q=-0.3 )
B. ( p=-1 ) and ( q=1.1 )
C. ( p=2 ) and ( q=-1 )
D. ( p=-2 ) and ( q=1.8 )
12
367( f )
( vec{a}, b, vec{c} ) be any three non-zero, non-
vecto topp
Happy Father’s Day
( c )
D.
AS
12
36843.
If (āx5x7=āxoxa) where a, 5 and ē are any three
vectors such that a b +0.5.C+0 then ā and į are
[2006]
(a) inclined at an angle of
between them
la wa
(b) inclined at an angle of
between them
() perpendicular
(d) parallel
12
369If ( vec{a}+vec{b}+vec{c}=0, ) then the angle ( theta ) between ( vec{b} ) and ( vec{c} ) is given by
( ^{mathrm{A}} cdot cos theta=frac{a^{2}-b^{2}-c^{2}}{2 b c} )
B. ( cos theta=frac{b^{2}+c^{2}-a^{2}}{2 b c} )
c. ( cos theta=frac{a^{2}+b^{2}-c^{2}}{2 b c} )
D. none of these
12
3706. The vector that must be added to the vector i – 3j +2k
and 3i +69-7k so that the resultant vector is a unit vector
along the y-axis is
(a) 4 +27 +5h (b) – 41 – 2 + 5 ha
(c) 3î +4j+5k (d) Null vector
12
371The velocity vector of a sphere after it hits a vertical wall which is parallel to ( hat{j} ) is ( (-hat{mathbf{i}}+mathbf{3} hat{mathbf{j}}) ) on a smooth horizontal
surface. The coefficient of restitution
between ball and wall is (1/2). Find the
velocity vector of sphere immediately before collision:
( mathbf{A} cdot hat{i}+3 hat{j} )
В. ( -2 hat{i}+3 hat{j} )
c. ( -hat{i}+3 hat{j} )
D. ( 2 hat{i}+3 hat{j} )
12
372Given, ( |vec{a}|=|vec{b}|=1 ) and ( |vec{a}+vec{b}|=sqrt{3} .1 )
( vec{c} ) be a vector such that ( vec{c}-vec{a}-2 vec{b}= ) ( mathbf{3}(overrightarrow{boldsymbol{a}} times overrightarrow{boldsymbol{b}}), ) then ( overrightarrow{boldsymbol{c}} . overrightarrow{boldsymbol{b}} ) is equal to
A ( cdot-frac{1}{2} )
в. ( frac{1}{2} )
( c cdot frac{3}{2} )
D.
12
373The ratio in which ( bar{i}+2 bar{j}+3 bar{k} ) divides the join of ( -2 bar{i}+3 bar{j}+5 bar{k} ) and ( 7 bar{i}-bar{k} ) is
( mathbf{A} cdot-3: 2 )
B. 1: 2
( c cdot 2: 3 )
D. -4: 3
12
37494. Let à = 3i+j and B = 2î – i+3k.
If B = B; -B2, where B; is parallel to ã and ß, is
perpendicular to ā , then B, xB2 is equal to:
[JEE M 2019-9 April (M)
(a) -3ỉ+9ị+sť (b) zî – 9j – 5k
© }(-3i+99+5k) (a) (31–9j+5k)
12
375If ( vec{b}=3 hat{i}+4 hat{j} ) and ( vec{a}=hat{i}-hat{j} ) the vector
having the same magnitude as that of ( vec{b} ) and parallel to ( vec{a} ) is
A ( cdot frac{5}{sqrt{2}}(hat{i}-hat{j}) )
B. ( frac{5}{sqrt{2}}(hat{i}+hat{j}) )
( mathbf{c} cdot 5(hat{i}+hat{j}) )
D. ( 5(hat{i}-hat{j}) )
12
376If ( 4 bar{i}+7 bar{j}+8 bar{k}, 2 bar{i}+3 bar{j}+4 bar{k} ) and ( 2 bar{i}+ )
( 5 bar{j}+7 bar{k} ) are the position vectors of the
vertices ( A, B ) an ( C ) of triangle ( A B C, ) the position vector of the point where the bisector of ( angle mathrm{A} ) meets BC is?
A ( cdot frac{2}{3}(-6 bar{i}-8 bar{j}-6 bar{k}) )
B ( cdot frac{2}{3}(6 bar{i}+8 bar{j}+6 bar{k}) )
c. ( frac{1}{3}(6 bar{i}+13 bar{j}+18 bar{k}) )
D. ( 2(bar{i}+bar{j}+bar{k}) )
12
377A vector of length ( l ) is turned through
the angle ( theta ) about its tail. What is the
change in the position vector of its head-
A. ( l cos (theta / 2) )
B. ( 2 l sin (theta / 2) )
c. ( 2 l cos (theta / 2) )
D. ( l sin (theta / 2) )
12
378( operatorname{Let} overrightarrow{boldsymbol{a}}=hat{boldsymbol{i}}+hat{boldsymbol{j}}+hat{boldsymbol{k}}, overrightarrow{boldsymbol{b}}=hat{boldsymbol{i}}-hat{boldsymbol{j}}+hat{boldsymbol{k}} ) and
( vec{c}=hat{i}-hat{j}-hat{k} ) be three vectors. A vector ( vec{v} ) in the plane of ( vec{a} ) and ( vec{b}, ) whose projection on ( vec{c} ) is ( frac{1}{sqrt{3}}, ) is given by
A ( cdot hat{i}-3 hat{j}+3 hat{k} )
B . ( -3 hat{i}-3 hat{j}-3 hat{k} )
( mathbf{c} cdot_{3 hat{i}-hat{j}+3 hat{k}} )
D ( cdot hat{i}+3 hat{j}-3 hat{k} )
12
379Let the vectors ( vec{a} ) and ( vec{b} ) be such that
( |vec{a}|=3 ) and ( |vec{b}|=frac{sqrt{2}}{3}, ) then ( vec{a} times vec{b} ) is a
is
A ( cdot frac{pi}{6} )
в.
( c cdot frac{pi}{3} )
D.
12
380A scooter going to the east at 10
( mathrm{m} / mathrm{s} ) turns right through an angle of ( 90^{circ} )
If the speed of the scooter remains unchanged in taking this turn, the change in the velocity of the scooter is:
A. 20 ( mathrm{m} / mathrm{s} ) in south-west direction
B. zero
c. ( 10.0 mathrm{m} / mathrm{s} ) in south direction
D. 14.14 m/s in south-western direction
12
381Find the vector equation of a plane at a distance of 5 units from the origin and has ( hat{i} ) as the unit vector normal to it.12
382If ( x, y ) and ( z ) are three unit vectors in
three-dimensional space, then the minimum value of ( |hat{boldsymbol{x}}+hat{boldsymbol{y}}|^{2}+mid hat{boldsymbol{y}}+ )
( left.hat{z}right|^{2}+|hat{z}+hat{x}|^{2} ) is :
A ( cdot frac{3}{2} )
B. 3
c. ( 3 sqrt{3} )
D. 6
12
383( ln Delta A B C, P, Q, R ) are points on
( B C, C A, A B ) respectively, dividing them in the ratio 1: 4,3: 2 and 3: 7
The point ( S ) divides ( A B ) in the ratio 1: 3
Then ( frac{|overrightarrow{boldsymbol{A P}}+overline{boldsymbol{B} Q}+overline{boldsymbol{C} boldsymbol{R}}|}{|overline{boldsymbol{C S}}|} ) is
A ( cdot frac{1}{5} )
B. ( frac{2}{5} )
( c cdot frac{5}{2} )
D. ( frac{7}{10} )
12
384If ( overrightarrow{boldsymbol{a}} cdot hat{boldsymbol{i}}=overrightarrow{boldsymbol{a}} cdot(hat{boldsymbol{i}}+hat{boldsymbol{j}})=overrightarrow{boldsymbol{a}} cdot(hat{boldsymbol{i}}+hat{boldsymbol{j}}+hat{boldsymbol{k}})= )
1 then ( vec{a}= )
A. ( hat{i}+hat{j} )
в. ( hat{i}-hat{k} )
( c )
D. ( hat{i}+hat{j}-hat{k} )
12
385For ( boldsymbol{O} ) being the origin and 3 points ( boldsymbol{P}, boldsymbol{Q} ) and ( R ) lie on a plane. If ( overrightarrow{P O}+overrightarrow{O Q}= ) ( overrightarrow{Q O}+overrightarrow{O R}, ) then ( P, Q, R ) are
A. the vertices of an equilateral triangle
B. the vertices of an isoceles triangle
c. collinear
D. none of these
12
386Let ( a=2 i+j-2 k ) and ( b=i+j . ) If ( c ) is
a vector such that ( quad boldsymbol{a} . boldsymbol{c}=|boldsymbol{c}| cdot|boldsymbol{c}-boldsymbol{a}|= )
( 2 sqrt{2} ) and the angle between ( (a times b) ) and
( c ) is ( 30^{circ}, ) then ( |(a times b) times c| ) is equal to
A ( cdot frac{2}{3} )
B. ( frac{3}{2} )
c. 2
D. 3
12
387If the vector ( b ) is collinear with the vector
( boldsymbol{a}=(mathbf{2} sqrt{mathbf{2}},-mathbf{1}, mathbf{4}) ) and ( |boldsymbol{b}|=mathbf{1 0}, ) then
A ( cdot a pm b=0 )
В . ( a+2 b=0 )
c. ( 2 a_{pm} b=0 )
D. None of these
12
388(
110 11
(u) 100 III
10. Unit vector parallel to the resultant of vectors A=4i – 3j
and B = 8î +8ſ will be
(a) 24 +5j
13
(c) 6i +5
(d) None of these
(b)
12+5 i
13
13
12
389The resultant of two forces ( P ) and ( Q )
acting at an angle a is equal to ( (2 m+1) sqrt{P^{2}+Q^{2}} . ) When the forces
act at an angle ( (90-alpha), ) the resultant
is ( (2 m-1) sqrt{P^{2}+Q^{2}} . ) Then value of
tanalpha is:
12
390Let p and q be the position vectors of P and o
respectively, with respect to O and p = P, 9 = 9. The
points R and S divide PQ internally and externally in the
ratio 2 : 3 respectively. If OR and OS are perpendicular
then
(1994)
(a) 9q2 = 492
(b) 4p2 = 992
(c) 9p=49
(d) 4p=9q
12
391The ( P . V .^{prime} s ) of the vertices of a ( triangle A B C )
( operatorname{are} bar{i}+bar{j}+bar{k}, 4 bar{i}+bar{j}+bar{k}, 4 bar{i}+5 bar{j}+bar{k} )
The ( P . V . ) of the circumcentre of ( triangle A B C )
is
A ( cdot frac{5}{2} bar{i}+3 bar{j}+bar{k} )
В ( cdot 5 bar{i}+frac{3}{2} bar{j}+bar{k} )
c. ( 5 bar{i}+3 bar{j}+frac{1}{2} bar{k} )
D. ( bar{i}+bar{j}+bar{k} )
12
392Find ( |overrightarrow{boldsymbol{b}}|, ) if ( (overrightarrow{boldsymbol{a}}+overrightarrow{boldsymbol{b}}) cdot(overrightarrow{boldsymbol{a}}-overrightarrow{boldsymbol{b}})=8 ) and
( |overrightarrow{boldsymbol{a}}|=mathbf{8}|overrightarrow{boldsymbol{b}}| )
12
393Classify the following measure as
scalar and vector:
10 meters south-east
A. a vector with magnitude 5
B. a vector with magnitude 10
c. a scalar with magnitude 10
D. none of the above
12
39423. Two vectors , and b are such that a+b = a-5. What
is the angle between ā and b?
a. 0° b. 90°
c. 60°
d. 180°
12
395( boldsymbol{A}(mathbf{1},-mathbf{1},-mathbf{3}), boldsymbol{B}(mathbf{2}, mathbf{1},-mathbf{2}) )
( & quad C(-5,2 )
are the position vectors of the vertices of a triangle ( A B C ), then the length of the bisector of its internal
angle ( A ) is
12
396( A B C ) is an equilateral triangle of side a The value of ( overrightarrow{A B} cdot overrightarrow{B C}+overrightarrow{B C} cdot overrightarrow{C A}+ )
( overrightarrow{C A} cdot overrightarrow{A B} ) is equal to
( A cdot frac{3 a^{2}}{2} )
B. ( 3 a^{2} )
( mathrm{c} cdot frac{3 a^{2}}{2} )
D. None of these
12
397If ( boldsymbol{A}=mathbf{3} boldsymbol{i}-mathbf{4} boldsymbol{j} ) and ( boldsymbol{B}=-boldsymbol{i}-boldsymbol{4} boldsymbol{j} )
calculate the direction of ( boldsymbol{A}-boldsymbol{B} )
A. Along positive x axis
B. Along negative x axis
c. Along positive y axis
D. Along negative y axis
12
398Enter 1 if true else 0 . If ( vec{a}=hat{i}+hat{j}+hat{k}, vec{b}=4 hat{i}-2 hat{j}+3 hat{k} ) and
( overrightarrow{boldsymbol{c}}=hat{boldsymbol{i}}-boldsymbol{2} hat{boldsymbol{j}}+hat{boldsymbol{k}} )
then ( 2 hat{i}-4 hat{j}+4 hat{k} ) a vector of
magnitude 6 units which is parallel to the vector ( 2 vec{a}-vec{b}+3 vec{c} )
12
399If ( vec{a} ) and ( vec{b} ) two collinear vectors then
which of the following are incorrect
A ( cdot vec{b}=lambda vec{a} ) for some scalar
B ( cdot vec{a}=pm vec{b} )
C. The respective components of ( vec{a} ) and ( vec{b} ) are proportional
D. Both the vectors ( vec{a} ) and ( vec{b} ) have same direction, but different magnitude.
12
400A blind person after walking 10 steps in one direction, each of length ( 80 mathrm{cm} ) turns randomly to the left or to the right
by ( 90^{0} ) each time moving 10 steps. After walking a total of 40 steps the maximum displacement of the person from his starting position could be :
A. 320 ( m )
B. 32 ( m )
c. ( 16 sqrt{2} mathrm{m} )
D. none
12
401(u) 3.2V2
31. ABC is a triangle, right angled at A. The resultant of the
forces acting along AB,BC with magnitudes and
– AB AC
respectively is the force along AD, where D is the foot of
the perpdicular from A onto BC. The magnitude of the
resultant is
[2006]
(a) AB² +42
(AB)? (AC)
(AB)(AC)
AB + AC
(d)
АВАС
AD
12
402If ( vec{a} ) and ( vec{b} ) are non – zero vectors
which are linearly dependent such that ( frac{|vec{a}+vec{b}|}{|vec{a}-vec{b}|}=2,|vec{b}|>|vec{a}| ) then
( begin{array}{l}text { A. } vec{b}=3 vec{a} \ vec{a}end{array}=3=3 vec{a} )
В . ( vec{b}=-3 vec{a} )
c. ( vec{b}=2 vec{a} )
D. ( vec{b}=-2 vec{a} )
12
403If ( vec{a}+2 vec{b}, 2 vec{a}+vec{b} ) be the position vectors
of the points ( A ) and ( B ), then the position
vector of the point ( C ) which divides ( A B )
internally in the ratio 2: 1 is
A ( cdot frac{5 vec{a}-4 vec{b}}{3} )
в. ( frac{5 vec{a}+4 vec{b}}{3} )
c. ( frac{5 vec{a}-2 vec{b}}{3} )
D. ( frac{5 vec{a}+2 vec{b}}{3} )
12
404Find the direction angles of vector ( -8 hat{i}+3 hat{j}+2 hat{k} )
A ( cdot 156^{circ}, 70^{circ}, 77^{circ} )
В. ( 155^{circ}, 72^{circ}, 80^{circ} )
c. ( 145^{circ}, 83^{circ}, 74^{circ} )
D. ( 150^{circ}, 76^{circ}, 83^{circ} )
12
405Find the direction cosines of the vector
joining the point ( A(1,2,-3) ) and ( B(-1,-2,1), ) directed from ( A ) and ( B )
12
406( ln Delta O A B, ) if ( O A=vec{a}, overrightarrow{O B}=vec{b} . L ) is mid
point of ( boldsymbol{O A} ) and ( boldsymbol{M} ) is point on ( boldsymbol{O B} ) such that ( boldsymbol{O} boldsymbol{M}: overrightarrow{boldsymbol{M}} boldsymbol{B}=boldsymbol{2}: 1 . ) If ( boldsymbol{P} ) is mid
point of ( L M ) then ( overrightarrow{A P}= )
A ( cdot frac{1}{3} vec{b}-frac{3}{4} vec{a} )
B ( cdot frac{1}{3} vec{b}+frac{3}{4} vec{a} )
c. ( frac{1}{3} vec{a}-frac{3}{4} vec{b} )
D. ( frac{1}{3} vec{a}+frac{3}{4} vec{b} )
12
407Find the vectors of magnitude 6 which are perpendicular to both the vectors ( 4 vec{i}-vec{j}+3 vec{k} ) and ( -2 vec{i}+vec{j}-2 vec{k} )12
408Let ( bar{a}, bar{b}, bar{c} ) be vectors of length 3,4,5
respectively. Let ( bar{a} ) be perpendicular to ( bar{b}+bar{c}, bar{b} ) is perpendicular to ( bar{c}+bar{a} & bar{c} ) is
perpendicular to ( bar{a}+bar{b} ). Then ( |bar{a}+bar{b}+bar{c}| )
is:
A ( .2 sqrt{5} )
B. ( 2 sqrt{2} )
c. ( 10 sqrt{5} )
D. ( 5 sqrt{2} )
12
409Any vector in an arbitrary direction can always be replaced by two (or three)
A. parallel vectors which have the original vector as their resultant
B. mutually perpendicular vectors which have the original vector as their resultant.
c. arbitrary vectors which have the original vector as their resultant
D. it is not possible to resolve a vector.
12
410Set the following vectors in the increasing order of their magnitudes.
a) ( 3 hat{i}+4 hat{j} )
b) ( 2 hat{i}+4 hat{j}+6 hat{k} )
c) ( 2 hat{i}+2 hat{j}+2 hat{k} )
( A cdot b, a, c )
B. ( c, a, b )
( c cdot a, c, b )
D. a, b, c
12
411stion 4.4 A particle moves in a semicircular path
ius R from 0 to A (Fig. 4.6.) Then it moves parallel to
scovering a distance R upto B. Finally it moves along
parallel to y-axis through a distance 2R. Find the ratio
of D/s.
Fig. 4.6
12
412Let ( G ) and ( G^{1} ) be the centroids of the
triangles ( A B C ) and ( A^{1} B^{1} C^{1} )
respectively, then ( boldsymbol{A} boldsymbol{A}^{1}+boldsymbol{B} boldsymbol{B}^{1}+boldsymbol{C} boldsymbol{C}^{1} )
is equal to
( mathbf{A} cdot 2 G G^{1} )
B. ( 3 G^{1} G )
( mathbf{c} cdot 3 G G^{1} )
D. ( frac{3}{2} G G^{1} )
12
413If vector ( overrightarrow{boldsymbol{v}}=hat{boldsymbol{i}}+hat{boldsymbol{j}}+hat{boldsymbol{k}}, ) then find their
direction angles.
( mathbf{A} cdotleft(cos ^{-1} frac{1}{sqrt{3}}, cos ^{-1} frac{1}{sqrt{3}}, cos ^{-1} frac{1}{sqrt{3}}right) )
B ( cdotleft(cos ^{-1}(1), cos ^{-1}(1), cos ^{-1}(1)right) )
( ^{c} cdotleft(cos ^{-1}(1), cos ^{-1} frac{1}{sqrt{3}}, cos ^{-1} frac{1}{sqrt{3}}right) )
D ( cdotleft(cos ^{-1} frac{1}{sqrt{3}}, cos ^{-1}(1), cos ^{-1} frac{1}{sqrt{3}}right) )
12
414If ( vec{a}=hat{i}+widehat{j}+widehat{k}, vec{b}=4 hat{i}+3 widehat{j}+4 widehat{k} ) and
( vec{c}=hat{i}+alpha hat{j}+beta widehat{k} ) are linearly dependent
vectors and ( |overrightarrow{boldsymbol{c}}|=sqrt{mathbf{3}}, ) then
B. ( a=1, b=pm 1 )
c. ( alpha=-1, beta=pm 1 )
D. ( alpha=pm 1, beta=1 )
12
415The point ( C=left(frac{12}{5}, frac{-1}{5}, frac{4}{5}right) ) divides the
line segment ( A B ) in the ratio ( 3: 2 . ) If
( B=(2,-1,2) ) then ( A ) is
A . (3,1,1)
В. (3,1,-1)
D. (-3,1,-1)
12
416The value of ( b ) such that scalar product of the vector ( (hat{i}+hat{j}+hat{k}) ) with the unit vector parallel to the sum of the vectors ( (2 hat{i}+4 hat{j}-5 hat{k}) ) and ( (b hat{i}+2 hat{j}+3 hat{k}) ) is 1
is
A . -2
B. –
( c .0 )
D.
12
417For the vectors ( overline{boldsymbol{x}}=(mathbf{1}, mathbf{2}, mathbf{1}) ; overline{boldsymbol{y}}= )
(2,-3,-1) Component of ( bar{y} ) along ( bar{x}= )
A ( frac{5}{sqrt{14}} )
в. ( -frac{5}{14} )
c. ( frac{5}{14} )
D. ( -frac{5}{sqrt{14}} )
12
418If a vector is multiplied by a real number, then which of the following statements is incorrect?
A. only magnitude of the vector may change
B. Only direction of the vector may change
c. Both magnitude and direction of the vector may change
D. Its direction will never change
12
419A force ( vec{F}=4 hat{i}+6 hat{j}+3 hat{k} ) acting on a particle produces a displacement of ( vec{S}= ) ( 2 hat{i}+3 hat{j}+5 hat{k} ) where ( F ) is expressed in
Newtonn and ( S ) in the metre. Find the
work done by the force.
12
420Assertion
If three points ( P, Q ) and ( R ) have position vectors ( vec{a}, vec{b} ) and ( vec{c}, ) respectively, and ( 2 vec{a}+3 vec{b}-5 vec{c}=0, ) then the points ( P, Q )
and ( R ) must be collinear.
Reason
If for three points ( A, B ) and ( C, A B= ) ( lambda overrightarrow{A C}, ) then points ( A, B ) and ( C ) must be
collinear.
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
421State the following statement is True or
False

If the starting and end points of a vector are collinear, it is known as a unit
vector
A. True
B. False

12
422The distance travelled by the car, if a car travels ( 4 k m ) towards north at an
angle of ( 45^{circ} ) to the east and then travels
a distance of ( 2 k m ) towards north at an
angle of ( 135^{circ} ) to the east, is
( mathbf{A} cdot 6 k m )
в. ( 8 k m )
( c .5 k m )
D. ( 2 mathrm{km} )
12
42313. If Ā+ B = A = Ë , then the angle between A and B is
a. 120° b. 60°c. 90 d. 0°
12
424Three forces are applied to a square
plate as shown in figure:

Find the point of application of the
resultant force, if this point is taken on
the side ( B C )
A. The force is applied at the midpoint of the side ( B C ).
B. At point B
C. At point ( c )
D. cannot be found

12
425For vectors ( vec{a} & vec{b} . ) Prove that ( |overrightarrow{boldsymbol{a}} times overrightarrow{boldsymbol{b}}|^{2}=|overrightarrow{boldsymbol{a}}|^{2}|overrightarrow{boldsymbol{b}}|^{2}-|overrightarrow{boldsymbol{a}} cdot overrightarrow{boldsymbol{b}}|^{2} )12
426In a quadrilateral ( P Q R S, overrightarrow{P Q}= ) ( vec{a}, overrightarrow{Q R}=vec{b}, overrightarrow{S P}=vec{a}-vec{b} . M ) is the mid-
point of ( Q R ) and ( X ) is a point on ( S M ) such that ( overrightarrow{S X}=frac{4}{5} S vec{M}, ) then ( overrightarrow{P X} ) is
A ( cdot frac{1}{5} overrightarrow{P R} )
В. ( frac{3}{5} overrightarrow{P R} )
c. ( frac{2}{5} overrightarrow{P R} )
D. None of these
12
42710. Let ū=i+j, v =î – and w=î +2ị +3ť . If în is a unit
vector such that ü.n= 0 and v.ñ= 0, then w.n is equal to
[2003]
(a) 3 (6) 0 (c) 1 (d) 2.
12
428Points ( X & Y ) are taken on the sides QR
& RS respectively of a parallelogram ( mathrm{PQRS}, ) so that ( boldsymbol{Q} overrightarrow{boldsymbol{X}}=4 overrightarrow{boldsymbol{X}} boldsymbol{R} & overrightarrow{boldsymbol{R} boldsymbol{Y}}= )
( 4 vec{Y} ) S. The line ( X Y ) cuts the line ( P R ) at ( Z ) Prove that ( overrightarrow{P Z}=left(frac{21}{25}right) overrightarrow{P R} )
12
429f ( a=2 i+5 j+k ) and ( b=4 i+m j+ )
( n k ) are collinear vectors, then find ( m+n )
12
4302. A vector ā is turned without a change in its length
through a small angle do. The value of Aă| and Aa are
respectively
(a) 0, a de
(b) a do, o
(c) 0,0
(d) None of these
12
431Let ( A(vec{a}), B(vec{b}), C(vec{c}) ) be the vertices of the triangle ( A B C ) and let ( D E F ) be the
mid points of the sides ( B C, C A, A B )
respectively. If ( boldsymbol{P} ) divides the median
( A D ) in the ratio 2: 1 then the position
vector of ( boldsymbol{P} ) is
A . 0
B ( cdot vec{a}+vec{b}+vec{c} )
( overbrace{3}^{text {c. }} frac{vec{a}+vec{b}+vec{c}}{3} )
( frac{2 vec{a}+vec{b}+vec{c}}{3} )
12
432Let ( G ) be the centroid of ( triangle A B C ). If ( A B=vec{a} ) and ( overrightarrow{A C}=vec{b}, ) then ( overrightarrow{A G}, ) in
terms of ( vec{a} ) and ( vec{b} ) is
A ( cdot frac{2}{3}(vec{a}+vec{b}) )
в. ( frac{1}{6}(vec{a}+vec{b}) )
c. ( frac{1}{3}(vec{a}+vec{b}) )
D. ( frac{1}{2}(vec{a}+vec{b}) )
12
433Show that the vector ( overrightarrow{boldsymbol{a}}=(hat{boldsymbol{i}}+hat{boldsymbol{j}}+hat{boldsymbol{k}}) ) is
equally inclined to the coordinate axes.
12
434If ( boldsymbol{A}(overrightarrow{boldsymbol{a}}), boldsymbol{B}(overrightarrow{boldsymbol{b}}), boldsymbol{C}(overrightarrow{boldsymbol{c}}) ) be the vertices
of a triangle whose circumcentre is the origin, then orthocenter is given by
( frac{vec{a}+vec{b}+vec{c}}{3} )
( ^{mathrm{B}} frac{vec{a}+vec{b}+vec{c}}{2} )
c. ( vec{a}+vec{b}+vec{c} )
D. None of these
12
435If the position vector of a point (-4,-3)
be ( vec{a}, ) find ( |vec{a}| )
12
436Let ( a=i+2 j+3 k ) and ( b=3 i+j . ) Find
the unit vector in the direction of ( a+b )
12
437Two forces of ( 1 N ) and ( P N ) act at a point so that the magnitude of resultant is ( mathbf{1} N ) and it is perpendicular to ( 1 N . ) The
value of ( boldsymbol{P} ) in newtons and the angle
between the ( 1 N ) and ( P N ) are:
A ( cdot sqrt{2}, 135^{circ} )
B . ( sqrt{2}, 120^{0} )
c. ( 1,45^{circ} )
D. ( 2,150^{circ} )
12
438Let ( bar{a}, bar{b}, bar{c} ) and ( bar{d} ) be position vectors of four points ( A, B, C ) and ( D ) lying in a
plane. If ( (bar{a}-bar{d}) cdot(bar{b}-bar{c})=0= )
( (bar{b}-bar{d}) cdot(bar{c}-bar{a}), ) then ( Delta A B C ) has ( D ) as
A . in-centre
B. circum-centre
c. ortho-centre
D. centroid
12
439Let ( bar{a}, bar{b}, bar{c} ) be three non-zero vectors, no two of which are collinear. If the vector
( bar{a}+2 bar{b} ) is collinear with ( bar{c} ) and ( bar{b}+3 bar{c} ) is
collinear with ( bar{a}, ) then ( bar{a}+bar{b}+3 bar{c}= )
( mathbf{A} cdot lambda bar{a} bar{a} )
в. ( lambda bar{b} )
( c cdot lambda bar{c} )
D.
12
440If ( overline{boldsymbol{p}}=hat{boldsymbol{i}}-boldsymbol{2} hat{boldsymbol{j}}+hat{boldsymbol{k}} ) and ( overline{boldsymbol{q}}=hat{boldsymbol{i}}+boldsymbol{4} hat{boldsymbol{j}}-boldsymbol{2} hat{boldsymbol{k}} )
are position vectors of points ( boldsymbol{P} ) and ( boldsymbol{Q} )
find the position vector of the point ( boldsymbol{R} ) which divides segment ( P Q ) internally in
the ratio 2: 1
12
441A vector is represented by ( 3 bar{i}+bar{j}+2 bar{k} )
It’s length in ( x y ) plane is:
( A cdot 2 )
B. ( sqrt{14} )
c. ( sqrt{10} )
D. ( sqrt{5} )
12
442A zero vector has
This question has multiple correct options
A. Any direction
B. Many directions
C. No direction
D. None of these
12
443If ( vec{a}=hat{i}+hat{j}+hat{k}, vec{b}=4 hat{i}+3 hat{j}+4 hat{k} ) and
( vec{c}=hat{i}+alpha hat{j}+beta hat{k} ) are linearly dependent
vectors and ( |vec{c}|=sqrt{3} ) then
A. ( alpha=1, beta=-1 )
В. ( alpha=1, beta=pm 1 )
c. ( alpha=-1, beta=pm 1 )
D. ( alpha=pm 1, beta=1 )
12
444If ( vec{a}=i+2 j+k, vec{b}=i-j+k ) and ( vec{c}= )
( i+j-k . ) A vector in the plane of ( vec{a} ) and ( vec{b} ) whose projection on ( vec{c} ) is ( 1 / sqrt{3}, ) is
A ( .4 i-j+4 k )
в. ( 3 i+j-3 k )
c. ( 2 i+j-2 k )
D. ( 4 i+j-4 k )
12
445If ( |widehat{a}-widehat{b}|=sqrt{3}, ) then ( |widehat{a}+widehat{b}| ) may be:
A .
B. ( frac{sqrt{3}}{2} )
c. Either (1) and (2)
D. None of these
12
446If ( vec{a}=-hat{i}-hat{j}+hat{k} ) and ( vec{b}=-hat{i}+hat{j}-hat{k} )
then find the value of ( |vec{a} cdot vec{b}| )
12
447If the unit vectors ( vec{e}_{1} ) and ( vec{e}_{2} ) are inclined
at an angle ( 2 theta ) and ( left|vec{e}_{1}-vec{e}_{2}right|<1 ), then
for ( boldsymbol{theta} in[mathbf{0}, boldsymbol{pi}], boldsymbol{theta} ) may lie in the interval
( mathbf{A} cdotleft[0, frac{pi}{6}right] )
В ( cdotleft[frac{pi}{6}, frac{pi}{2}right] )
c. ( left[frac{5 pi}{6}, piright] )
D. ( left[frac{pi}{2}, frac{5 pi}{6}right] )
12
44824.
If a =(i + i + k), ā.5 = 1 and à x D = ì – k, then ő is
(2) i – }
(b) 2j-Ã (2004)
(d) zi
12
449Let ( O B=hat{i}+2 hat{j}+2 hat{k} ) and ( O A=4 hat{i}+ )
( 2 hat{j}+2 hat{k} . ) The distance of the point ( B ) from the straight line passing through ( A ) and parallel to the vector ( 2 hat{i}+3 hat{j}+ ) ( 6 hat{k} ) is
A ( cdot frac{7 sqrt{5}}{9} )
в. ( frac{5 sqrt{7}}{9} )
c. ( frac{3 sqrt{5}}{7} )
D. ( frac{9 sqrt{5}}{7} )
E ( cdot frac{9 sqrt{7}}{5} )
12
4501.
The scalar À (B+C)*(À+B+C) equals :
(1981 – 2 Marks)
(a) O
(b) À B Č]+[? Å
C) À C] (d) None of these
12
451Find the projection of ( vec{b}+vec{c} ) on ( vec{a}, ) where ( vec{a}=2 hat{i}-2 hat{j}+hat{k}, vec{b}=hat{i}+2 hat{j}-2 hat{k} ) and
( vec{c}=2 hat{i}-hat{j}+4 hat{k} )
12
452If ( vec{a} ) and ( vec{b} ) are unit vectors and ( alpha ) is the
angle between them, then ( vec{a}+vec{b} ) is a
unit vector when ( cos alpha= )
A ( cdot-frac{1}{2} )
B. ( frac{1}{2} )
( c cdot-frac{sqrt{3}}{2} )
D. ( frac{sqrt{3}}{2} )
12
453Which one of the following is the unit vector perpendicular to both ( overrightarrow{boldsymbol{a}}=-hat{mathbf{i}}+ ) ( hat{boldsymbol{j}}+hat{boldsymbol{k}} ) and ( overrightarrow{boldsymbol{b}}=hat{boldsymbol{i}}-hat{boldsymbol{j}}+hat{boldsymbol{k}} ? )
A ( cdot frac{hat{i}+hat{j}}{sqrt{2}} )
в. ( hat{k} )
( c cdot frac{hat{j}+hat{k}}{sqrt{2}} )
D. ( frac{hat{i}-hat{jmath}}{sqrt{2}} )
12
454If ( overline{boldsymbol{a}}=(2 overline{boldsymbol{i}}-mathbf{1 0} overline{boldsymbol{j}}+mathbf{6} overline{boldsymbol{k}}) ; overline{boldsymbol{b}}= )
( (5 bar{i}-3 bar{j}+bar{k}) . ) The ratio of projection of
( bar{a} ) on ( bar{b} ) to projection of ( bar{b} ) on ( bar{a} ) is
A . 2: 1
B. 1: 2
( c cdot 2: 3 )
D. 3: 2
12
45511. The three vectors Ā=3î – 2j+Â , B=î – 39 +5k and
Č=2ỉ + – 4k form
(a) An equilateral triangle
(b) Isosceles triangle
(c) A right angled triangle
(d) No triangle
On
12
456Given: ( vec{a} ) and ( vec{b} ) are unit vector, and ( theta ) be
the angle between them. Then ( frac{1-vec{a} cdot vec{b}}{1+vec{a} cdot vec{b}}= )
( A cdot sin ^{2} frac{theta}{2} )
B. ( cos ^{2} frac{theta}{2} )
( c cdot tan ^{2} frac{theta}{2} )
( D cdot cot ^{2} frac{theta}{2} )
12
457Let ( boldsymbol{A}=mathbf{2} hat{mathbf{i}}+mathbf{4} hat{mathbf{j}}-hat{boldsymbol{k}}, boldsymbol{B}=mathbf{4} hat{mathbf{i}}+mathbf{5} hat{mathbf{j}}+hat{boldsymbol{k}} . ) If
the centroid ( G ) of the triangle ( A B C ) is ( mathbf{3} hat{mathbf{i}}+mathbf{5} hat{mathbf{j}}-hat{boldsymbol{k}}, ) then the position vector of
( C ) is
( mathbf{A} cdot 3 hat{i}-6 hat{j}+3 hat{k} )
B . ( 3 hat{i}-6 hat{j}-3 hat{k} )
c. ( 3 hat{i}-6 hat{j}+2 hat{k} )
D. ( 3 hat{i}+6 hat{j}-3 hat{k} )
12
458If ( |vec{a}|=5, mid vec{b}=13 ) and ( |vec{a} times vec{b}|=25 ), find
( vec{a} cdot vec{b} )
12
459If ( vec{a}+vec{b}+vec{c}=overrightarrow{0} ) such that ( |vec{a}|=3,|vec{b}|= )
( mathbf{5} ) and ( |overrightarrow{boldsymbol{c}}|=mathbf{7} )
What is ( vec{a} cdot vec{b}+vec{b} cdot vec{c}+vec{c} cdot vec{a} ) equal to.
A . -83
B. ( frac{-83}{2} )
c. 75
D. ( frac{-75}{2} )
12
460If the vertices ( A, B, C ) of a triangle ( A B C ) are the points with position vectors
( boldsymbol{a}_{1} hat{boldsymbol{i}}+boldsymbol{a}_{2} hat{boldsymbol{j}}+boldsymbol{a}_{3} hat{boldsymbol{k}}, boldsymbol{b}_{1} hat{boldsymbol{i}}+boldsymbol{b}_{2} hat{boldsymbol{j}}+boldsymbol{b}_{3} hat{boldsymbol{k}}, boldsymbol{c}_{1} hat{boldsymbol{i}}+ )
( c_{2} hat{j}+c_{3} hat{k} ) respectively, what are the
vectors determined
by its sides? Find the length of these
vectors.
12
461Find the altitude of a parallelopiped determined by ( vec{a}, vec{b}, vec{c} ) if yhe base ( vec{a}, & vec{b} ) and if ( vec{a}=hat{i}+hat{j}+hat{k} ; vec{b}=2 hat{i}+4 hat{j}- )
( hat{boldsymbol{k}} & overrightarrow{boldsymbol{c}}=hat{boldsymbol{i}}+hat{boldsymbol{j}}+mathbf{3} hat{boldsymbol{k}} )
12
462Let ( vec{a}, vec{b}, vec{c} ) be the vectors of length 3,4 5 respectively. Let ( vec{a} ) be perpendicular to ( vec{b}+vec{c}, vec{b} ) to ( vec{c}+vec{a} ) and ( vec{c} ) to ( vec{a}+vec{b} )
Then ( |vec{a}+vec{b}+vec{c}| ) is equal to
A ( cdot 2 sqrt{5} )
B. ( 2 sqrt{2} )
с. ( 10 sqrt{5} )

D. ( 5 sqrt{2} )
12
463Illustration 3.29 Find the value of m so that the
vector 3i -2j+k may be perpendicular to the vector
zi +67+mk.
u
andislariedbeiend..
12
464If ( vec{a}=hat{i}+2 hat{j} ) and ( vec{b}=3 hat{j}, ) then ( vec{a} cdot vec{b}= )
A. 3
B . – –
( c cdot 6 )
D. – –
12
465Show that each of the three given vectors is a unit vectors. also Show that
they are mutually ( perp ) to each other ( vec{a}=frac{1}{7}(2 hat{i}+3 hat{j}+6 hat{k}) )
( vec{b}=frac{1}{7}(3 hat{i}-6 hat{j}+2 hat{k}) )
( overrightarrow{boldsymbol{c}}=frac{1}{7}(6 hat{i}+2 hat{j}-3 hat{k}) )
12
466The area ( (text { in } s q . text { units }) ) of the
parallelogram whosed diagonals along the vectors ( 8 hat{i}-6 hat{j} ) and ( 3 hat{i}+4 hat{j}-12 hat{k} )
is:
A . 20
B. 65
( c .52 )
D. 26
12
467If the vectors c, a = xi + yj + zk and b = ſ are such that
ă, c and form a right handed system then c is :[2002]
(a) zi – xk
(6) J
(c) vị .
(d) -z + xh
12
468The vectors ( overrightarrow{A B}=3 hat{i}+4 hat{k} ) and ( overrightarrow{A C}= ) ( mathbf{5} hat{mathbf{i}}-mathbf{2} hat{mathbf{j}}+mathbf{4} hat{boldsymbol{k}} ) are the sides of a triangle
( A B C, ) then the length of the median
through ( boldsymbol{A} ) is:
A. ( sqrt{72} )
B. ( sqrt{33} )
c. ( sqrt{45} )
D. ( sqrt{18} )
12
469The position vectors of four points ( boldsymbol{P}, boldsymbol{Q}, boldsymbol{R}, boldsymbol{S} ) are ( boldsymbol{2} overline{boldsymbol{a}}+boldsymbol{4} overline{boldsymbol{c}}, boldsymbol{5} overline{boldsymbol{a}}+boldsymbol{3} sqrt{boldsymbol{3} boldsymbol{b}}+ )
( 4 bar{c}),-2 sqrt{3 b}+bar{c} ) and ( 2 bar{a}+bar{c} )
respectively, then
A. ( overline{P Q} ) is parallel to ( overline{R S} )
B. ( overline{P Q} ) is not parallel to ( overline{R S} )
c. ( overline{P Q} ) is equal to ( overline{R S} )
D. ( overline{P Q} ) is parallel and equal to ( overline{R S} )
12
470Calculate the scalar product of the following vectors. Two points ( A ) and ( B ) are given in the rectangular Cartesian system of coordinates Oxy on the curve ( boldsymbol{y}=mathbf{2}^{boldsymbol{x}+mathbf{2}} )
the points being such that ( overrightarrow{boldsymbol{O A}} cdot boldsymbol{i}= )
-1 and ( overrightarrow{O B} cdot i=2, ) where 1 is a unit
vector of the ( 0 x ) axis. Find the length of the vector ( overline{-4 O A}+overrightarrow{O B} )
12
471If ( boldsymbol{A}=mathbf{1} hat{mathbf{1}}+mathbf{2} hat{boldsymbol{j}}-boldsymbol{R} ) and ( hat{boldsymbol{B}}=mathbf{3} hat{mathbf{1}}+hat{boldsymbol{j}}+hat{boldsymbol{k}} )
then
Find unit vector along ( overrightarrow{boldsymbol{A}} )
A. ( vec{A}+vec{B} )
В ( cdot|vec{A}+vec{B}| )
C . Unit vector along the direction of ( |hat{A}+hat{B}| )
D. None of the above
12
47210. If
la a²
b 52
o c?
1+ a²
1+b = 0
1+c3|
and
vectors
(1,a,a?),
(1,5,62) and (1,c,c?) are non- coplanar, then the product
abc equals
[2003]
(a) 0 (6) 2 (1) 1 (d) 1
12
473Calculate the scalar product of the following vectors. Find the vector a which is collinear with
the vector ( b={2,-1,0} ) if ( a cdot b=10 )
12
4746.
If A, B, C, D are any four points in space, prove that –
(1987 – 2 Marks)
AB CD + BC * AD + CA BD = 4 (area of triangle
ABC)
12
475Find ( vec{a} . vec{b}, ) when ( vec{a}=hat{j}+2 hat{k} ) and ( vec{b}=2 hat{i}+ )
( hat{boldsymbol{k}} )
12
476If for three non-zero vectors ( vec{a}, vec{b} ) and ( vec{c}, vec{a} cdot vec{b}=vec{a} cdot vec{c} ) and ( vec{a} times vec{b}=vec{a} times vec{c}, ) then
show that ( vec{b}=vec{c} )
12
477If ( 2 vec{a}+3 vec{b}-5 vec{c}=overrightarrow{0}, ) then ratio in which
( vec{c} ) divides ( overrightarrow{A B} ) is
A. 3: 2 internally
B. 3: 2 externally
c. 2: 3 internally
D. 2: 3 externally
12
478Illustration 3.35 By vector method, prove that if the
diagonals of a parallelogram intersect perpendicularly, then
the parallelogram is a rhombus.
12
479If ( bar{a} ) and ( bar{b} ) are non-collinear unit vectors
and ( |bar{a}+bar{b}|=sqrt{3} ) then ( (2 bar{a}+5 bar{b}) )
( (3 bar{a}-bar{b})=? )
A ( cdot frac{15}{4} )
в. ( frac{15}{2} )
c. 15
D. 16
12
480Define unit vector.12
481A line passes through the points whose position vectors are ( hat{boldsymbol{i}}+hat{boldsymbol{j}}-2 hat{boldsymbol{k}} ) and ( hat{boldsymbol{i}}- )
( 3 hat{j}+hat{k} . ) The position vector of a point on it at unit distance from the first point
is
12
482Let ( a=2 hat{i}+widehat{j}-2 widehat{k} ) and ( b=hat{i}+widehat{j} . ) If ( c ) is
a vector such that ( boldsymbol{a} . boldsymbol{c}=|boldsymbol{c}|,|boldsymbol{c}-boldsymbol{a}|= )
( 2 sqrt{2} ) and the angle between ( a times b ) and ( c )
is ( 30^{circ} . ) Then, ( [(a times b) times c] ) is equal to
A ( cdot frac{2}{3} )
B. ( frac{3}{2} )
( c cdot 2 )
D. 3
12
48313.
If the vectors 5.c.ā, are not coplanar, then prove that the
vector
(ã xb) x (ẽ xả) + (ã xº) x (4 x 6)+(ã xd) x (b xº) is
parallel to a
(1994 – 4 Marks)
12
484Let ( vec{p} ) and ( vec{q} ) be the position vectors of ( boldsymbol{P} )
and ( Q ) respectively with respect to ( O ) and ( |vec{p}|=p,|vec{q}|=q . R, S ) divide ( P, Q )
Internally and externally in the ratio 2:
3 respectively. If ( O R ) and ( O S ) are
perpendicular, then
( mathbf{A} cdot 9 p^{2}=4 q^{2} )
B. ( 4 p^{2}=9 q^{2} )
c. ( 9 p=4 q )
D. ( 4 p=9 q )
12
4855.
If lā= 5, 10 = 4, 10 = 3 thus what will be the value of
Tā.b+b.c +c.al, given that a + b + c = 0
(a) 25 (6) 50 (c) -25
[2002]
(d) -50
12
486Let ( a, b ) and ( c ) be non-zero vectors such
that no two are collinear and ( (a times b) times ) ( c=frac{1}{3}|b||c| a . ) If ( theta ) is the acute angle
between the vectors b and ( c, ) then ( sin theta )
equals
A ( cdot frac{2 sqrt{2}}{3} )
B. ( frac{sqrt{2}}{3} )
( c cdot frac{2}{3} )
D.
12
4875. Let à =î+j+k, b =î – j+and © = xỉ + (x – 2)° –
If the vectors ē lies in the plane of a and 5, then x equals
[2007]
(a) – 4 (6) -2 (c) 0 (d) 1.
12
48864. Ifă = Tio (3* + k) and 5 = (2î +39 – 6k), then the value
of (27–6)[(āxb)x(ā +26)] is
(a) -3 (b) 5 C) 3. (d) -5
[2011]
12
4896.
For three vectors u, v, w which of the following expression
is not equal to any of the remaining three? (1998 – 2 Marks)
(a) u (v x w)
(b) (v x w) u
(c) v.(ux w)
(d) (ux v). w
11.1. fthe fallowing expressions are meaningfu12
12
490( boldsymbol{a}=mathbf{2} overline{mathbf{i}}+overline{boldsymbol{k}}, overline{boldsymbol{b}}=mathbf{3} overline{boldsymbol{j}}+mathbf{4} overline{boldsymbol{k}}, overline{boldsymbol{c}}=mathbf{8} overline{mathbf{i}}-mathbf{3} overline{boldsymbol{j}} )
and ( bar{a} ) is expressed as a linear combination of ( bar{b} ) and ( bar{c} ) then ( bar{a}= )
A ( cdot frac{1}{2}(bar{b}+bar{c}) )
B ( cdot frac{1}{3}(bar{b}+bar{c}) )
c. ( frac{1}{4}(bar{b}+bar{c}) )
D. ( frac{1}{4}(bar{b}-bar{c}) )
12
491Find ( (overrightarrow{boldsymbol{a}} times overrightarrow{boldsymbol{b}}) cdot(overrightarrow{boldsymbol{c}} times overrightarrow{boldsymbol{d}}) ) if ( overrightarrow{boldsymbol{a}}=overrightarrow{boldsymbol{i}}+ )
( overrightarrow{boldsymbol{j}}+overrightarrow{boldsymbol{k}}, overrightarrow{boldsymbol{b}}=2 overrightarrow{boldsymbol{i}}+overrightarrow{boldsymbol{k}}, overrightarrow{boldsymbol{c}}=2 overrightarrow{boldsymbol{i}}+overrightarrow{boldsymbol{j}}+overrightarrow{boldsymbol{k}} )
and ( overrightarrow{boldsymbol{d}}=overrightarrow{boldsymbol{i}}+overrightarrow{boldsymbol{j}}+boldsymbol{2} overrightarrow{boldsymbol{k}} )
12
492( bar{a} ) and ( bar{b} ) are non-zero vectors such that
( bar{a} times bar{b}=bar{b} times bar{a} ) then ( bar{b}, bar{a} ) are
A. this result is always true
B. this result is impossible
c. parallel to each other
D. perpendicular to each other
12
493Let ( bar{a}, bar{b}, bar{c} ) be non coplanar vectors. The vectors ( 3 bar{a}-2 bar{b}-4 bar{c},-bar{a}+2 bar{c},-2 bar{a}+ )
( bar{b}+3 bar{c} ) are
A. Linearly dependent
B. Linearly independent
c. Collinear
D. Non collinear
12
49417. If Ā= B + C , and the magnitudes of A, B, C are 5,4,
and 3 units, then the angle between A and C is
c.
sin-1
12
49521.
If ū,v and w are three non- coplanar vectors, then
(ū+v-w).(ū – v)x(v – w) equals
[2003]
(a) 3ūvx W
(6) o
(c) ūvxã
(d) ū.wxv
IE
12
496If ( vec{a}=4 hat{i}-hat{j}+hat{k} ) and ( vec{b}=2 hat{i}-2 hat{j}+hat{k} )
then find a unit vector parallel to the vector ( overrightarrow{boldsymbol{a}}+overrightarrow{boldsymbol{b}} )
12
497( f(vec{a}, vec{b} text { are the position vectors of } A, B ) respectively, find the position vector of a point ( C ) in ( A B ) produced such that ( A C= ) ( 3 mathrm{AB} ) and that a point ( mathrm{D} ) in BA produced such that ( mathrm{BD}=2 mathrm{BA} )12
49812. The vectors AB = 3î +4k & AČ = 5ỉ – 2j + 4k
are the sides of a triangle ABC. The length of the median
through A is
[2003]
(a) V288 (6) V18 @ 72 (d) 133
12
4994.
If a,b,c are vectors show that a+b+ c = 0 and
| a = 7,1 b = 5, c = 3 then angle between vector b and
cis
(a) 60°
(b) 300
(c) 45°
[2002]
(d) 90°
12
500If ( vec{a}, vec{b}, vec{c}, vec{d} ) are the position vectors of the points ( A, B, C, D ) respectively such that ( 3 vec{a}+5 vec{b}-3 vec{c}-5 vec{d}=0 ) then ( A B )
intersects ( C D ) in the ratio
A .2: 3
B. 3: 2
c. 3: 5
D. 5: 3
12
501A vector ( a=(x, y, z) ) of length ( 2 sqrt{3} )
which makes equal angles with the vectors ( b=(y,-2 z, 3 x) ) and ( c= )
( (2 z, 3 x,-y) ) and is perpendicular to
( boldsymbol{d}=(1,-1,2) ) and makes an obtuse angle with y-axis is
B. ( (1,1, sqrt{10}) )
D. none of these
12
502If ( bar{a} ) and ( bar{b} ) include an angle of ( 120^{circ} ) and their magnitudes are 2 and ( sqrt{3} ) then ( bar{a} . bar{b} ) is
( A cdot 3 )
B. ( -sqrt{3} )
( c cdot sqrt{3} )
D. – 3
12
503If ( vec{a}=i+j+k, vec{b}=4 i+3 j+4 k, vec{c}=i+alpha i+ )
( beta mathrm{k} ) are linearly dependent and ( |c|=sqrt{3} )
then
B. ( alpha=2, beta=1 )
c. ( alpha=3, beta=1 )
D. ( alpha=4, beta=1 )
12
5043.
Find all values of a such that x, y, z,+ (0,0,0) and
(i + j +3k )x+ (3i – 3j+k)y +( 4 i +5 j)z
= 2(xixi y+k z) where i, j, k are unit vectors along
the coordinate axes.
(1982 – 3 Marks)
12
50515. If the vectors a, b and c form the sides BC, CA and AB
respectively of a triangle ABC, then
(2000S)
→ → → → → →
(a) a b + b c +c.a =0 (b) ax b = bx c = cx a
→ → → →
(C) a.b = b. c = c.a
(d) ax b+ bx c + cx a = 0
12
50628.
A particles is acted upon by constant forces 4i + 1-3k
and 3î + j – k which displace it from a point î +2j +3ť to
the point 5î +4j+ k . The work done in standard units by
the forces is given by
[2004]
(a) 15 (b) 30 (0) 25 : (d) 40
12
507If position vectors of four vertices of a square are ( boldsymbol{A}=mathbf{3} boldsymbol{i}-boldsymbol{lambda} boldsymbol{j}, boldsymbol{B}=mathbf{9} boldsymbol{i}-boldsymbol{6} boldsymbol{k} )
( boldsymbol{C}=mathbf{6} boldsymbol{j}-mathbf{5} boldsymbol{k} ) and ( boldsymbol{D}=boldsymbol{alpha} boldsymbol{i}-boldsymbol{3} boldsymbol{j}+boldsymbol{4} boldsymbol{k} )
then value of ( lambda-alpha ) is :
A . 10
B. 15
c. 25
D. None of these
12
508( A(1,-1,-1), B(2,1,-2) ) and ( C(-5,2,-6) ) are the
position vectors of the vertices of triangle ( A B C ) The length of the bisector
of its internal angle at ( A ) is:
A. ( frac{sqrt{10}}{4} )
B. ( frac{3 sqrt{10}}{4} )
c. ( sqrt{10} )
D. none
12
509Find a vector ( overrightarrow{boldsymbol{v}} ) which is co-planar with the vectors ( vec{a}=hat{i}-2 hat{j}+hat{k} ) and ( vec{b}=hat{i}- )
( hat{boldsymbol{j}}+2 hat{boldsymbol{k}} ) and is orthogonal to the vector ( vec{c}=-2 hat{i}+hat{j}+hat{k} . ) It is given that the projection of ( overrightarrow{boldsymbol{v}} ) along the vector ( hat{boldsymbol{i}}-hat{boldsymbol{j}}+ )
( hat{k} ) is equal to ( 16 sqrt{3} )
( mathbf{A} cdot 6 hat{j}-12 hat{k} )
B . ( 12 hat{i}-6 hat{j}+30 hat{k} )
c. ( 9(hat{i}-hat{j}+hat{k}) )
D. None of these
12
510If ( vec{a}, vec{b}, vec{c} ) are mutually perpendicular vectors of equal magnitudes, then the angle between the vectors ( vec{a} ) and ( vec{a}+ ) ( vec{b}+vec{c} ) is
A ( cdot frac{pi}{3} )
в.
( ^{mathrm{c}} cdot cos ^{-1}left(frac{1}{sqrt{3}}right) )
D.
12
511If the vectors ( bar{a}=2 i+j-k ) and ( bar{b}= )
( i+j+k, ) then find the angle between
its.
A ( cdot sin ^{-1} frac{sqrt{2}}{3} )
B. ( cos ^{-1} frac{sqrt{2}}{3} )
c. ( cos ^{-1} frac{2}{sqrt{3}} )
D. ( sin ^{-1} frac{2}{sqrt{3}} )
12
512For any vector ( vec{a}, ) the value of ( (vec{a} times hat{i})^{2}+ )
( (vec{a} times hat{j})^{2}+(vec{a} times hat{k})^{2} ) is equal to
A ( cdot 3 vec{a}^{2} )
в. ( vec{a}^{2} )
c. ( 2 vec{a}^{2} )
D. ( 4 vec{a}^{2} )
12
513Find the direction cosines of the vector ( vec{a}=hat{i}+hat{j}-2 hat{k} )
( ^{mathbf{A}} cdotleft(frac{1}{sqrt{6}}, frac{1}{sqrt{6}},-frac{2}{sqrt{6}}right) )
B ( cdotleft(frac{1}{sqrt{3}}, frac{1}{sqrt{6}}, frac{-2}{sqrt{6}}right) )
( ^{mathbf{c}} cdotleft(frac{1}{sqrt{6}}, frac{1}{sqrt{6}}, frac{-2}{sqrt{3}}right) )
D. None of these
12
514( operatorname{Let} mathbf{P}=(mathbf{1}, mathbf{0},-mathbf{1}) boldsymbol{Q}=(-mathbf{1}, mathbf{2}, mathbf{0}) mathbf{R}= )
( (2,0,-3) S=(3,-2,-1), ) then the
length of the component of ( R S ) on ( P Q ) is
A . ( 1 / 3 )
B . ( 2 / 3 )
c. ( 4 / 3 )
D. ( 5 / 3 )
12
515If the vector ( O P ) in ( X Y ) plane whose
magnitude is ( sqrt{3} ) makes an angle ( 60^{circ} )
with ( Y- ) axis, the length of the
component of the vector in direction of
( X- ) axis is :
A . 1
B. ( sqrt{3} )
( c cdot frac{1}{2} )
D. ( frac{3}{2} )
12
516( mathbf{f} overrightarrow{boldsymbol{A}}=widehat{boldsymbol{j}}+widehat{boldsymbol{k}} ) and ( overrightarrow{boldsymbol{B}}=2 hat{boldsymbol{i}}-widehat{boldsymbol{j}}+boldsymbol{4} widehat{boldsymbol{k}} ) then
the unit vector of ( vec{A}+vec{B} ) is
( A cdot frac{3 i+5 hat{k}}{sqrt{24}} )
B. ( frac{3 hat{i}-5 vec{k}}{sqrt{34}} )
C. ( frac{3 hat{i}+5 vec{k}}{sqrt{34}} )
D. None of these
12
517The component of a vector ( vec{a} ) along and perpendicular to a nonzero vector ( overrightarrow{boldsymbol{b}} ) is:12
518If ( 4 hat{i}+7 widehat{j}+8 widehat{k}, 2 hat{i}+3 widehat{j}+4 widehat{k} ) and ( 2 hat{i}+ )
( mathbf{5} widehat{boldsymbol{j}}+mathbf{7} widehat{boldsymbol{k}} ) are the position vectors of the
vertices ( A, B ) and ( C ) respectively, of
triangle ( A B C, ) then the position vector of the point where the bisector of angle
( A ) meets ( B C ) is
A ( cdot frac{2}{3}(-6 hat{i}-8 hat{j}-6 widehat{k}) )
B ( cdot frac{2}{3}(6 hat{i}+8 hat{j}+6 widehat{k}) )
c. ( frac{1}{3}(6 hat{i}+13 hat{j}+18 widehat{k}) )
D ( cdot frac{1}{3}(5 hat{j}+12 widehat{k}) )
12
519If ( hat{a} ) and ( hat{b} ) are unit vectors inclined at an
angle ( theta ), then prove that
( cos frac{boldsymbol{theta}}{mathbf{2}}=frac{mathbf{1}}{mathbf{2}}|hat{boldsymbol{a}}+hat{boldsymbol{b}}| )
12
520If ( vec{a}, vec{b} ) and ( vec{c} ) are three non-zero vectors such that ( vec{a} cdot vec{b}=vec{a} cdot vec{c}, ) then
( A cdot vec{b}=vec{c} )
В . ( vec{a} perp vec{b}, vec{c} )
c. ( vec{a} perp vec{b}-vec{c} )
D. either ( vec{a} perp(vec{b}-vec{c}) ) or ( vec{b}=vec{c} )
12
521( (a . c) c+(c . b) a )
( A cdot-2 k )
B . ( 2 k )
c. ( i-j )
D. ( i+j-2 k )
12
522Calculate the scalar product of the following vectors. Given three forces ( M= )
( {3,-4,2}, W= )
( {2,3,-5}, ) and ( P={-3,-2,4} )
applied to the same point. Calculate the work performed by the resultant of these three forces when the point of application of the resultant moves along a straight line and is displaced
from the position ( M_{1}{5,3,-7} ) to the
position ( M_{2}{4,-1,-4} )
12
523Find the position vector of the mid point of the vector joining the points ( boldsymbol{P}(mathbf{2}, mathbf{3}, mathbf{4}) ) and ( boldsymbol{Q}(mathbf{4}, mathbf{1}, mathbf{2}) )12
524Illustration 3.33 Prove that (A+2B)-(2A-3B) = 2A? +
AB cos 0 – 6B2
12
5258. If a vector 2ỉ +3j +8k is perpendicular to the vector
49-4î +ak. Then the value of a is
(d) 1
7-
rand
12
52620. Given P = 3î – 49. Which of the following is perpendicular
to P?
a. 3 b. 4 c. 4i + 3 d. 47 -37
12
527If vectors ( (x-2) vec{a}+vec{b} ) and
( (2 x+1) vec{a}-vec{b} ) are parallel then ( x )
A ( cdot 1 / 3 )
B. 3
( c cdot-3 )
D. ( -1 / 3 )
12
528ABCD is a parallelogram and ( A C, B D ) are its diagonals Express ( overrightarrow{A B} ) and ( overrightarrow{A D} ) in terms of ( overrightarrow{boldsymbol{A C}} ) and ( overrightarrow{boldsymbol{B D}} )
A ( cdot overrightarrow{A B}=frac{1}{2} overrightarrow{A C}-frac{1}{2} overrightarrow{B D} quad ; overrightarrow{A D}=frac{1}{2} overrightarrow{A C}+frac{1}{2} overrightarrow{B D} )
B ( cdot overrightarrow{A B}=frac{1}{2} overrightarrow{A C}+frac{1}{2} overrightarrow{B D} quad ; overrightarrow{A D}=frac{1}{2} overrightarrow{A C}+frac{1}{2} overrightarrow{B D} )
c ( cdot overrightarrow{A B}=frac{1}{2} overrightarrow{A C}-frac{1}{2} overrightarrow{B D} quad ; overrightarrow{A D}=frac{1}{2} overrightarrow{A C}-frac{1}{2} overrightarrow{B D} )
D・ ( overrightarrow{A B}=frac{1}{2} overrightarrow{A C}+frac{1}{2} overrightarrow{B D} quad ; overrightarrow{A D}=frac{1}{2} overrightarrow{A C}-frac{1}{2} overrightarrow{B D} )
12
52912.
In a triangle ABC, D and E are points on BC and AC
respectively, such that BD=2 DC and AE = 3EC. Let P be
the point of intersection of AD and BE. Find BP/PE using
vector methods.
(1993 – 5 Marks)
12
530If ( theta ) is the angle between ( bar{a} ) and ( bar{b} ), then ( frac{|bar{a} times bar{b}|}{bar{a} cdot bar{b}} ) equals
A ( cdot tan theta )
B. – ( tan theta )
( c cdot cot theta )
D. – cot ( theta )
12
531A unit vector perpendicular to ( hat{mathbf{i}}-mathbf{2} hat{mathbf{j}}+ ) ( hat{k} ) and ( hat{3} i-hat{j}+2 hat{k} ) is
( frac{A hat{i}+3 hat{j}+7 hat{k}}{sqrt{83}} )
B. ( frac{-3 hat{i}+hat{j}+5 hat{k}}{sqrt{35}} )
c. ( frac{5 hat{i}+3 hat{j}-7 hat{k}}{sqrt{83}} )
D. ( frac{3 hat{i}-5 hat{j}+7 hat{k}}{sqrt{83}} )
12
5326. If AXB=BXA, then the angle between A and B is
(a) 12 (b) 3 (c) T (d) 1/4
12
533U.
TWT
12. The projection of a vector r = 3î + + 2k on the x-y
plane has magnitude
a. 3 b. 4 c. √14 d. /10
12
5343. A vector perpendicular to i + i + Â is
a. i – } t bo i – i – Ľ
c. – i – i – r o d. 3i +27 – 5h
From Eic 276 the corroot
12
53516. Three vectors A, B, C satisfy the relation AB = 0 and
AC = 0. The vector A is parallel to
a. b. c. B. Ĉ d. BxČ
12
536f ( |vec{a}|=7,|vec{b}|=11,|vec{a}+vec{b}|=10 sqrt{3} )
then ( |vec{a}-vec{b}|= )
A . 10
B. ( sqrt{10} )
c. ( 2 sqrt{10} )
D. 20
12
537The sine of the angle between the vectors ( bar{i}+3 bar{j}+2 bar{k} ) and ( 2 bar{i}-overline{4 j}+bar{k} ) is
A. ( sqrt{frac{155}{1.56}} )
В. ( sqrt{frac{115}{116}} )
c. ( sqrt{frac{115}{147}} )
D. ( sqrt{frac{sqrt{5}}{188}} )
12
53853.
The non-zero vectors are ä , and c are related by a = 86
and c = -75. Then the angle between a and ĉ is [2008]
12
539If ( (2 bar{i}+4 bar{j}+2 bar{k}) times(2 bar{i}-x bar{j}+5 bar{k})= )
( 16 bar{i}-6 bar{j}+2 x bar{k}, ) then the value of ( x ) is
( A cdot 2 )
B. -2
( c . )
D. 3
12
540Two forces of ( 10 mathrm{N} ) and6 ( mathrm{N} ) act upon a
body. The direction of the forces are
unknown.the resultant force on the body
may be
A . 15 N
B. 3 N
c. 17
D. 2 N
12
54120. Find 3-dimensional vectors 71, 72, 73 satisfying
v.v = 4, 71.12 = -2, V1.13 = 6,12.12
= 2, 72.73 = -5, V3.13 = 29
(2001 – 5
12
542If the position vector ( vec{a} ) of the point ( (5, n) ) is such that ( |vec{a}|=13 ), then the
value/values of ( n ) be
( mathbf{A} cdot pm 8 )
B. ±12
c. 8 only
D. 12 only
12
543If ( overrightarrow{D A}=vec{a}, overrightarrow{A B}=vec{b}, overrightarrow{C B}=overrightarrow{k a} ) where
( k>0 mathrm{X}, mathrm{Y} ) are the mid points of ( mathrm{DB} ) and
AC respectively, such that ( |vec{a}|=17 ) and ( |vec{X} Y|=4, ) then k equal to ( ? )
12
544Enter 1 if true else 0 .
The direction cosines of the vector ( 3 hat{i}- ) ( 4 hat{k} ) are
( frac{3}{5}, 0, frac{-4}{5} )
12
545( vec{a} ) and ( vec{b} ) form the consecutive sides of a
regular hexagon ABCDEF. ( overrightarrow{A B}=vec{a}, overrightarrow{B C}=vec{b} )
12
546( I^{prime} ) is the incentre of triangle of ( A B C )
whose corresponding sides are ( a, b, c ) respectively, ( a overrightarrow{I B}+b overrightarrow{I B}+c overrightarrow{I C} ) is
always equal to
A. 0
в. ( (a+b+c) overrightarrow{B C} )
c. ( (vec{a}+vec{b}+vec{c}) overrightarrow{A C} )
D. ( (a+b+c) overrightarrow{A B} )
12
547For which values of ‘ ( a ) ‘ the different
vectors ( bar{x}=(2 a, 3 a, 0) ) and ( bar{y}= )
( (0,0,4 a) ) are orthogonal vectors
( mathbf{A} cdot a in R )
B . ( a in N cup{0} )
( mathbf{c} cdot a in R-{0} )
( mathbf{D} cdot a=0 )
12
548Illustration 3.5 The resultant of two vectors A and B is per-
pendicular to the vector Ā and its magnitude is equal to half of
the magnitude of the vector B. Find out the angle between A and
B.
D B sin
B
B cos e
90°
E
——–
B sin @ 0
Fig. 3.19
12
549The vector ( boldsymbol{alpha} hat{boldsymbol{i}}+boldsymbol{2} hat{boldsymbol{j}}+boldsymbol{beta} hat{boldsymbol{k}} ) lies in the
plane of vectors ( vec{b}=hat{i}+hat{j} ) and ( vec{c}=hat{j}+hat{k} ) and bisects the angle between ( vec{b} ) and ( vec{c} ) Which one of the following gives
possible values of ( boldsymbol{alpha} ) and ( boldsymbol{beta} )
A. ( alpha=2, beta=2 )
2
в. ( alpha=1, beta=2 )
c. ( alpha=2, beta=1 )
D. ( alpha=1, beta=1 )
12
55037. For any vector a, the value of
(a xi)2 +lā ~ ;)2 +(a xk)2 is equal to
(a) 32² 6 a² (6) 22²
1200
(2) 4a
12
551The position vector of the foci of an ellipse are ( vec{b} ) and ( -vec{b}, ) and the length of
major axis is ( 2 a ), then the equation of
ellipse
A ( cdot a^{4}-a^{2}left(vec{r}^{2}+vec{b}^{2}right)+left(vec{b}^{2} times vec{r}^{2}right)=0 )
( ^{text {В }} a^{4}-a^{2}left(|vec{r}|^{2}+|vec{b}|^{2}right)+left(|vec{b}|^{2} cdot vec{r}^{2}right)=0 )
c. ( a^{4}+vec{a}^{2}left(vec{r}^{2}+vec{b}^{2}right)+left(vec{b}^{2} cdot vec{r}^{2}right)=0 )
D・ ( a^{4}-a^{2}left(vec{r}^{2}-vec{b}^{2}right)+left(vec{b}^{2} cdot vec{r}^{2}right)=0 )
12
552If ( |vec{a}|=|vec{b}|=|vec{c}|=1, vec{a} cdot vec{b}=vec{b} cdot vec{c}= )
( vec{c} cdot vec{a}=frac{1}{2} operatorname{then}[vec{a}, vec{b}, vec{c}]= )
A . 1
B. ( frac{1}{2} )
c. ( frac{1}{sqrt{2}} )
D. ( frac{sqrt{3}}{2} )
12
553If ( vec{a} ) and ( vec{b} ) are unit vectors such that ( vec{a}+3 vec{b} ) is perpendicular to ( 7 vec{a}-5 vec{b} ) and ( vec{a}-4 vec{b} ) is perpendicular to ( 7 vec{a}-2 vec{b} ) then
the angle between ( vec{a} ) and ( vec{b} ) is
A ( cdot frac{pi}{6} )
B.
( c cdot frac{pi}{2} )
D. ( frac{pi}{3} )
12
554Show that the vectors ( 2 vec{i}-vec{j}+vec{k}, vec{i}- ) ( 3 vec{j}-5 vec{k} ) and ( -3 vec{i}+4 vec{j}+4 vec{k} ) are the
sides of a right angled triangle.
12
555Let ( vec{u}, vec{v}, vec{w} ) be such that ( |vec{u}|=1,|vec{v}|= )
( mathbf{2},|overrightarrow{boldsymbol{w}}|=mathbf{3} . ) If the projection of ( overrightarrow{boldsymbol{v}} ) along ( overrightarrow{boldsymbol{u}} )
is equal to that of ( vec{w} ) along ( vec{u} ) and ( vec{v}, vec{w} ) are perpendicular to each other then ( |vec{u}-vec{v}+vec{w}| ) equals
A. ( sqrt{14} )
B. ( sqrt{7} )
( c cdot 2 )
D. 14
12
55658.
If ū, , ū are non-coplanar vectors and p, q are real numbers,
then the equality [3ū pū pā]-[pū o qū]-[2o qū qū]=0
holds for :
[2009]
(a) exactly two values of (p, q)
(b) more than two but not all values of (p, q)
(c) all values of (p, q)
(d) exactly one value of (p, q)
12
557If ( vec{r}=3 hat{i}+2 hat{j}-5 hat{k}, vec{a}=2 hat{i}-hat{j}+hat{k} )
( overrightarrow{boldsymbol{b}}=hat{boldsymbol{i}}+boldsymbol{3} hat{boldsymbol{j}}-boldsymbol{2} hat{boldsymbol{k}}, overrightarrow{boldsymbol{c}}=-boldsymbol{2} hat{boldsymbol{i}}+hat{boldsymbol{j}}-boldsymbol{3} hat{boldsymbol{k}} )
such that ( vec{r}=lambda vec{a}+mu vec{b}+v vec{c}, ) then ( mu, frac{lambda}{2} )
( boldsymbol{v} ) are in
A. ( H . P )
в. ( G . P )
c. ( A . G . P )
D. A.P
12
55835. In an equilateral triangle ABC, AL, BM, and CN are
medians. Forces along BC and BA represented by them
will have a resultant represented by
a. 2AL b. 2BM c. 2CN d. AC
36
The vector gum of the forma in nemandianlo.
12
559Illustration 3.37 Calculate the area of the triangle determined
by the two vectors A = 3 + 4 j and B =-3i +7j.
T
1
1
1
.
1
.
..
file non product of
12
560In Figure, identify the following vectors.
(i) Coinitial
(ii) Equa
(iii) Collinear but not equal
12
561( f(vec{a}, vec{b}, vec{c} ) are unit vectors such that ( vec{a}+vec{b}+vec{c}=0, ) find the value of ( vec{a} cdot vec{b}+vec{b} cdot vec{c}+vec{c} cdot vec{a} )12
562If ( vec{e}=l hat{i}+m hat{j}+n hat{k} ) is a unit vector,
then the maximum value of ( l m+ )
( m n+n l ) is
A. ( -frac{1}{2} )
B.
c. 1
D. ( frac{3}{2} )
12
563If ( l(bar{b} times bar{c})+m(bar{c} times bar{a})+n(bar{a} times bar{b})=0 )
And at least one of ( l, boldsymbol{m}, boldsymbol{n} ) is not zero then the vectors ( bar{a}, bar{b}, bar{c} ) are
A. Parallel
B. Coplanar
c. Mutually perpendicular
D. None of these
12
564The plane ( 2 x-3 y+z+6=0 ) divides
the line segment joining (2,4,16) and (3,5,-4) in the ratio
A .4: 5
B. 4: 7
c. 2: 1
D. 1: 2
12
565If ( vec{a}, vec{b}, vec{c} ) are unit vectors and the angles between ( vec{a}, vec{b} ) and ( vec{b}, vec{c} ) and ( vec{c}, vec{a} ) are ( frac{pi}{6}, frac{pi}{3} )
and ( frac{pi}{4} ) respectively, then ( |vec{a}+vec{b}+vec{c}| )
A . 3
B. ( 4+sqrt{3}+sqrt{2} )
c. ( sqrt{4+3+2} )
D. None of these
12
566If ( vec{x} ) and ( vec{y} ) are two non-collinear vectors
and ( a, b ) and ( c ) represent the sides of a
( Delta A B C ) satisfying ( (a-b) vec{x}+(b- )
( c) overrightarrow{boldsymbol{y}}+(boldsymbol{c}-boldsymbol{a})(overrightarrow{boldsymbol{x}} times overrightarrow{boldsymbol{y}})=0, ) then ( Delta boldsymbol{A} boldsymbol{B C} )
is ( where ( vec{x} times vec{y} ) is perpendicular to the
plane of ( vec{x} text { and } vec{y}) )
A. An acute-angle triangle
B. An obtuse-angled triangle
c. Right-angled triangle
D. A scalene triangle
12
567If ( overrightarrow{boldsymbol{A}}=boldsymbol{i}-boldsymbol{j}, overrightarrow{boldsymbol{B}}=boldsymbol{i}+boldsymbol{j}+boldsymbol{k} ) are two
vectors and ( vec{C} ) is another vector such that ( vec{A} times vec{C}+vec{B}=overrightarrow{0} ) and ( vec{A} cdot vec{C}=0, ) then
( |vec{C}|^{2}= )
A ( cdot frac{3}{2} )
в. 8
c. ( frac{19}{2} )
D. ( frac{17}{2} )
12
568Show that ( |vec{a}| vec{b}+|vec{b}| vec{a} ) is perpendicular to ( |vec{a}| vec{b}-|vec{b}| vec{a}, ) for any two nonzero vectors ( vec{a} ) and ( vec{b} )12
569If the vectors ( 2 x hat{i}-y hat{j}-4 hat{k} ) and ( 10 hat{i}+ ) ( hat{boldsymbol{y}}+5 hat{k} ) are perpendicular, then the locus of ( (x, y) ) is
A. A Parabola with vertex at (1,0) and latus rectum 10
B. A Parabola with vertex at (0,1) and latus rectum 20
c. A Parabola with vertex at (1,0) and latus rectum 20
D. A Parabola with vertex at (0,1) and latus rectum 10
12
570Let ( overrightarrow{boldsymbol{alpha}}=mathbf{3} hat{boldsymbol{i}}+hat{boldsymbol{j}} ) and ( overrightarrow{boldsymbol{beta}}=mathbf{2} hat{boldsymbol{i}}-hat{boldsymbol{j}}+boldsymbol{3} hat{boldsymbol{k}} . ) If
( overrightarrow{boldsymbol{beta}}=overrightarrow{boldsymbol{beta}}_{1}-boldsymbol{beta}_{2}, ) where ( overrightarrow{boldsymbol{beta}}_{1} ) is parallel to ( overrightarrow{boldsymbol{alpha}} )
and ( vec{beta}_{2} ) is perpendicular to ( vec{alpha}, ) then ( vec{beta}_{1} times )
( vec{beta}_{2} ) is equal to?
( mathbf{A} cdot-3 hat{i}+9 hat{j}+5 hat{k} )
B . ( 3 hat{i}-9 hat{j}-5 hat{k} )
c. ( frac{1}{2}(-3 hat{i}+9 hat{j}+5 hat{k}) )
D・ ( frac{1}{2}(3 hat{i}-9 hat{j}+5 hat{k}) )
12
5713.
The volume of the parallelopiped whose sides are given by
OA = 2i – 2j, OB = i + j – k, OC = 3i – k, is
(1983 – 1 Mark)
(6) 4
(d) none of these
12
5724. From Fig. 3.76, the correct
relation is
a. Ā+ B + Ē = b.
C – D = – Ā
+
d. All of the above
Fig. 3.76
12
573A man travelling east at ( 8 mathrm{km} ) per hour
find that the wind seems to blow
directly from the north. On doubling the speed, he finds that it appears to come from N-E. Find the velocity of the wind and it’s direction.
A. ( 8 sqrt{2} ), its direction is from ( N-S ). . Its direction is
B. ( 8 sqrt{2} ), its direction is from ( N-w )
c. ( 8 sqrt{2} ), its direction is from s-w.
D. None of these
12
574If co-ordinates of a point ( P ) is (1,2,2) then unit vector along the position vector of the point ( P ) will be
A ( cdot frac{1}{3}(hat{i}+2 hat{j}+2 hat{k}) )
B . ( hat{i}+2 hat{j}+2 hat{k} )
c ( cdot frac{1}{3}(hat{i}-2 hat{j}-2 hat{k}) )
D. ( frac{1}{sqrt{3}}(hat{i}+hat{j}+hat{k}) )
12
575Given two vectors ( a=2 i-3 j+6 k ) on
( b=2 i+3 j-k ) and ( lambda= )
( frac{text {the projection of } a text { on } b}{text {the projection of bon } a}, ) then the
value of ( lambda ) is
A. 3/
в. ( 7 / 3 )
( c cdot 3 )
D. 7
12
57642. Let à =î + i + ł, =í – î +â and c =i- j – k be three
vectors. A vector ū in the plane of a and b, whose
projection on ĉ is T, is given by
(2011)
(a) î – 3j+zł
(b) -3i – 39 –
(C) Zî – +3Ã (d) î+393k
12
577If ( |bar{a}+bar{b}|=|bar{a}-bar{b}| ) then
A ( . bar{a} | bar{b} )
B. ( bar{a} perp bar{b} )
( mathbf{c} cdot bar{a}=bar{b} )
D. None of these
12
578(i) If ( vec{a} times vec{b}=vec{c} times vec{d} ) and ( vec{a} times vec{c}=vec{b} times vec{d} )
show that ( vec{a}-vec{d} ) and ( vec{b}-vec{c} ) are paralle
(ii) Find the direction cosines of the line
joining (2,-3,1) and (3,1,-2)
12
579If ( vec{b}=3 hat{i}+4 hat{j} ) and ( vec{a}=hat{i}-hat{j}, ) the vector
having the same magnitude as that of ( vec{b} ) and parallel to ( vec{a} ) is
A ( cdot frac{5}{sqrt{2}}(hat{i}-hat{j}) )
B. ( frac{5}{sqrt{2}}(hat{i}+hat{j}) )
c. ( 5(hat{i}-hat{j}) )
D. ( 5(hat{i}+hat{j}) )
12
5809. Vector A is 2 cm long and is 60° above the x-axis in the
first quadrant. Vector B is 2 cm long and is 60° below the
x-axis in the fourth quadrant. The sum A + B is a vector
of magnitude
a. 2 cm along positive y-axis
b. 2 cm along positive x-axis
c. 2 cm along negative y-axis
d. 2 cm along negative x-axis
12
581Find vector of magnitude 9 equality
inclined to the coordinate axes?
12
582Illustration 3.26 Given that A+B+C =0. Out of three
vectors, two are equal in magnitude and the magnitude of the
third vector is 12 times that of either of the two having equal
magnitude. Find the angles between the vectors.
12
583If ( vec{a} ) and ( vec{b} ) are the vectors determined by
two adjacent sides of regular hexagon, then vector ( boldsymbol{E} boldsymbol{F} ) is
A ( cdot(vec{a}+vec{b}) )
B . ( (vec{a}-vec{b}) )
( c cdot 2 vec{a} vec{a} vec{a} )
( D cdot 2 vec{b} )
12
584P
5. A body is at rest under the action of three forces, two of
which are Ē;=4ỉ, Fz=6, the third force is
(a) 4 +6j
(b) 4 – 69
(c) – 4i +6
(d) -41 – 67
12
585If the vectors ( a ) and ( b ) are linearly independent satisfying ( (sqrt{3} tan theta+ )
1) ( vec{a}+(sqrt{3} sec theta-2) vec{b}=0, ) then the
most general values of ( theta ) are
A ( cdot n pi-frac{pi}{6}, n in Z )
B. ( 2 n pi pm frac{11 pi}{6}, n in Z )
c. ( n pi pm frac{pi}{6}, n in Z )
D. ( 2 n pi+frac{11 pi}{6}, n in Z )
12
58611. The angle between A + B and Āx B is
a.
o b . t/4 c. 12 d. a
12
587Find the sum of the following vectors:
( overrightarrow{boldsymbol{a}}=hat{boldsymbol{i}}-boldsymbol{3} hat{boldsymbol{k}}, overrightarrow{boldsymbol{b}}=2 hat{hat{boldsymbol{j}}}-hat{boldsymbol{k}}, overrightarrow{boldsymbol{c}}=boldsymbol{2} hat{boldsymbol{i}}-boldsymbol{3} hat{boldsymbol{j}}+ )
( mathbf{2} hat{boldsymbol{k}} )
12
58841. Let ā = i-k, = xi+ị + (1 – x) Â and
ē =yî +xì +(1 +x – y) Â. Then [a ,b,c] depends on
[2005]
(a) only y
(c) both x and y
(b) onlyx
(d) neither x nor y
12
58914. Let x, y and z be three vectors each of magnitude 72
and the angle between each pair of them is If á is a
non-zero vector perpendicular to x and yx z and
non-zero vector perpendicular to y and zx x, then
(JEE Adv. 2014)
(a)
b
=
b.z ||z – x
(6) 3-(3.5)3)
12
590Find the projection of vector ( 2 hat{i}+hat{j} ) on the vector ( hat{mathbf{i}}+mathbf{2} hat{mathbf{j}} )12
591A unit vector ( a ) makes an angle ( frac{pi}{4} ) with z-axis.. If ( a+i+j ) is a unit vector, then
( a ) is equal to
A ( cdot frac{i}{2}+frac{j}{2}+frac{k}{sqrt{2}} )
В ( cdot frac{i}{2}+frac{j}{2}-frac{k}{sqrt{2}} )
c. ( -frac{i}{2}-frac{j}{2}+frac{k}{sqrt{2}} )
D. None of these
12
592If the diagonals of a parallelogram are ( bar{i}+5 bar{j}-2 bar{k} ) and ( -2 bar{i}+bar{j}+3 bar{k}, ) then the
lengths of its sides are
A ( cdot frac{sqrt{38}}{2}, frac{sqrt{50}}{2} )
B. ( frac{sqrt{35}}{2}, frac{sqrt{45}}{2} )
c. ( sqrt{38}, sqrt{50} )
D. ( sqrt{35}, sqrt{45} )
12
593Find the value of ( lambda ) in the unit vector
( mathbf{0 . 4} hat{mathbf{i}}+mathbf{0 . 8} hat{mathbf{j}}+boldsymbol{lambda} hat{boldsymbol{k}} )
12
59426.
The unit vector which is orthogonal to the vector
3ỉ + 2 j +6k and is coplanar with the vectors 2î + i +k
and î – į+is
(2004)
zî – 67 –
27 -3
141
4i +37 – 3
(b)
J13
i – h
(d)
134
V10
aftha ana unit from the
12
595Mo Tuus u exactly three values
73. If the vectors AB = 3 + 4k and AC = 5î -2; +4k are
the sides of a triangle ABC, then the length of the median
through A is
(JEEM 2013]
(a) V18
(b) 72
(c) V33
(d) 45
12
59630.
If the incident ray on a surface is along the unit vector ,
the reflected ray is along the unit vector w and the normal
is along unit vector â outwards. Express û in terms of a
and û.
(2005 – 4 Marks)
12
597. A person moves 30 m north and then 201
finally 30/2 m in south-west direction. The
of the person from the origin will be
(a) 10 m along north (b) 10 m long south
(c) 10 m along west (d) Zero
s 30 m north and then 20 m towards east and
south-west direction. The displacement
12
598The points (2,3,4),(-1,-2,1),(1,2,5)
and (4,7,8) are the vertices of a
A. rhombus
B. square
c. a parallelogram
D. none of these
12
599→ ^ ^
? ?
a = 3i-5; and b = 6i+3 ; are two vectors and c is a
vector such that c = ax b then
a 1:1 b
🙂
(a) V34 : N45 : 139
(C) 34:39:45
(b) 134: 145:39 [2002]
(d) 39:35:34
12
600( vec{A} ) is a vector which when added to the resultant of vectors ( (2 hat{i}-3 hat{j}+4 widehat{k}) ) and ( (widehat{boldsymbol{i}}+boldsymbol{5} widehat{boldsymbol{j}}+boldsymbol{2} widehat{boldsymbol{k}}) ) yields a unit vector along
the y-axis. Then vector ( vec{A} ) is:
A . ( -3 hat{i}-widehat{j}-6 widehat{k} )
B. ( 3 hat{i}+widehat{j}-6 widehat{k} )
c. ( 3 hat{i}-widehat{j}+6 widehat{k} )
D. ( 3 hat{i}+widehat{j}+6 widehat{k} )
12
601If ( vec{a} ) and ( vec{b} ) are position vectors of point ( A ) and ( B ) respectively, then the position vector of points of trisection of AB12
602The position vectors of ( mathbf{A}, mathbf{B} ) are ( mathbf{a}, mathbf{6} ) respectively. The position vector of ( mathbf{C} ) is ( frac{5 bar{a}}{3}-bar{b} . ) Then 3
( mathbf{A} cdot mathrm{C} ) is inside the ( Delta O A B )
B. ( C ) is outside the ( Delta O A B ) but inside the angle OAB
C. ( mathrm{C} ) is outside the ( Delta O A B ) but inside the angle ( mathrm{OBA} )
D. None of these
12
603( boldsymbol{M B} )
( mathbf{A} cdot overrightarrow{M B}=frac{1}{2}(bar{a}-2 bar{b}) )
B ( cdot overrightarrow{M B}=frac{1}{2}(2 bar{a}-bar{b}) )
C ( cdot overrightarrow{M B}=frac{1}{2}(bar{a}+2 bar{b}) )
D ( cdot overrightarrow{M B}=frac{1}{2}(2 bar{a}+bar{b}) )
12
604A straight line ( r=a+lambda b ) meets the
plane ( r . n=0 ) in ( P . ) The position vector
of ( boldsymbol{P} ) is
A ( cdot a+frac{a cdot n}{b cdot n} b )
B. ( a+frac{b . n}{a . n} b )
c. ( _{a-} frac{a . n}{b . n} b )
D. None of these
12
605Assertion
If vectors ( a ) and ( c ) are non collinear then
the lines ( r=6 a-c+lambda(2 c-a) ) and
( boldsymbol{r}=boldsymbol{a}-boldsymbol{c}+boldsymbol{mu}(boldsymbol{a}+boldsymbol{3} boldsymbol{c}) ) are coplanar
Reason
There exist ( lambda ) and ( mu ) such that the two
values of ( r ) become same.
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
606Calculate the scalar product of the following vectors.

Resolve the vector ( d={1,1,1} ) into
components with respect to three
noncoplanar vectors ( boldsymbol{a}= )
( {1,1,-2}, b={1,-1,0}, ) and ( c= )
{0,2,3}

12
60776. If [axh bxc cxa]=a[abel’ then a is equal to
(a)
(b) 1
(c) 2
[JEE M 2014]
(d) 3
12
608A vector of magnitude 10 has its rectangular components as 8 and 6 along ( x ) and ( y ) axes. Find the angle it make with these axes.12
60970. Let ABCD be a parallelogram such that AB =9, AD =ń
and ZBAD be an acute angle. If † is the vector that coincide
with the altitude directed from the vertex B to the side AD
then ř is given by :
[2012]
30
(a) = đã 3
(a)
7 = 39 –
p
(b) = -4 (P.4) –
(b)
=-9+
©) 7=ă – (3:);
(d) 7=-3-3 (3.;
(P.P
12
610If ( vec{a}, vec{b}, vec{c} ) are non-coplanar vector and ( lambda ) is
a real number, then the vectors ( vec{a}+ ) ( 2 vec{b}+3 vec{c}, lambda vec{b}+mu vec{c} a n d(2 lambda-1) vec{c} ) are
coplanar when
( mathbf{A} cdot mu epsilon R )
B ( cdot lambda=frac{1}{2} )
c. ( lambda=0 )
D. All value are correct
12
611Two unit vector when added give a unit
vector. Then choose the correct
statement.
A. Magnitude of their difference is ( sqrt{3} )
B. Magnitude of their difference is 1
c. Angle between the vectors is ( 90^{circ} )
D. Angle between the sum and the difference of the two vectors is ( 90^{circ} )
12
612An airplane is heading north east at a speed of ( 141 m s^{-1} ). The northward
component of its velocity is:
A. ( 141.4 m s^{-1} )
B. ( 100 mathrm{ms}^{-1} )
c. zero
D. ( 50 m s^{-1} )
12
613If ( vec{a}=2 hat{i}+l hat{j}-3 hat{k} ) and ( vec{b}=4 hat{i}-3 hat{j}- )
( 2 hat{k} ) are perpendicular to each other then
find the value of scalar ( lambda )
12
614Answer the following as true or false.
(i) ( vec{a} ) and ( -vec{a} ) are collinear
(ii) Two collinear vectors are always equal in magnitude.
(iii) Two vectors having same magnitude are collinear
(iv) Two collinear vectors having the
same magnitude are equal.
12
615Show that vector area of a quadrilateral ( A B C D ) is ( frac{1}{2}(A C times B D), ) where ( A C )
and ( B D ) are its diagonals.
12
616The values of ( lambda ) such than ( (x, y, z) neq ) (0,0,0)
and ( (i hat{+} hat{j}+3 hat{k}) x+(3 hat{i}-3 hat{j}+hat{k}) y+ )
( (-4 hat{i}+5 hat{j}) z=lambda(x hat{i}+y hat{j}+z hat{k}) ) are
12
61714. The position vectors of the vertices A, B and
tetrahedron ABCD are î+ i + k, i and 3î , respectively. The
altitude from vertex D to the opposite face ABC meets the
median line through A of the triangle ABC at a point E. the
length of the side AD is 4 and the volume of the tetrahedron
Is <12
, find the position vector of the point E for all its
possible positions.
(1996 – 5 Marks)
12
618( operatorname{Let} overrightarrow{boldsymbol{A}}=hat{boldsymbol{i}}+2 hat{boldsymbol{j}}+boldsymbol{3} hat{boldsymbol{k}}, overrightarrow{boldsymbol{B}}=boldsymbol{4} hat{boldsymbol{i}}+2 hat{boldsymbol{j}}, overrightarrow{boldsymbol{C}}= )
( mathbf{2} hat{mathbf{i}}+mathbf{2} hat{mathbf{j}}+mathbf{2} hat{k} . ) Then the ratio in which ( boldsymbol{C} )
divides ( A B ) is
A . 3: 4
B. 1: 3
c. 1: 2
D. 1: 1
12
619( boldsymbol{P C} )
( mathbf{A} cdot overrightarrow{P C}=2 b-a )
B ( cdot overrightarrow{P C}=2 a+b )
( mathbf{c} cdot overrightarrow{P C}=2 a-b )
D. none of these
12
6204. The magnitudes of vectors A, B and C are 3, 4 and 5 units
respectively. If A + B = C, the angle between A and B is
(b) cos-(0.6)
12
621If the position vectors of ( A, B, C, D ) are ( vec{a}, vec{b} cdot 2 vec{a}+3 vec{b}, vec{a}-2 vec{b} ) respectively, then
( overrightarrow{boldsymbol{A C}}, overrightarrow{boldsymbol{D B}}, overrightarrow{boldsymbol{B A}}, overrightarrow{boldsymbol{D A}} ) are?
A ( cdot vec{a}+3 vec{b}, 3 vec{b}-vec{a}, vec{a}-vec{b}, 2 vec{b} vec{b} )
B . ( 2 vec{b}, vec{b}-2 vec{a}, 3 vec{b}+vec{a}, vec{b}-vec{a} )
c. ( vec{a}-3 vec{b}, 3 vec{b}-vec{a}, vec{a}+vec{b}, 2 vec{b} )
D. ( -2 vec{b}, vec{b}-2 vec{a}, 3 vec{b}-vec{a}, vec{b}-vec{a} )
12
622The two vectors ( left(x^{2}-1right) hat{i}+(x+2) hat{j}+ )
( x^{2} hat{k} & 2 hat{i}-x hat{j}+3 hat{k} ) are orthogonal
A. For no real value of ( x )
B. For ( x=-1 )
( mathrm{c} cdot operatorname{For} x=frac{1}{2} )
D. For ( x=-frac{1}{2} & x=1 )
12
62323.
The value of a’ so that the volume of parallelopiped formed
by î + aj + k, j + ak and aî +becomes minimum is
(2003)
(2) 3 (6) 3 (c) 1/3 (d) √3
12
62410.
If a, b and i are unit vectors, then
(a) 4
(6)
9
es NOT exceed (20015)
(c) 8 (d) 6
12
625Which of the following expressions are meaningful?
(1998 – 2 Marks)
(a) u (v x w)
(b) (u.v). w
(c) (u.v) w
(d) ux(
vw)
171
11:
12
626If ( vec{a} ) and ( vec{b} ) are two non-zero and non collinear vectors, then ( vec{a}+vec{b} ) and ( vec{a}-vec{b} )
are
A. linearly dependent vectors
B. linearly independent vectors
c. linearly dependent and independent vectors
D. none of the above
12
627If the points ( boldsymbol{A}(overline{boldsymbol{a}}), boldsymbol{B}(overline{boldsymbol{b}}), boldsymbol{C}(overline{boldsymbol{c}}) ) satisfy
the relation ( 3 bar{a}-8 bar{b}+5 bar{c}=0 ) then the
points are
A. vertices of an equilateral triangle
B. collinear
c. vertices of a right angled triangle
D. vertices of an isosceles triangle
12
628If ( vec{a}=hat{i}+2 hat{j}-3 hat{k}, vec{b}=2 hat{i}+hat{j}-hat{k} ) and
( vec{u} ) is a vector satisfying ( overrightarrow{boldsymbol{a}} times overrightarrow{boldsymbol{u}}=overrightarrow{boldsymbol{a}} times overrightarrow{boldsymbol{b}} )
and ( vec{a} cdot vec{u}=0, ) then ( 2|vec{u}|^{2} ) is equal to
A. 5
B. 4
( c cdot 8 )
D. None of these
12
629A unit vector ( vec{a} ) makes angles ( frac{pi}{4} ) and ( frac{pi}{3} ) with ( hat{i} ) and ( hat{j} ) respectively and an acute angle ( theta ) with ( hat{k} ). Find the angle ( theta ) and
components of ( overrightarrow{boldsymbol{a}} )
12
630State whether True or False:
Motion in 3-Dimension can’t be
represented with the help of vectors.
A. True
B. False
12
631If ( theta ) be the angle between the vectors ( i+ ) ( j ) and ( j+k, ) then ( theta ) is
( mathbf{A} cdot mathbf{0} )
в. ( pi / 4 )
c. ( pi / 2 )
D . ( pi / 3 )
12
632The triangle ( A B C ) is defined by the vertices ( boldsymbol{A}=(mathbf{0}, mathbf{7}, mathbf{1 0}), boldsymbol{B}=(-mathbf{1}, mathbf{6}, mathbf{6}) )
and ( C=(-4,9,6) . ) Let ( D ) be the foot of the attitude from ( B ) to the side ( A C ) then
( B D ) is
A ( . bar{i}+2 bar{j}+2 bar{k} )
B. ( -bar{i}+2 bar{j}+2 bar{k} )
c. ( bar{i}+2 bar{j}-2 bar{k} )
D. ( bar{i}-2 bar{j}+2 bar{k} )
12
6334. Following forces start acting on a particle at rest at the
origin of the co-ordinate system simultaneously
Ē=-4î – 5j +5k F2 = 5ỉ +8j+6k Fz=-3ỉ +49 – 7k
and F = 2î – 3j – 2Â then the particle will move
(a) In x-y plane (b) In y–z plane
(c) In x-z plane (d) Along x-axis
12
634If ( bar{a}, bar{b}, bar{c} ) are unit vectors satisfying the relation ( bar{a}+bar{b}+sqrt{3} bar{c}=0, ) then the
angle between ( bar{a} ) and ( bar{b} ) is
( ^{A} cdot frac{pi}{6} )
в.
( c cdot frac{pi}{3} )
D.
12
6352.
A1, A2,…………………. A, are the vertices of a regular plane
polygon with n sides and O is its centre. Show that
n-1
(Oá ¡+1) = (1 – n)(OA2 ⓇOẢ1) (1982 – 2 Marks)
i=1
12
636Illustration 3.31 If the sum of two unit vectors is a unit vector,
then find the magnitude of their difference.
12
6377. The ratio of maximum and minimum magnitudes of the
resultant of two vectors ā and b is 3:1. Now, lal is equal to
a. 161. b. 2161 c. 3161 d. 416)
12
638If ( a, b, c ) form a system of linearly independent vectors then show that the
system of vectors ( a-2 b+c, 2 a-b+c )
and ( 3 a+b+2 c ) is also linearly
independent.
12
639The unit vector perpendicular to both the vectors ( vec{a}=vec{i}+vec{j}+vec{k} ) and ( vec{b}=2 vec{i}- )
( vec{j}+3 vec{k} ) and making an acute angle with the vector ( vec{k} ) is
A ( cdot-frac{1}{sqrt{26}}(4 vec{i}-vec{j}-3 vec{k}) )
B. ( frac{1}{sqrt{26}}(4 vec{i}-vec{j}-3 vec{k}) )
c. ( frac{1}{sqrt{26}}(4 vec{i}-vec{j}+3 vec{k}) )
D. None of these
12
640If ( alpha, beta, gamma ) be the direction angles of a vector and ( cos alpha=frac{14}{15}, cos beta=frac{1}{3}, ) then
( cos gamma= )
This question has multiple correct options
( A cdot pm frac{2}{15} )
в. ( frac{1}{5} )
( c cdot pm frac{1}{15} )
D. None of these
12
641A vector of magnitude ( |Psi| ) is turned
through angle ( frac{phi}{2} . ) The magnitude of
change in the vector is given by
A ( cdot 2 mid Psi | cos phi / 4 )
B . ( |Psi| mid sin phi / 4 )
( mathbf{C} cdot|Psi|^{2}|sin phi / 4|^{2} )
D. ( 2|Psi | sin phi / 4| )
12
642If ( cos alpha, cos beta ) and ( cos gamma ) are direction
cosines of a vector, then they satisfy which of the following? Prove it.
A ( cdot cos ^{2} alpha+cos ^{2} beta+cos ^{2} gamma=0 )
B . ( cos ^{2} alpha+cos ^{2} beta=cos ^{2} gamma )
c. ( cos ^{2} alpha+cos ^{2} beta+cos ^{2} gamma=-1 )
D. ( cos ^{2} alpha+cos ^{2} beta+cos ^{2} gamma=1 )
12
643If a unit vector is represented by ( mathbf{0 . 5} overline{mathbf{i}}+mathbf{0 . 8} overline{mathbf{j}}+mathbf{c overline { k }} )
then the value of ‘c’ is
( A )
B. ( sqrt{0.8} )
c. ( sqrt{0.11} )
D. ( sqrt{0.01} )
12
644If ( vec{a}=a_{1} hat{i}+a_{2} hat{j}+a_{3} hat{k}, vec{b}=b_{1} hat{i}+b_{2} hat{j}+ )
( b_{3} hat{k}, vec{c}=c_{1} hat{i}+c_{2} hat{j}+c_{3} hat{k} ) and ( vec{c} ) is a unit
vector orthogonal to the plane of ( vec{a}, vec{b} ) and the angle between ( vec{a} ) and ( vec{b} ) is ( frac{pi}{3} )
then ( |vec{c}|^{2}= )
( mathbf{A} cdot mathbf{0} )
B.
c. ( frac{3}{4}left(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}right)left(b_{1}^{2}+b_{2}^{2}+b_{3}^{2}right) )
D ( cdot frac{3}{4}left(C_{1}^{2} a_{1}^{2}+b_{1}^{2}+mathrm{X} a_{2}^{2}+b_{2}^{2}+c_{2}^{2}right) )
12
645Five equal forces of ( 10 mathrm{N} ) each are
applied at one point and all are lying in one plane. If the angles between any two adjacent forces are equal, the resultant of these forces will be :
A . zero
B. 10N
c. 20N
D. ( 10 sqrt{2} N )
12
646A man goes 100 m North then 100 m
East and then 20 m North and then ( 100 sqrt{2} ) m South West. Find the displacement
( mathbf{A} cdot 20 m ) west
B. ( 20 mathrm{m} ) East
c. ( 20 m ) North
D. ( 20 m ) south
12
647If ( 3 p+2 q=i+j+k ) and ( 3 p-2 q= )
( i-j-k, ) then the angle between ( p ) and
( q ) is
A ( cdot frac{pi}{6} )
в.
c. ( frac{pi}{3} )
D. ( frac{pi}{2} )
( E )
12
648If ã and ū are two unit vectors such that a +25 and
5ā – 45 are perpendicular to each other then the angle
between ã and ő is
(2002)
(6) 60°
(a) 45°
(c) cos”)
12
64916. a,b,c are 3 vectors, such that a+b + c = 0
lal = 1 15 = 2/7 = 3, , then āb+be+cā is equal to [2003]
(a) 1 (b) 0 (c) -7 (d) 7.
12
650( (a . b)+(b . c) div(c . a) )
( mathbf{A} cdot mathbf{0} )
B.
( c cdot frac{1}{3} )
D.
12
651rume is the thupuric vi vectur ( cup boldsymbol{r} )
and point ( Q ) is the endpoint of vector
OQ as shown in the above figure. When
the vectors ( overline{O P} ) and ( overline{O Q} ) are added,
calculate the length of the resultant
vector.
A . 1.41
B. 2.24
c. 2.65
D. 3.00
E . 8.6
12
652Find ( ^{prime} lambda^{prime} ) when the projection of ( vec{a}= ) ( lambda hat{i}+hat{j}+4 hat{k} ) on ( vec{b}=2 hat{i}+6 hat{j}+3 hat{k} ) is 4
units.
12
653If ( boldsymbol{a}+boldsymbol{b}+boldsymbol{c}=mathbf{0} ) and ( |boldsymbol{a}|=mathbf{3} ;|boldsymbol{b}|= )
( mathbf{5} ;|boldsymbol{c}|=mathbf{7}, ) find the angle between a
vector and
A ( cdot 60^{circ} )
B. ( 30^{circ} )
( c cdot 45^{circ} )
D. none of these
12
654If ( vec{a}=2 hat{i}+2 hat{j} ) and ( vec{b}=3 hat{i}+4 hat{j} ) then ( vec{a}+ )
( vec{b} ) is
( mathbf{A} cdot hat{i}+2 hat{j} )
B. ( 5 hat{i}+6 hat{j} )
c. ( 5 hat{i}+2 hat{j} )
D. ( hat{i}+6 hat{j} )
12
65512. Two adjacent sides of a parallelogram are represented by
the two vectors î + 2î + 3k and 3î – 2ſ + k. What is the
area of parallelogram?
(a) 8 (b) 8/3 (c) 3/8 (d) 192
12
656If the position vector ( vec{a} ) of a point ( (12, n) ) is such that ( |vec{a}|=13 ), find the value of
( mathbf{n} )
12
657Find the magnitude of two vectors ( vec{a} ) and ( vec{b}, ) having the same magnitude and such that the angle between them is ( 60^{circ} ) and their scalar product is ( frac{1}{2} )12
658If ( |a|=5 .|vec{b}|=4, ) and ( |c|=3 . ) then what will be the value of ( vec{a} . vec{b}+vec{b} . vec{c}+vec{c} . vec{a} ) given that ( vec{a}+vec{b}+vec{c}=0 )
A . 25
B. 50
c. -25
D. -50
12
659If the angle between the unit vectors ( vec{a} ) and ( vec{b} ) is ( 60^{circ}, ) then ( |vec{a}-vec{b}| ) is
A. 0
B. 2
( c )
D. 4
12
660If ( vec{a}, vec{b}, vec{c} ) are three vectors such that ( vec{a}, vec{b}, vec{c}, ) then prove that ( vec{a} times vec{b}=vec{b} times vec{c}= )
( vec{c} times vec{a}, ) and hence show that ( [overrightarrow{boldsymbol{a}} overrightarrow{boldsymbol{b}} overrightarrow{boldsymbol{c}}]=mathbf{0} )
12
661If ( vec{a} ) and ( vec{b} ) are two unit vectors such that ( vec{a}+2 vec{b} ) and ( 5 vec{a}-4 vec{b} ) are perpendicular
to each other then the angle between ( vec{a} ) and ( vec{b} ) is
A . 45
B. ( 60^{circ} )
( ^{mathbf{c}} cdot cos ^{-1}left(frac{1}{3}right) )
D. ( cos ^{-1}left(frac{2}{7}right) )
12
662If ( vec{a}=hat{i}+hat{j}+hat{k}, vec{b}=hat{j}-hat{k}, ) then find a
vector ( vec{c} ) such that ( vec{a} times vec{c}= )
( overrightarrow{boldsymbol{b}} quad ) and ( quad overrightarrow{boldsymbol{a}} cdot overrightarrow{boldsymbol{c}}=mathbf{3} )
12
663If vector ( overline{O P} ) in ( X Y ) plane whose
magnitude is ( sqrt{3} ) makes an angle ( 60^{circ} ) with ( Y ) -axis, the length of the component of the vector in the direction of X-axis is
A .
B. ( sqrt{3} )
( c cdot frac{1}{2} )
D. ( frac{3}{2} )
12
664A vector ( vec{V} ) is given in a rectangular coordinates system as ( boldsymbol{V}=boldsymbol{6} overrightarrow{boldsymbol{i}}+boldsymbol{8} overrightarrow{boldsymbol{j}} )
where ( vec{i} ) and ( vec{jmath} ) are the unit vectors along ( x ) and ( y ) axes, then the angle ( theta ) which the vector ( vec{V} ) makes with the ( x- )
axis is
A. ( tan theta=8 / 6 )
B. ( tan theta=6 / 8 )
( mathbf{c} cdot tan theta=6 )
( mathbf{D} cdot tan theta=8 )
12
665If ( bar{a}, bar{b}, bar{c} ) are position vectors of the noncollinear points ( A, B, C ) respectively, the shortest distance of ( A ) and ( B C ) is?
A. ( bar{a} cdot(bar{b}-bar{c}) )
a)
в. ( bar{b} cdot(bar{c}-bar{a}) )
c. ( |bar{b}-bar{a}| )
D ( cdot sqrt{|vec{b}-vec{a}|^{2}-left[frac{(vec{a}-vec{b} cdot(vec{b}-vec{c})}{|vec{b}-vec{c}|}right)^{2}} )
12
666If ( bar{a}=hat{i}-hat{j}+hat{k} ) and ( bar{b}=2 hat{i}+hat{j}+3 hat{k} )
then the vector along ( bar{a}+bar{b} ) is
A ( cdot frac{3 hat{i}+4 hat{k}}{5} )
B. ( frac{-3 hat{i}+4 hat{k}}{5} )
c. ( frac{-3 hat{i}-4 hat{k}}{5} )
D. none
12
667A system of vectors is said to be coplanar, if
I. Their scalar triple product is zero.
II. They are linearly dependent. Which of the following is true?
A. Onlyı
B. Only II
c. Both I and II
D. None of these
12
668Component of ( vec{a}=hat{i}-hat{j}-hat{k} )
perpendicular to the vector ( overrightarrow{boldsymbol{b}}=2 hat{boldsymbol{i}}+ ) ( hat{boldsymbol{j}}-hat{boldsymbol{k}} ) is?
A ( cdot frac{1}{3}(hat{i}+2 hat{j}+2 hat{k}) )
B ( cdot frac{1}{3}(hat{i}-4 hat{j}-2 hat{k}) )
c ( cdot frac{1}{3}(hat{i}+4 hat{j}+2 hat{k}) )
D ( cdot frac{1}{3}(hat{i}+2 hat{j}+hat{k}) )
12
669Position vectors of mid point of the
vector joining the points ( boldsymbol{P}(mathbf{2}, mathbf{3}, mathbf{4}) ) and ( Q(4,1,-2) ) is
( mathbf{A} cdot 2 hat{i}+widehat{j}+widehat{k} )
B. ( 3 hat{i}+2 hat{j}+widehat{k} )
c. ( -3 hat{i}-hat{j}+widehat{k} )
D. None
12
670The area of the parallelogram whose adjacent sides are ( 2 mathrm{i}-3 mathrm{k} ) and ( 4 mathrm{j}+2 mathrm{k} ) is
A ( cdot 2 sqrt{(14)} )
в. ( 4 sqrt{(14)} )
C ( .16 sqrt{(14)} )
D. ( sqrt{(14)} )
12
671If ( a, b ) are nonzero vectors and ( a ) is
perpendicular to ( b ), then ( a ) has nonzero
vector ( r ) satisfying ( r cdot a=alpha, ) for some
scalar ( boldsymbol{alpha}, boldsymbol{a} times boldsymbol{r}=boldsymbol{b} ) is
A ( cdot frac{alpha a+(a+b)}{|a|^{2}} )
в. ( frac{alpha a+a times b}{|b|^{2}} )
c. ( frac{alpha a-(a times b)}{|a|^{2}} )
D. ( frac{alpha a-(a times b)}{|b|^{2}} )
12
672If the sides of a parallelogram are ( 2 hat{i}+ ) ( 4 hat{j}-5 hat{k} ) and ( hat{i}+2 hat{j}+3 hat{k}, ) then the unit
vector parallel to one of the diagonals, is
A ( cdot frac{1}{7}(3 hat{i}+6 hat{j}-2 hat{k}) )
B ( cdot frac{1}{7}(3 hat{i}-6 hat{j}-2 hat{k}) )
c ( cdot frac{1}{7}(-3 hat{i}+6 hat{j}-2 hat{k}) )
D ( cdot frac{1}{7}(3 hat{i}+6 hat{j}+2 hat{k}) )
12
67348. The resultant of three vectors 1, 2, and 3 units whose
directions are those of the sides of an equilateral triangle
is at an angle of
a. 30° with the first vector
b. 15° with the first vector
c. 100° with the first vector
d. 150° with the first vector
12
674( operatorname{Let} overrightarrow{boldsymbol{A}}=mathbf{2} hat{mathbf{i}}+mathbf{7} hat{boldsymbol{j}}, overrightarrow{boldsymbol{B}}=hat{boldsymbol{i}}+mathbf{2} hat{boldsymbol{j}}+ )
( mathbf{4} hat{boldsymbol{k}}, overrightarrow{boldsymbol{C}}=frac{mathbf{9} hat{mathbf{i}}+mathbf{3 0} hat{mathbf{j}}+mathbf{4} hat{boldsymbol{k}}}{mathbf{5}} )
The ratio in which ( vec{C} ) divides ( overrightarrow{A B} )
internally is?
A . 1: 4
B . 2: 3
c. 3: 2
D. 5: 1
12
675(a) Find the unit vector which is parallel to the vector ( overrightarrow{mathrm{A}}=2 hat{mathrm{i}}+2 hat{mathrm{j}}-2 widehat{mathrm{k}} )
(b) Find the unit vector which is
perpendicular to both of the vector ( overrightarrow{mathrm{A}}=2 hat{mathrm{i}} ) and ( vec{B}=3 hat{mathrm{i}}+4 hat{mathrm{j}}+12 widehat{mathrm{k}} )
12
676Let a, ß, y be distinct real numbers. The points with position
vectors aî + Bị + yk, pî + vj + ak, yî + aj +Bî (1994)
(a) are collinear
(b) form an equilateral triangle
(c) form a scalene triangle
(d) form a right angled triangle
12
677A vector ( a ) is collinear with vector ( b= ) ( left(6,-8,-frac{15}{2}right) ) and make an acute
angle with the positive direction of ( z^{-} ) axis. If ( |a|=50, ) then ( a= )
в. (24,-32,30)
c. (-24,32,30)
D. None of these
12
67886. Let u be a vector coplanar with the vectors
a = 2î +3j-k and 5 = i+k. If ù is perpendicular to a
and ū.6-24, then uſ2 is equal to: [JEE M 2018]
(a) 315 (6) 256 (c) 84 (d) 336
12
679T of a particle is determined by the
9. The position vector of a particle is deu
expression i = 31?î + 472; + 7k
The distance traversed in first 10 sec is
(a) 500 m
(b) 300 m
(c) 150 m
(d) 100 m
12
680The vectors ( hat{boldsymbol{i}}+boldsymbol{2} hat{boldsymbol{j}}+boldsymbol{3} hat{boldsymbol{k}}, boldsymbol{2} hat{boldsymbol{i}}-hat{boldsymbol{j}}+hat{boldsymbol{k}} ) and
( mathbf{3} hat{mathbf{i}}+hat{boldsymbol{j}}+mathbf{4} hat{boldsymbol{k}} ) are so placed that the end
point of one vector is the starting point of the next vector. Then the vectors are :
A. Not coplanar
B. Coplanar but cannot form a triangle
c. coplannar but can form a triangle
D. coplanar and can form a right angled triangle
12
681If ( A, B, C, D ) be any four points and ( E ) and ( F ) be the mid-points of ( A C ) and ( B D ) respectively, then ( overrightarrow{boldsymbol{A B}}+overrightarrow{boldsymbol{C B}}+overrightarrow{boldsymbol{C D}}+ )
( overrightarrow{A D} ) is equal to
A ( .3 overrightarrow{E F} )
B. ( 4 overrightarrow{E F} )
c. ( 4 overrightarrow{F E} )
D. ( 3 overrightarrow{F E} )
12
682If the vectors ( 2 vec{a}-vec{b}+vec{c}, vec{a}+2 vec{b}- )
( overrightarrow{3} c, 3 vec{a}+m vec{b}+5 vec{c} ) are linearly
dependent, then ( boldsymbol{m}= )
( A cdot 2 )
B. –
( c cdot 4 )
D. -4
12
683Find the distance between the pairs of points whose cartesian coordinates are (2,3,-1),(2,6,2)
A. ( 3 sqrt{2} )
B. ( 2 sqrt{3} )
( c cdot 5 sqrt{2} )
D. ( 2 sqrt{5} )
12
6841. If a vector A is parallel to another vector B, then the
resultant of the vector AX B will be equal to
(a) A
(b) A
(c) Zero vector
(d) Zero
12
685Expressing ( vec{a} ) and ( vec{b} ) as linear combinations of ( vec{c} ) and ( overrightarrow{boldsymbol{d}} ) we ( operatorname{get} overrightarrow{boldsymbol{a}}= )
( k vec{c}+m vec{d} ) and ( vec{b}=h vec{c}+r vec{d} )
Find integer part of ( boldsymbol{k}+boldsymbol{m}+boldsymbol{h}+boldsymbol{r} ) ?
( -boldsymbol{a}+boldsymbol{b}=-boldsymbol{c} ; mathbf{2} boldsymbol{a}-boldsymbol{b}=boldsymbol{c}+boldsymbol{d} )
A ( cdot frac{21}{4} )
в. ( frac{27}{8} )
c. ( frac{25}{8} )
D. ( frac{33}{8} )
E ( frac{31}{8} )
F. ( frac{29}{8} )
G. ( frac{23}{8} )
12
686The vectors ( 3 i+j-5 k ) and ( a i+b j- )
( 15 k ) are collinear, if
A ( . a=3, b=1 )
в. ( a=9, b=1 )
c. ( a=3, b=3 )
D. ( a=9, b=3 )
12
68722. Let L, and L, denote the lines
=i+2-1+ 2) + 2h), 2 ER
and 7 = u(2i – + zł), JER
respectively. IfL, is a line which is perpendicular to both L
and L, and cuts both of them, then which of the following
options describe(s) LZ ?
(JEE Adv. 2019)
(a) 7 = 2 (4ì + j + Â) +t(2*+23 – ),ter
(b) = ? (2î– + zł)+t(2î+2– k),te R
(c) 7=t(2î + 2) –Ã)TER
(d) 7 = {(24 + ) ++(2î + 2ğ –),TER
12
688The vector ( boldsymbol{T}+mathbf{2} overline{boldsymbol{y}}+boldsymbol{2} boldsymbol{k} ) restated
through an angle ( theta ) and doubled in
magnitude then it becomes ( 2 T+ )
( (2 x+2)}+(6 x-2) k ) values of ( x ) are
A ( cdot 1, frac{1}{3} )
B. ( -1, frac{1}{3}= )
c. ( 1, frac{-1}{3} )
D. (0,3)
() (0,3)
12
6898.
If vectors a, b, c are coplanar, show that
(19.
a. a. a. 5. a. c. = 0
5. ā. Ī. Ī. . č.
12
690If ( vec{V}_{1}=hat{i}-2 hat{j}+3 hat{k} ) and ( vec{V}_{2}=a hat{i}+b hat{j}+ )
( boldsymbol{c} hat{boldsymbol{k}} ) where ( boldsymbol{a}, boldsymbol{b}, boldsymbol{c} in{-mathbf{1}, boldsymbol{0}, boldsymbol{1}, boldsymbol{2}, boldsymbol{3}} ) then
find the numbers of non-zero vectors ( vec{V}_{2} )
which are normal to ( vec{V}_{1} )
12
691If ( bar{a} ) and ( bar{b} ) are any two non-zero and non-
collinear vectors, then prove that any vector ( bar{r} ) coplanar with ( bar{a} ) and ( bar{b} ) can be
uniquely expressed as ( overline{boldsymbol{r}}=boldsymbol{t}_{1} boldsymbol{a}+boldsymbol{t}_{2} boldsymbol{b} )
where ( t_{1} ) and ( t_{2} ) are scalars.
12
692Find the distance between the points ( A(2,3,1), B(-1,2,3), ) using vector method12
693= Ti + 24j, the vector having the
7. If A=3i +49 and B = 7+24ị, the vector
same magnitude as B and parallel to A IS
(a) 5i + 20ị
(c) 20î +15j (d) 15ỉ + 20ị
(b) 15 +10
12
6944. Given two vectors A = 3î + i + and B =î – j – k . Find
the
a. Area of the triangle whose two sides are represented
by the vectors A and B
b. Area of the parallelogram whose two adjacent sides
are represented by the vectors A and B
c. Area of the parallelogram whose diagonals are
represented by the vectors A and B
hominental Ant around anerson is standing at a point
12
695( P ) is any point on the circumcircle of
( triangle A B C ) other than the vertices. ( H ) is the
orthocenter of ( triangle A B C, M ) is the mid-
point of ( boldsymbol{P H} ) and ( boldsymbol{D} ) is the mid-point of
( B C . ) Then
A. ( A P ) is opposite side of ( D M )
B. ( D M ) is parallel to ( A P )
c. ( D M ) is perpendicular to ( A P )
D. None of these
12
696(2010)
38. Let P, Q, R and S be the points on the plane with position
vectors -2î – j,4î, 3î +3j and _3ỉ +2j respectively. The
quadrilateral PQRS must be a
(a) parallelogram, which is neither a rhombus nor a rectan
(b) square
(c) rectangle, but not a square
(d) rhombus, but not a square
12
69748. Let O be the origin and let PQR be an arbitrary triangle. The
point S is such that
OP.OQ+OR.OS = OR.OP+OQ.OS = OQ.OR + OP.OS
Then the triangle PQR has S as its (JEE Adv. 2017)
(a) Centroid
(b) Circumcentre
(c) Incentre
(d) Orthocenter
12
698Get the area of ( triangle A B C ) for
( boldsymbol{A}(mathbf{1}, mathbf{1}, mathbf{2}), boldsymbol{B}(mathbf{2}, mathbf{3}, mathbf{5}), boldsymbol{C}(mathbf{1}, mathbf{3}, mathbf{4}) ) by use of
vectors.
12
699The points ( P, Q, R ) have regular cartesian coordinates
(1,1,-1),(4,1,2) and (-2,1,2)
respectively. Which of the following is
true for the triangle ( P Q R ? )
A. All the sides have the same length.
в. ( Q R=2 P Q )
c. Angle ( P Q R ) is a right angle.
D. The area of the triangle is 18 square units.
12
700A unit vector which make ( 45^{0} ) with
vector ( 2 i+2 j-k ) and angle of ( 60^{0} ) with
vector ( boldsymbol{j}-boldsymbol{k} ) is
A. ( frac{1}{sqrt{2}} i-frac{1}{sqrt{2}} k )
в. ( frac{1}{sqrt{2}} i-frac{1}{sqrt{2}} j )
c. ( frac{1}{sqrt{2}} i-frac{1}{sqrt{2}} j+frac{1}{sqrt{2}} j )
D. none of the above
12
701Illustration 3.15 Find the unit vector of (A + B) where
Ā=2ỉ – ; +3k and B = 3î – 2j-2k.
12
702If ( overrightarrow{boldsymbol{A}}=boldsymbol{3} hat{boldsymbol{i}}+boldsymbol{2} hat{boldsymbol{j}} ) and ( overrightarrow{boldsymbol{B}}=boldsymbol{2} hat{boldsymbol{i}}+boldsymbol{3} hat{boldsymbol{j}}-hat{boldsymbol{k}} )
then find a unit vector along ( (vec{A}-vec{B}) )
12
7034.
The vector (2î – 2ị +k) is
(199
(a) a unit vecotr
(b) makes an angle with the vector (2ỉ – 4+3£)
(@) parallel to the vector (-i + 3 =1#)
(d) perpendicular to the vector 3î +2j – 2î
12
704Let ( vec{r}=vec{a}+lambda vec{l} ) and ( vec{r}=vec{b}+mu vec{m} ) be two
lines in space where ( overrightarrow{boldsymbol{a}}=mathbf{5} hat{mathbf{i}}+hat{boldsymbol{j}}+mathbf{2} hat{boldsymbol{k}} )
( overrightarrow{boldsymbol{b}}=-hat{boldsymbol{i}}+boldsymbol{7} hat{boldsymbol{j}}+boldsymbol{8} hat{boldsymbol{k}}, overrightarrow{boldsymbol{l}}=-boldsymbol{4} hat{boldsymbol{i}}+boldsymbol{7} hat{boldsymbol{j}}-hat{boldsymbol{k}} )
and ( overrightarrow{boldsymbol{m}}=-2 hat{boldsymbol{i}}-boldsymbol{5} hat{boldsymbol{j}}-boldsymbol{7} hat{boldsymbol{k}} ) then the
position vector of a point which lies on both of these lines, is
( mathbf{A} cdot hat{i}+2 hat{j}+hat{k} )
B. ( 2 hat{i}+hat{j}+hat{k} )
c. ( hat{i}+2 hat{j}+2 hat{k} )
D. Non existent as the lines are skew
12
705If ( overline{A B}=2 vec{i}-3 vec{j}+vec{k}, overline{C B}=vec{i}+vec{j}+vec{k} )
( C D=4 vec{i}-7 vec{j} ) then ( overline{A D}= )
A ( .5 vec{i}+11 vec{j}-vec{k} )
B . ( 5 vec{i}-11 vec{j} )
c. ( 5 vec{i}+11 vec{j} )
D. ( -5 vec{i}+11 vec{j} )
12
706The magnitude of two vectors ( a ) and ( b )
having the same magnitude and such
that the angle between them is ( 60^{circ} ) and their scalar product is ( frac{1}{2}, ) are
A ( cdot|a|=frac{1}{2},|b|=1 )
в. ( |a|=|b|=1 )
c. ( |a|=1,|b|=frac{1}{2} )
D. None of these
12
7077.
Let OA CB be a parallelogram with O at the origin and OC a
diagonal. Let D be the midpoint of OA. Using vector methods
prove that BD and CO intersect in the same ratio. Determine
this ratio.
(1988 – 3 Marks)
12
7089. Unit vectors Ê and ê are inclined at an angle 8. Prove
that f-0 = 2 sin (0/2).
12
709( operatorname{Let} overrightarrow{boldsymbol{a}}=2 hat{hat{boldsymbol{i}}}+boldsymbol{lambda}_{1} hat{boldsymbol{j}}+boldsymbol{3} hat{boldsymbol{k}}, overrightarrow{boldsymbol{b}}=boldsymbol{4} hat{boldsymbol{i}}+(boldsymbol{3}- )
( left.lambda_{2}right) hat{j}+6 hat{k} ) and ( vec{c}=3 hat{i}+6 hat{j}+left(lambda_{3}-1right) hat{k} )
be three vectors such that ( vec{b}=2 vec{a} ) and ( vec{a} )
is perpendicular to ( vec{c} ). Then a possible
value of ( left(lambda_{1}, lambda_{2}, lambda_{3}right) ) is?
( ^{mathbf{A}} cdotleft(frac{1}{2}, 4,-2right) )
в. ( left(-frac{1}{2}, 4,0right) )
c. (1,3,1)
D. (1,5,1)
12
710The adjacent sides of a parallelogram ( operatorname{are} 2 hat{i}+4 hat{j}-5 hat{k} ) and ( hat{i}+2 hat{j}+3 hat{k} ) then
the unit vector parallel to a diagonal is
A ( cdot frac{-bar{i}-2 bar{j}+8 bar{k}}{sqrt{69}} )
в. ( frac{3 bar{i}+6 bar{j}-2 bar{k}}{7} )
c. ( frac{bar{i}+2 bar{j}-8 bar{k}}{sqrt{69}} )
D. All the above
12
711If ( bar{a}-bar{b} ) is perpendicular to ( bar{a},|b|= ) ( sqrt{2}|bar{a}|, ) then
A ( .2 bar{a}+b ) parallel to ( bar{a} )
B . ( (2 bar{a}+bar{b}) ) is perpendicular to ( bar{b} )
C . ( (2 bar{a}-bar{b}) ) is perpendicular to ( bar{b} )
D. ( (2 bar{a}-bar{b}) ) is parallel to ( bar{a} )
12
712If ( overrightarrow{boldsymbol{a}}=boldsymbol{x} hat{boldsymbol{i}}+boldsymbol{2} hat{boldsymbol{j}}-boldsymbol{z} hat{boldsymbol{k}} ) and ( overrightarrow{boldsymbol{b}}=boldsymbol{3} hat{boldsymbol{i}}- )
( y hat{j}+hat{k} ) are two equal vectors, then write
the value of ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z} )
12
713If vectors ( bar{b}, bar{c}, bar{d} ) are not coplanar then
prove that ( (overline{boldsymbol{a}} times overline{boldsymbol{b}}) times(overline{boldsymbol{c}} times overline{boldsymbol{d}})+(overline{boldsymbol{a}} times overline{boldsymbol{c}}) times(overline{boldsymbol{d}} times )
( vec{b})+(bar{a} times bar{d}) times(bar{b} times bar{c}) ) is parallel to ( bar{a} )
12
714( A, B, C ) and ( D ) have position vectors ( vec{a}, vec{b}, vec{c} ) and ( vec{d}, ) respectively, such that ( vec{a}- ) ( vec{b}=2(vec{d}-vec{c}), ) then
A. ( A B ) and ( C D ) bisect each other
B. ( A B ) and ( C D ) trisect each other
c. ( B D ) and ( A C ) bisect each other
D. ( B D ) and ( A C ) trisect each other
12
715If the projection of ( vec{a} ) on ( vec{b} ) and the projection of ( vec{b} ) on ( vec{a} ) are equal then the angle between ( vec{a}+vec{b} ) and ( vec{a}-vec{b} ) is
( A cdot frac{pi}{3} )
B. ( frac{pi}{2} )
( c cdot frac{pi}{4} )
D. ( frac{2 pi}{3} )
12
716Let ( vec{p} ) is the position vector of the
orthocentre ( & vec{g} ) is the position vector of the centroid of the triangle ( A B C ) where circumcentre is the origin. If ( overrightarrow{boldsymbol{p}}=boldsymbol{K} overrightarrow{boldsymbol{g}} )
then ( boldsymbol{K}= )
A . 3
B. 2
c. ( frac{1}{3} )
D. ( frac{2}{3} )
12
717A body is moving under the action of two forces ( vec{F}_{1}=hat{2} i-hat{5} j ; vec{F}_{2}=3 i-4 j . ) Its
velocity will become uniform under a third force ( vec{F}_{3} ) given by
A ( .5 i-9 j )
B. ( -5 i-9 j )
c. ( 5 i+9 j )
D. ( -5 i+9 j )
12
718If ( 0.4 hat{i}+0.7 widehat{j}+c hat{k} ) is a unit vector, then
the value of ( c ) is:
A ( cdot sqrt{0.67} )
the 0.66
B. ( sqrt{0.12} )
c. ( sqrt{1.44} )
D. ( sqrt{0.35} )
12
7198. If | +21=171-72 and V, is finite, then
(a) V, is parallel to V2
(b) 7 =72
(c) V, and V, are mutually perpendicular
(d) 11 =172
12
720The angle subtended by the vector ( overline{boldsymbol{A}}= ) ( 4 hat{i}+3 hat{j}+12 hat{k} ) with the ( x ) -axis is
A ( cdot sin ^{-1}left(frac{3}{13}right) )
B ( cdot sin ^{-1}left(frac{4}{13}right) )
c. ( cos ^{-1}left(frac{4}{13}right) )
D ( cdot cos ^{-1}left(frac{3}{13}right) )
12
7212.20
viecllarliCS I
Illustration 3.39
a. Prove that the vectors – 34-2i+k. B=i-3; +SK,
and C = 2î + -4â form a right-angled triangle.
b. Determine the unit vector parallel to the cross product of
the vectors Ā= 3 – 5j+10 & B = 6î +5j+2.
12
722u.
30 N, tall (574) N0LW
40. The resultant of the three vectors
OA, OB, and OC shown in Fig.
3.79 is
a. r
b.2r
c. r(1+ V2) d. r(12 – 1)
“459
45°
A1-
12
723If unit vectors ( hat{boldsymbol{i}} & hat{boldsymbol{j}} ) are at right angles to each other and ( overrightarrow{boldsymbol{p}}=mathbf{3} hat{boldsymbol{i}}+boldsymbol{4} hat{boldsymbol{j}}, overrightarrow{boldsymbol{q}}= )
( mathbf{5} hat{mathbf{i}}, mathbf{4} overrightarrow{boldsymbol{r}}=overrightarrow{boldsymbol{p}}+overrightarrow{boldsymbol{q}} ) and ( mathbf{2} overrightarrow{boldsymbol{s}}=overrightarrow{boldsymbol{p}}-overrightarrow{boldsymbol{q}} ) then
This question has multiple correct options
A ( cdot|vec{r}+k vec{s}|=|vec{r}-k vec{s}| ) for all real ( mathrm{k} )
B. ( vec{r} ) is perpendicular to ( vec{s} )
c. ( vec{r}+vec{s} ) is perpendicular to ( vec{r}-vec{s} )
D・|vec| = |vec| = | ( vec{p}|=| vec{q} )
12
724Calculate the scalar product of the following vectors ( boldsymbol{a}=mathbf{3} boldsymbol{i}+mathbf{2} boldsymbol{j}+mathbf{2} boldsymbol{k} boldsymbol{a} boldsymbol{n} boldsymbol{d} boldsymbol{b}=mathbf{1} mathbf{8} boldsymbol{i}- )
( 22 j-5 k )
12
725MUL DE PIQUICWU
. If a parallelogram is formed with two sides represented
by vectors a and b, then a + b represents the
a. Major diagonal when the angle between vectors is
acute
b. Minor diagonal when the angle between vectors is
obtuse
c. Both of the above
d. None of the above
12
726The condition for the vectors ( vec{a}, vec{b}, vec{c}, vec{d} ) to be the sides of a parallelogram taken in order is
A . ( vec{a}+vec{b}=vec{c}+vec{d} )
В . ( vec{a}+vec{b}=vec{c}+vec{d}=0 )
c. ( vec{a}+vec{c}=vec{b}+vec{d} )
D . ( vec{a}+vec{c}=vec{b}+vec{d}=0 )
12
72729. If 8, 6, ē are non-coplanar vectors and is a real number,
then the vectors a +26+38, 16+48 and (22. – 1)e are
non coplanar for
120041
(a) no value of
(b) all except one value of a
(c) all except two values of
(d) all values of
12
7288.
+ + + + + +
If ax b = bx c = c x a then a + b + c =
(a) abc (b) -1 (C) O.
[2002]
(d) 2
12
729( vec{r}=vec{x} hat{i}+vec{y} hat{j} ) is the equation of:
A ( . ) yoz plane
B. a straight line joining the points ( vec{i} ) and ( vec{j} )
c. zox plane
D. ( x ) oy plane
12
730The adjacent sides of a parallelogram ( operatorname{are} 2 i+4 bar{j}-5 bar{k} ) and ( i+2 bar{j}+3 bar{k} ) then
the unitvector parallel to a diagonal is
A ( cdot frac{-i+2 j+8 k}{sqrt{69}} )
B. ( frac{3 i+6 bar{j}-2 bar{k}}{7} )
C. ( frac{-i+2 bar{j}-8 bar{k}}{sqrt{69}} )
D. ( frac{-i-2 bar{j}+8 bar{k}}{sqrt{69}} )
12
731With reference to a right handed system of mutually perpendicular unit vectors ( i, j, k, alpha=3 i-j ) and ( beta=2 i+j-3 k . )
( beta=beta_{1}+beta_{2}, ) where ( beta_{1} ) is parallel to ( alpha )
and ( beta_{2} ) is perpendicular to ( alpha, ) then This question has multiple correct options
A ( , beta_{1}=frac{3}{2} i+frac{1}{2} j )
в. ( beta_{1}=frac{3}{2} i-frac{1}{2} j )
c. ( _{beta_{2}}=frac{1}{2} i+frac{3}{2} j-3 k )
D. ( _{beta_{2}}=frac{1}{2} i-frac{3}{2} j-3 k )
12
732If ( vec{x} ) and ( vec{y} ) are two non-collinear vectors and ( A B C ) is a triangle with sides ( a, b, c ) satisfying ( (20 a-15 b) vec{x}+(15 b- )
( 12 c) vec{y}+(12 c-20 a)(vec{x} times vec{y})=overrightarrow{0}, ) then
the triangle ( A B C ) is
A. An acute angle triangle
B. An obtuse angle triangle
c. A right angle triangle
D. An isosceles triangle
12
733If ( a hat{i}+b hat{j}+c hat{k} ) is the position vector of
the mid – point of the vector joining the points ( boldsymbol{P}(mathbf{2} hat{boldsymbol{i}}-boldsymbol{3} hat{boldsymbol{j}}+boldsymbol{4} hat{boldsymbol{k}}) ) and ( boldsymbol{Q}(boldsymbol{4} hat{boldsymbol{i}}+hat{boldsymbol{j}}- )
( 2 hat{k}) ) then ( a+b+c ) is equal to
12
734Find the unit vectors perpendicular to both ( vec{a} ) and ( vec{b}, ) when ( vec{a}=3 hat{i}+hat{j}-2 hat{k} ) and ( overrightarrow{boldsymbol{b}}=mathbf{2} hat{boldsymbol{i}}+boldsymbol{3} hat{boldsymbol{j}}-hat{boldsymbol{k}} )12
73516. Prove, by vector methods or otherwise, that the point of
intersection of the diagonals of a trapezium lies on the line
passing through the mid-points of the parallel sides. (You
may assume that the trapezium is not a parallelogram.)
(1998 -8 Marks)
12
736If ( vec{a} ) and ( vec{b} ) are unit vectors then what is
the angle between ( vec{a} ) and ( vec{b} ) for ( vec{a}-sqrt{2 vec{b}} )
to be a unit vector?
12
737EFGH is a rhombus such the angle
EFG is ( 60^{circ} . ) The magnitude of vectors ( overrightarrow{F H} ) and ( {m overrightarrow{E G}} ) are equal where ( m ) is a scalar. What is the value of ( m ? )
( A cdot 3 )
в. 1.5
( c cdot sqrt{2} )
D. ( sqrt{3} )
12
738The work done by the force ( overrightarrow{boldsymbol{F}}=mathbf{2} hat{mathbf{i}}- )
( hat{j}-hat{mathbf{k}} ) in moung an object along the vector ( 3 hat{i}+2 j-5 hat{k} ) is
A. -9 units
B. 9 units
c. -15 units
D. None of these
12
739The points ( boldsymbol{O}, boldsymbol{A}, boldsymbol{B}, boldsymbol{C} ) are the vertices of
a pyramid and ( P, Q, R, S ) are the mid-
points of ( boldsymbol{O} boldsymbol{A}, boldsymbol{O} boldsymbol{B}, boldsymbol{B} boldsymbol{C}, boldsymbol{A} boldsymbol{C} )
respectively. If ( overrightarrow{boldsymbol{O A}}=boldsymbol{a}, overrightarrow{boldsymbol{O B}}=boldsymbol{b}, overrightarrow{boldsymbol{O C}}= )
( c, ) express in terms of ( a, b, c ) the vectors ( boldsymbol{O P}, overrightarrow{boldsymbol{O Q}}, overrightarrow{boldsymbol{O R}} ) and ( overrightarrow{boldsymbol{O S}} )
A ( cdot overrightarrow{O P}=frac{1}{2} a, overrightarrow{O Q}=frac{1}{2} b, overrightarrow{O R}=frac{1}{2}(b+c), overrightarrow{O S}=frac{1}{2}(a+c) )
B ( cdot overrightarrow{O P}=frac{1}{2} c, overrightarrow{O Q}=b, overrightarrow{O R}=frac{1}{2}(b+c), overrightarrow{O S}=frac{1}{2}(a+c) )
c・ ( overrightarrow{O P}=frac{1}{2} c, overrightarrow{O Q}=b, overrightarrow{O R}=frac{1}{2}(a+c), overrightarrow{O S}=frac{1}{2}(b+c) )
D・ ( overrightarrow{O P}=frac{1}{2} a, overrightarrow{O Q}=b, overrightarrow{O R}=frac{1}{2}(a+c), overrightarrow{O S}=frac{1}{2}(b+c) )
12
740For non-zero vectors a,b,
holds if and only if
(1982 – 2 Marks)
(a) ā.5 = 0, .=0 (b) ] c = 0, c.a=0
(c) č. ā=0 à 5 = 0 (d) ä .] =5.č =č.a=0
12
741If ( vec{a}=hat{i}-2 hat{j}+3 hat{k}, vec{b}=2 hat{i}+3 hat{j}-hat{k} ) and
( overrightarrow{boldsymbol{c}}=boldsymbol{r} hat{boldsymbol{i}}+hat{boldsymbol{j}}+(boldsymbol{2} boldsymbol{r}-boldsymbol{1}) hat{boldsymbol{k}} ) are three
vectors such that ( vec{c} ) is parallel to the plane of ( vec{a} ) and ( vec{b}, ) then ( r ) is equal to?
( mathbf{A} cdot mathbf{0} )
B . 2
c. -1
( D )
12
742If lā=4,1b = 2 and the angle between ã and bis
then (axb)2 is equal to
(a) 48
[2002]
(b)
16
(c)
a
(d) none of these
12
743The non zero vector ( vec{a}, vec{b}, vec{c} ) related by ( vec{a}=8 vec{b} ) and ( vec{c}=-7 vec{b}, ) then angle
between ( vec{a} & vec{c} ) is
A . ( pi )
в. ( frac{pi}{2} )
c. 0
D. ( frac{pi}{4} )
12
7448. If the sum of two unit vectors is a unit vector, then
magnitude of difference is
(a) √
(c) 1/2
(d) √5
(b) 13
logo
12
745If ( a=hat{i}+hat{j}+hat{k}, b=4 hat{i}+3 hat{j}+4 hat{k} ) and
( c=hat{i}+alpha hat{j}+beta hat{k} ) are linearly dependent
vectors and ( |c|=sqrt{3}, ) then the value of
( alpha ) and ( beta ) are respectively
A. ±1,1
в. ±2,1
c. 0,±1
D. None of these
12
746Four point charges ( boldsymbol{q}_{boldsymbol{A}}=boldsymbol{2 mu} boldsymbol{C}, boldsymbol{q}_{B}= )
( mathbf{5} mu C, boldsymbol{q}_{C}=2 mu C, boldsymbol{q}_{D}=5 mu C ) are located
at the four corners of a square ABCD of side ( 10 mathrm{cm} . ) The force on a charge of
( 1 mu C ) placed at the centre of the square
is
A. zero
B. Towards AB
c. Towards BC
D. Towards AD
12
747Prove that ( [overrightarrow{boldsymbol{a}} . overrightarrow{boldsymbol{b}} cdot overrightarrow{boldsymbol{c}}+overrightarrow{boldsymbol{d}}]=[overrightarrow{boldsymbol{a}} cdot overrightarrow{boldsymbol{b}} cdot overrightarrow{boldsymbol{c}}]+ )
( [overrightarrow{boldsymbol{a}} cdot overrightarrow{boldsymbol{b}} cdot overrightarrow{boldsymbol{d}}] )
12
74818.
Let u and y be unit vectors. If w is a vector such that
w+wXu)=v, then prove that (ux v). w< 1/2 and that the
equality holds if and only if u is perpendicular to v.
(1999 – 10 Marks)
12
749If ( vec{b} ) and ( vec{c} ) are two non colinear vectors such that ( vec{a} |(vec{b} vec{c}) ) then prove that ( (vec{a} vec{b}) cdot(vec{a} vec{c}) ) is equal to ( |vec{a}|^{2}(vec{b} cdot vec{c}) )12
750Let ( vec{a} ) and ( vec{b} ) two unit vectors such that ( |vec{a}+vec{b}|=sqrt{3} cdot operatorname{lf} vec{c}=vec{a}+2 vec{b}+ )
( 3(vec{a} times vec{b}), ) then ( 2 vec{c} ) is equal to
A . ( sqrt{55} )
B. ( sqrt{37} )
c. ( sqrt{51} )
D. ( sqrt{43} )
12
751Find the components along the coordinate axes of the position vector of each of the following point:
( S(4,-3) )
12
75230. Choose the wrong statement.
a. Three vectors of different magnitudes may be com-
bined to give zero resultant.
b. Two vectors of different magnitudes can be combined
to give a zero resultant.
c. The product of a scalar and a vector is a vector quan-
tity.
d. All of the above are wrong statements.
12
753If ( a=hat{i}+hat{j}, b=2 hat{j}-hat{k} ) and ( r times a= )
( boldsymbol{b} times boldsymbol{a}, boldsymbol{r} times boldsymbol{b}=boldsymbol{a} times boldsymbol{b}, ) then a unit vector
in the direction of ( r ) is?
A ( cdot frac{1}{sqrt{11}}(hat{i}+3 hat{j}-hat{k}) )
B. ( frac{1}{sqrt{11}}(hat{i}-3 hat{j}+hat{k}) )
c. ( frac{1}{sqrt{3}}(hat{i}+hat{j}+hat{k}) )
D. None of these
12
754The ( x ) -component of the resultant of several vectors :-
(a) Is equal to the sum of the ( x ) components of the vectors
(b) May be smaller than the sum of the magnitudes of the vectors
(c) May be greater than the sum of the
magnitudes of the vectors
(d) May be equal to the sum of the magnitudes of the vectors
( A cdot a, c, d )
B. ( a, b, c )
( c cdot a, b, d )
D. b, c, d
12
75518. Let ā=i-k, b = xi +j + (1 – x)k and
c = yi + xj + (1+ x – y)k. Then ſabe depends on
(20015)
(a) only x
(b) only y
(c) Neither x Nor y
(d) both x and y
12
756( A B C D ) is a quadrilateral, ( E ) is the point of intersection of the line joining the
middle points of the opposite sides. If ( boldsymbol{O} ) is any point, then ( hat{O} A+hat{O} B+hat{O C}+ ) ( hat{O} D= )
A ( cdot 4 O E )
в. 3 О
( mathrm{c} cdot 2 O hat{E} )
D. ( O E )
12
757Evaluate the dot product of the given two vectors ( (3 vec{a}-5 vec{b}) cdot(2 vec{a}-7 vec{b}) )12
758let ABCD is trapezium such that ( overrightarrow{A B}=3 overrightarrow{D E}, ) E divides line segement
AB internally in the ratio 2: 1 and ( F ) is mid point of DC. if position vector of ( A, B ) and ( mathrm{C} ) are ( vec{a}, vec{b} ) and ( vec{c} ) respectively then find the vector ( overrightarrow{boldsymbol{F E}} ).
12
759Illustration 3.17 Determine that vector which when added to
the resultant of Ā= 3 – 5j+7k and B = 2î +49-3ỉ gives
unit vector along y-direction.
12
7602.
If C is the mid point of AB and Pis any point outside AB,
then
[2005]
(a) PA + PB = 2 PC (b) PA + PB = PC
(c) PA + PB + 2 PC = 7 (d) PA + PB + PC = 7
12
761Assertion
If ( vec{a}=5 hat{i}-hat{j}+2 widehat{k}, ) then a vector having magnitude of 8 units along ( vec{a} ) is ( frac{8}{sqrt{30}}(5 hat{i}-widehat{j}+2 widehat{k}) )
Reason
Vector having modulus m along a given
vector ( ^{prime} boldsymbol{a}^{prime} ) is given by, ( boldsymbol{m} times widehat{boldsymbol{a}} )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
76215. Given lÃ,1 = 2, IĀ,l= 3 and A, +Ā, = 3. Find the value
of (Ā, + 2A2). (3Ā, – 4A2).
a. 64 b. 60 c. -60 d. 64
12
763Let ( bar{a}, bar{b}, bar{c} ) be vectors of length 3,4,5 respectively. Let ( bar{a} ) be perpendicular to ( bar{b}+bar{c}, bar{b} t o bar{c}+bar{a} ) and ( bar{c} ) to ( bar{a}+ )
b. ( operatorname{Then}|bar{a}+bar{b}+bar{c}| ) is equals to:
A ( cdot 2 sqrt{5} )
B. ( 2 sqrt{2} )
с. ( 10 sqrt{5} )

D. ( 5 sqrt{2} )
12
764If the position vector ( vec{a} ) of a point ( (12, n) ) is such that ( |vec{a}|=13, ) find the value of
( n )
12
765TEX B=C, then which of the following statements is
wrong
(a) CIA onda
(b) CIB
() ČI(A+B) (d) Č I (AXB)
12
766Express ( -overline{mathbf{i}}-mathbf{3} overline{mathbf{j}}+mathbf{4} overline{mathbf{k}} ) as a linear
combination of ( 2 bar{imath}+bar{jmath}-4 bar{k}, 2 bar{imath}-bar{jmath}+3 bar{k} )
and ( 3 bar{imath}+bar{jmath}-2 bar{k} )
12
767If a line has direction ratios 2,-1,-2 determine its direction cosines.
A ( cdot frac{1}{3},-frac{1}{3},-frac{1}{3} )
B ( cdot_{3,-frac{1}{3},-2} )
c. ( frac{2}{3},-frac{1}{3},-frac{2}{3} )
D. ( 2,-frac{1}{3},-frac{2}{3} )
12
768In a trapezium, the vector ( overline{B C}=lambda overline{A D} )
and ( bar{P}=overline{A C}+overline{B D}=mu overline{A D}, ) then
A. ( mu=lambda+1 )
в. ( lambda=mu+1 )
c. ( lambda+mu=1 )
D. ( mu=2+lambda )
12
7692. What is the angle between A and B , if A and B are the
adjacent sides of a parallelogram drawn from a common
point and the area of the parallelogram is AB/2?
a. 15° b. 30° C . 45° d. 60°
12
770Wurks)
25. If a,b,c and à are distinct vectors such than
āxc=b xã and āxb = cxd. Prove that
(a-d).(5 –c)+ 0 i.e. ā.b + dc #db +ãč
12
771Two vectors ( vec{A} ) and ( vec{B} ) of magnitude 2
units and 1 unit, respectively are directed along the ( x ) -axis and ( y ) axis. Their resultant ( vec{A}+vec{B} ) is directed
along the line:
A. ( y-2 x=0 )
в. ( 2 y-x=0 )
c. ( y+x=0 )
D. ( y-x=0 )
12
77247. The angle between two vectors A and B is . The
resultant of these vectors R makes an angle of 0/2 with
A. Which of the following is true?
a. A = 2B
b. A = B/2
c. A=B
d. AB =1
12
773Vectors ( boldsymbol{a} times(boldsymbol{b} times boldsymbol{c}), boldsymbol{b} times(boldsymbol{c} times boldsymbol{a}) ) and
( boldsymbol{c} times(boldsymbol{a} times boldsymbol{b}) ) are
A. linearly dependent vectors
B. equal vectors
c. parallel vectors
D. none of these
12
774The vectors ( bar{a}, bar{b}, bar{c} ) are of the same length and taken pair wise, they form equal angles. If ( bar{a}=bar{i}+bar{j} ) and ( bar{b}=bar{j}+bar{k} )
then the components of ( bar{c} ) are?
A. (1,0,1)
в. (1,2,3)
c. (-1,1,2)
D. (-1,4,1)
12
77540. If a,b,c are non coplanar vectors and 1 is a real number
then [a(a + b)^27 ac]=[ā +ē 5 ] for [200
(a) exactly one value of a
(b) no value of a
© exactly three values of a
(d) exactly two values of a
12
776Given a parallelogram ( boldsymbol{A B C D} ). If ( |overrightarrow{A B}|=a,|overrightarrow{A D}|=b &|overrightarrow{A C}|=c, ) then
( overrightarrow{D B} cdot overrightarrow{A B} ) has the value
( ^{text {A } cdot} frac{3 a^{2}+b^{2}-c^{2}}{2} )
B. ( frac{a^{2}+3 b^{2}-c^{2}}{2} )
c. ( frac{a^{2}+b^{2}-3 c^{2}}{2} )
D. None
12
777In triangle ( A B C, angle A=30^{circ}, H ) is the
orthocentre and ( D ) is the midpoint of
BC. Segment ( H D ) is produced to ( T )
such that ( boldsymbol{H} boldsymbol{D}=boldsymbol{D} boldsymbol{T} . ) The length ( boldsymbol{A} boldsymbol{T} ) is
equal to
A. ( 2 B C )
в. ( 3 B C )
c. ( frac{4}{3} B C )
D. None of these
12
778Which statement is/are true?
This question has multiple correct options
A. Two non-zero non-collinear vectors are linearly dependent
B. Any two collinear vectors are linearly independent.
C. Two non-zero non-collinear vectors are always linearly independent
D. ( A ) and ( B ) both are true.
12
779A force ( F=3 hat{i}+2 hat{j}+c hat{j} N ) causes a displacement ( vec{r}=c hat{i}+4 hat{j}+c hat{k} m . ) The
work done is ( 36 J . ) Find the value(s) of ( c )
A ( . c=3,7 )
В. ( c=4,-7 )
c. ( c=5,-7 )
D. ( c=6,7 )
12
780Let ( hat{a}, hat{b}, hat{c} ) be unit vectors such that ( hat{a}+ ) ( hat{boldsymbol{b}}+hat{boldsymbol{c}}=hat{mathbf{0}} ) and ( mathbf{x}, y, mathbf{z} ) be distinct integers
then minimum value of ( |boldsymbol{x} hat{boldsymbol{a}}+boldsymbol{y} hat{boldsymbol{b}}+boldsymbol{z} hat{boldsymbol{c}}| )
is
( A cdot 2 )
B. ( sqrt{2} )
( c cdot sqrt{3} )
D.
12
781If ( vec{r} . hat{boldsymbol{i}}=overrightarrow{boldsymbol{r}} cdot hat{boldsymbol{j}}=overrightarrow{boldsymbol{r}} cdot hat{boldsymbol{k}} ) and ( |overrightarrow{boldsymbol{r}}|=boldsymbol{3}, ) then
( vec{r}= )
( mathbf{A} cdot pm 3(hat{i}+hat{j}+hat{k}) )
B. ( pm frac{1}{3}(hat{i}+hat{j}+hat{k}) )
c. ( pm frac{1}{sqrt{3}}(hat{i}+hat{j}+hat{k}) )
D. ( pm sqrt{3}(hat{i}+hat{j}+hat{k}) )
12
78267. Let ā and ū be two unit vectors. If the vectors
c = â+26 and à = 5â – 46 are perpendicular to each other.
then the angle between â and b is:
[2012]
12
783Given ( boldsymbol{a}=boldsymbol{i}+boldsymbol{j}-boldsymbol{k}, boldsymbol{b}=-boldsymbol{i}+boldsymbol{2} boldsymbol{j}+boldsymbol{k} )
and ( c=-i+2 j-k ) a unit vector
perpendicular to both ( a+b ) and ( b+c ) is
( mathbf{A} cdot i )
B.
c. ( k )
D. ( (i+j+k) / sqrt{3} )
12
784If ( bar{u}, bar{v}, bar{w} ) are non-coplanar vectors and p, ( q ) are real numbers, then the equality ( [mathbf{3} overline{boldsymbol{u}} boldsymbol{p} overline{boldsymbol{v}} boldsymbol{p} overline{boldsymbol{w}}]-[boldsymbol{p} overline{boldsymbol{v}} overline{boldsymbol{w}} boldsymbol{q} overline{boldsymbol{u}}]-[mathbf{2} overline{boldsymbol{w}} boldsymbol{q} overline{boldsymbol{v}} boldsymbol{q} overline{boldsymbol{u}}]= )
0 hold for
A. Exactly one value of ( (p, q) )
B. Exactly two value of ( (p, q) )
c. More than two but not all values of ( (p, q) )
D. All value of ( (p, q) )
12
785The projection of ( overrightarrow{boldsymbol{a}}=boldsymbol{2} hat{boldsymbol{i}}+boldsymbol{3} hat{boldsymbol{j}}-boldsymbol{2} hat{boldsymbol{k}} ) on
( overrightarrow{boldsymbol{b}}=hat{boldsymbol{i}}+boldsymbol{2} hat{boldsymbol{j}}+boldsymbol{3} hat{boldsymbol{k}} ) is:
A ( cdot frac{1}{sqrt{14}} )
B. ( frac{2}{sqrt{14}} )
c. ( frac{-1}{sqrt{14}} )
D. ( frac{-2}{sqrt{14}} )
12
786The value of ( hat{y} ). ( hat{z} ) is equal to
A. ( frac{-1}{4} )
B. ( frac{3}{4} )
c. 0
D. ( frac{1}{2} )
12
787ABCD is a parallelogram and ( A C, B D ) are its diagonals Express ( overrightarrow{boldsymbol{A C}} ) and ( overrightarrow{boldsymbol{B D}} ) in terms of ( overrightarrow{boldsymbol{A B}} ) and ( overrightarrow{boldsymbol{A D}} ) only
A ( cdot overrightarrow{A C}=overrightarrow{A B}+overrightarrow{A D} quad ; overrightarrow{B D}=-overrightarrow{A B}+overrightarrow{A D} )
в. ( overrightarrow{A C}=overrightarrow{A B}-overrightarrow{C B} quad ; overrightarrow{B D}=-overrightarrow{A B}+overrightarrow{A D} )
c. ( overrightarrow{A C}=overrightarrow{A B}+overrightarrow{A D} quad ; overrightarrow{B D}=overrightarrow{B C}+overrightarrow{C D} )
D. all of the above
12
78810. Let A = 27 + h , B = 1 +1 +k , and
C = 4i -3; +7k Determine a vector R. Satisfying
Ř x B = © x B and à =O
(1990 – 3 Marks)
12
789ff ( vec{a}=2 hat{i}-hat{j}+2 hat{k}, ) then the value of
( (a times i)^{2}+(a times j)^{2}+(a times k)^{2}= )
( A cdot 2 )
B. 4
( c .6 )
D. 18
12
790If ( bar{a}, bar{b} ) and ( bar{c} ) are non-coplanar vectors and If ( bar{d} ) is such that ( overline{boldsymbol{d}}=frac{mathbf{1}}{boldsymbol{x}}(overline{boldsymbol{a}}+overline{boldsymbol{b}}+overline{boldsymbol{c}}) )
and ( bar{a}=frac{1}{y}(bar{b}+bar{c}+bar{d}) ) where ( x ) and ( y ) are
non-zero real numbers, then ( frac{1}{x y}(bar{a}+ ) ( overrightarrow{boldsymbol{b}}+overline{boldsymbol{c}}+overline{boldsymbol{d}})= )
A . ( 3 bar{c} bar{c} bar{c} )
в. ( -bar{a} )
c. ( overline{0} )
D. ( 2 bar{a} bar{a} bar{a} )
12
791The unit vector along ( 2 i-3 j+k ) is
A. ( frac{2 i-3 j+k}{sqrt{14}} )
в. ( frac{2 i-3 j+k}{sqrt{5}} )
c. ( frac{2 i-3 j+k}{sqrt{15}} )
D. None of these
12
792Express ( overrightarrow{A B} ) in terms of unit vectors ( hat{i} ) and ( hat{j}, ) when the points are:
( A(4,-1), B(1,3) )
Find ( |overrightarrow{A B}| ) in each case.
A ( cdot overrightarrow{A B}=-3 hat{i}-4 hat{j},|overrightarrow{A B}|=5 )
B・方 ( =+3 hat{i}+4 hat{j},|A vec{B}|=5 )
c. ( overrightarrow{A B}=-3 hat{i}+4 hat{j},|overrightarrow{A B}|=5 )
D. none of these
12
793A man travels 1 mile due east, 5 mile
due south, 2 mile due east and finally
9 miles due north. How far is the
starting point?
A. 3 miles
B. 5 miles
c. 4 miles
D. between 5 miles and 9 miles
12
794If ( vec{a}=3 hat{i}-6 hat{i}-hat{k}, vec{b}=-1 hat{i}+4 hat{i}- )
( mathbf{3} hat{boldsymbol{k}}, overrightarrow{boldsymbol{c}}=mathbf{3} hat{boldsymbol{i}}-mathbf{4} hat{boldsymbol{i}}-mathbf{1 2} hat{boldsymbol{k}}, ) then the length
of projection of ( vec{a} times vec{b} ) on ( bar{c} ) is
A ( cdot frac{250}{13} )
B. ( 5 sqrt{2} )
c. ( frac{250}{sqrt{13}} )
D. 14
12
79511.
Let u, v and ū be vectors such that ū + V + i = 0.
|ū l = 3, |ū| = 4 and lüt = 5, then ūý + v.W + Wū is
(1995)
(a) 47 (6) – 25 (c) 0 (d) 25
12
If = T
1
12
796The non-zero vectors ( vec{a}, vec{b} ) and ( vec{c} ) are related by ( vec{a}=8 vec{b} ) and ( vec{c}=-7 vec{b}, ) then the
angle between ( vec{a} ) and ( vec{c} ) is
A ( cdot frac{pi}{2} )
в. ( pi )
( c cdot 0 )
D.
12
797Find the scaler product of vectors ( 3 hat{i}+ ) ( 4 hat{j}-7 hat{k} ) and ( 5 hat{i}+3 hat{j}+4 hat{k} )12
798Find the unit vector perpendicular to each of the vectors ( 6 hat{i}+2 hat{j}+3 hat{k} ) and ( mathbf{3} hat{mathbf{i}}-mathbf{2} hat{boldsymbol{k}} )12
799If the position vector of a point ( boldsymbol{R} ) which divides the line segment joining points ( P(hat{i}+2 hat{j}+hat{k}) ) and ( Q(-hat{i}+hat{j}+hat{k}) ) in the
ratio 2: 1 internally is ( -frac{1}{3} hat{i}+frac{a}{3} hat{j}+hat{k} )
What is the value of ( a ? )
12
800ff ( a ) and ( b ) unit vectors inclined at an
angle ( theta ), then prove that :
( tan frac{boldsymbol{theta}}{mathbf{2}}=frac{|hat{boldsymbol{a}}-hat{boldsymbol{b}}|}{|hat{boldsymbol{a}}+hat{boldsymbol{b}}|} )
12
801Find the components along the coordinate axes of the position vector of each of the following point:
( boldsymbol{P}(boldsymbol{3}, boldsymbol{2}) )
12
802Using vector method, prove that the following points are collinear:
( A(6,-7,-1), B(2,-3,1) ) and
( boldsymbol{C}(mathbf{4},-mathbf{5}, mathbf{0}) )
12
803If ( vec{x} & vec{y} ) are two non collinear vectors
and ( a, b, c ) represent the sides of a
( Delta A B C ) satisfying ( (a-b) vec{x}+ )
( (b-c) vec{y}+(c-a)(vec{x} times vec{y})=0 ) then
( Delta A B C ) is
A. an acute angle triangle
B. an obtuse angle triangle
c. a right angle triangle
D. a scalene triangle
12
804find the coordinate of the tip of the position vector which is equivalent to ( overrightarrow{A B}, ) where the coordinates of ( A ) and ( B ) ( operatorname{are}(-1,3) ) and (-2,1) respectively
( A cdot(+1,+2) )
B . (+1,-2)
c. (-1,+2)
D. (-1,-2)
12
805Le
If û and û are unit vectors and is the acute angle between
them, then 2 û x3 û is a unit vector for
[2007]
(a) no value of
(b) exactly one value of
(c) exactly two values of
(d) more than two values of e
12
806Let ( a, beta, gamma ) be distinct real numbers. The
points with position vectors ai ( +beta j+ ) ( gamma k, beta i+gamma j+a k, gamma i+a j+beta k )
A. are collinear
B. form an equilateral triangle
c. form an scalene triangle
D. form a right angled triangle
12
807If the position vectors of the points ( boldsymbol{A}(mathbf{3}, mathbf{4}), boldsymbol{B}(mathbf{5},-mathbf{6}) ) and ( boldsymbol{C}(mathbf{4},-mathbf{1}) ) are
( vec{a}, vec{b}, vec{c} ) respectively, compute ( vec{a}+2 vec{b}- )
( mathbf{3} overrightarrow{boldsymbol{c}} )
B. ( hat{i}-5 hat{j} )
c. ( hat{i}+5 hat{j} )
D. none of these
12
808If ( boldsymbol{a}=boldsymbol{x} boldsymbol{i}+(boldsymbol{x}-mathbf{1}) boldsymbol{j}+boldsymbol{k} ) and ( boldsymbol{b}=(boldsymbol{x}+ )
1) ( i+j+a k ) always make an acute
angle with each other for every value of
( boldsymbol{x} boldsymbol{epsilon} boldsymbol{R}, ) then
A ( . a epsilon(-infty, 2) )
В. ( a epsilon(2,-infty) )
c. ( a epsilon(-infty, 1) )
D. ( a epsilon(1,-infty) )
12
809If ( p ) and ( q ) are non-collinear unit vectors
and ( |boldsymbol{p}+boldsymbol{q}|=sqrt{mathbf{3}}, ) then ( (mathbf{2} boldsymbol{p}-boldsymbol{3} boldsymbol{q}) )
( (3 p+q) ) is equal to
( mathbf{A} cdot mathbf{0} )
в. ( frac{1}{3} )
( c cdot-frac{1}{3} )
D. ( frac{1}{2} )
( E cdot-frac{1}{2} )
12
810Let ( A ) and ( B ) be points with position vectors ( bar{a} ) and ( bar{b} ) with respect to origin ( O )
If the point ( C ) on ( O A ) is such that ( mathbf{2} overline{boldsymbol{A} boldsymbol{C}}=overline{boldsymbol{C} boldsymbol{O}}, overline{boldsymbol{C} boldsymbol{D}} ) is parallel to ( overline{boldsymbol{O} boldsymbol{B}} ) and
( |overline{C D}|=3|overline{O B}| ) then ( A D ) is
A ( cdot bar{b}-frac{a}{9} )
в. ( 3 bar{b}-frac{a}{3} )
C ( cdot bar{b}-frac{a}{3} )
D. ( bar{b}+frac{a}{3} )
12
81130. Let ū, v, ū be such that ū =1, = 2, W = 3. If the
projection ū along ū is equal to that of w along ū and
ū, ū are perpendicular to each other then
|ū – ū+W equals
[2004]
(a) 14 (b) (c) V14 (d) 2
12
812Find the position vector of the midpoint of the vector joining the points ( P(2,3,4) ) and ( Q(4,1,-2) )12
813( A: ) The angle between ( 2 hat{i}-m hat{j}+3 m hat{k} )
and ( (1+m) hat{i}-2 m hat{j}+hat{k} ) is acute angle
for ( boldsymbol{m} notinleft[-mathbf{2},-frac{mathbf{1}}{mathbf{2}}right] )
( boldsymbol{R}: ) If ( (boldsymbol{a}, boldsymbol{b}) ) is acute, then ( boldsymbol{a} cdot boldsymbol{b}>0 )
A. Both ( A ) and ( R ) are true and ( R ) is the correct explanation of ( A )
B. Both ( A ) and ( R ) are true and ( R ) is not correct explanation of ( A )
c. ( A ) is true but ( R ) is false
D. ( A ) is false but ( R ) is true
12
814Which is a unit vector?
A. ( (cos alpha, 2 sin alpha) )
B. ( (sin alpha, cos alpha) )
c. (1,-1)
D. ( (2 cos alpha, sin alpha) )
12
815Let ( hat{a} ) and ( hat{b} ) be two unit vectors. If the vectors ( vec{c}=hat{a}+2 hat{b} ) and ( vec{d}=5 hat{a}-4 hat{b} ) are
perpendicular to each other, then the angle between ( hat{a} ) and ( hat{b} ) is
A. ( frac{pi}{6} )
в.
( c cdot frac{pi}{3} )
D. ( frac{pi}{4} )
12
816A zero vector has
A. any direction
B. no direction
C . many direction
D. None of these
12
817Which of the following is correct?
A ( cdot vec{a} cdot(vec{b}+vec{c}) ) is equal to ( vec{a} cdot vec{b}+vec{d} cdot vec{c} )
В . ( vec{a} .(vec{b}+vec{c}) ) is equal to ( vec{a} cdot vec{b}-vec{d} . vec{c} ).
c. ( vec{a} times(vec{b}+vec{c}) ) is equal to ( vec{a} cdot vec{b}+vec{a} cdot vec{c} )
D. All of the above
12
818If ( bar{a} ) is collinear with ( bar{b}=3 bar{i}+6 bar{j}+6 bar{k} )
and ( bar{a} cdot bar{b}=27 . ) Then ( bar{a} ) is equal to?
A ( .3(bar{i}+bar{j}+bar{k}) )
B . ( bar{i}+3 bar{j}+3 bar{k} )
c. ( bar{i}+2 bar{j}+2 bar{k} )
D. ( 2 bar{i}+2 bar{j}+2 bar{k} bar{k}+2+2+2+2+2 bar{k}+2 bar{i}+2 bar{i} bar{i} )
12
819The dot products of a vector with the vectors ( (hat{i}+hat{j}-3 hat{k}),(hat{i}+3 hat{j}-2 hat{k}) ) and
( (2 hat{i}+hat{j}+4 hat{k}) ) are 0,5,8 respectively
Find the vector.
12
82034. Let two non-collinear unit vectors â and ŷ form an acute
angle. A point P moves so that at any time t the position
vector OP (where 0 is the origin) is given by
â cost +b sint. When Pis farthest from origin 0, let Mbe
the length of OP and û be the unit vector along OP.
Then,
(2008)
(a) û=4 to and M = (1+â. 6)1/2
âtb
a lãtbl
à-
Tâ-6 ana
and M = (1+â.)1/2
ât
ia and M =(1+2â: )1/2
â
=
and M = (1+2â: ) 1/2
12
821y
44. ABCDEF is a regular hexagon with point O as center. The
value of AB + AC + AD + AE + AF is
a. 220 b. 4A0 c. 6ÃO d. O
12
822Show that the vectors ( 2 hat{i}-3 hat{j}+4 hat{k} ) and ( -4 hat{i}-6 hat{j}+-8 hat{k} ) are collinear12
823Four forces of magnitude ( P, 2 P, 3 P ) and ( 4 P ) act along the four sides of a square ABCD in cyclic order. Use the vector method to find the resultant force.
A ( cdot 2 sqrt{2} P )
В. ( 3 sqrt{2} P )
c. ( sqrt{2} P )
D. ( -2 sqrt{2} P )
12
824If ( (x, y, z) neq(0,0,0) ) and ( (hat{i}+hat{j}+ )
( mathbf{3} hat{boldsymbol{k}}) boldsymbol{x}+(mathbf{3} hat{boldsymbol{i}}-mathbf{3} hat{boldsymbol{j}}+hat{boldsymbol{k}}) boldsymbol{y}+(-mathbf{4} hat{mathbf{i}}+ )
( mathbf{5} hat{boldsymbol{j}}) z=boldsymbol{lambda}(boldsymbol{x} hat{boldsymbol{i}}+boldsymbol{y} hat{boldsymbol{j}}+boldsymbol{z} hat{boldsymbol{k}}), ) then the value
of ( lambda ) will be
A ( .-2,0 )
B. 0,-2
c. -1,0
D. 0,-1
12
825( (a . i) i+(a . j) j+(a . k) k= )
( A cdot 0 )
B. ( a )
( c .3 a )
D. None of these
12
826If the vectors ( 3 vec{p}+vec{q} ; 5 vec{p}-3 vec{q} ) and ( 2 vec{p}+ )
( vec{q} ; 4 vec{p}-2 vec{q} ) are pairs of mutually perpendicular vectors.
12
82745. If à and ✓ are vectors such that a+b = 29 and
ax(2î +3ſ + 4k) = (2î +39 + 4k)<b, then a possible
value of (a+b). (-7ỉ +29 + 3k) is
(a) 0 (6) 3 (c) 4 (d) 8
(2012)
12
828Match the following12
829Three vectors of magnitudes ( a, 2 a, 3 a ) meeting a point and three directions are along the diagonals of three adjacent faces of a cube. The magnitude of their resultant is
( mathbf{A} cdot 3 a )
B. ( 5 a )
( c cdot 2 a )
D. ( 4 a )
12
830If ( vec{a}=hat{i}+2 hat{j}+hat{k}, vec{b}=2 hat{i}+hat{j} ) and ( vec{c}= )
( mathbf{3} hat{mathbf{i}}-mathbf{4} hat{mathbf{j}}-mathbf{5} hat{k}, ) then find a unit vector
perpendicular to both of the vectors ( (vec{a}-vec{b}) ) and ( (vec{c}-vec{b}) )
12
831If vectors ( i+2 j+2 k ) is rotated through
an angle of ( 90^{circ} ) so as to cross positive
direction of ( y ) -axis, then the vector in the new positive is?
A ( cdot-frac{2}{sqrt{5}} i+sqrt{5} j-frac{4}{sqrt{5}} k )
B. ( frac{2}{sqrt{5}} i-sqrt{5} j+frac{4}{sqrt{5}} k )
c. ( 4 i-j-k )
D. None of these
12
832If ( vec{a}, vec{b}, vec{c} ) are position vectors of the vertices ( A, B, C ) of a triangle ( A B C )
show that the area of the triangle ( A B C ) is ( frac{1}{2}[vec{a} times vec{b}+vec{b} times vec{c}+vec{c} times vec{a}] . ) Also deduc
the condition for collinearity of the
points ( A, B ) and ( C )
12
833Find a unit vector perpendicular to the plane ( A B C, ) where the coordinates of
( A, B ) and ( C ) is
( boldsymbol{A}(mathbf{3},-mathbf{1}, mathbf{2}), boldsymbol{B}(mathbf{1},-mathbf{1},-mathbf{3}) ) and
( boldsymbol{C}(mathbf{4},-mathbf{3}, mathbf{1}) )
12
834If ( vec{a}=i+j-k, vec{b}=1-j+k, vec{c} ) is a
unit vector such that ( vec{c} . vec{a}=0,[vec{c} vec{a} vec{b}]=0 )
then a unit vectors perpendicular to both ( vec{a} ) and ( vec{c} ) is
A ( cdot frac{1}{sqrt{6}}(2 i-j+k) )
B. ( frac{1}{sqrt{2}}(j+k) )
c. ( frac{1}{sqrt{2}}(i+j) )
D. ( frac{1}{sqrt{2}}(i+k) )
12
835The magnitude of a position vector in a XY plane is ( 4 . ) Its slope is ( frac{1}{sqrt{3}}, ) then the position vector is:
A. ( sqrt{3 i}+3 )
B . ( 2 sqrt{3} hat{i}-2 hat{j} )
c. ( 2 sqrt{3 i}+2 hat{j} )
D. ( 2 hat{i}+2 sqrt{3} hat{j} )
12
836Consider ( Delta A B C ) with ( A equiv(vec{a}), B equiv(vec{b}) )
and ( C=(vec{c}), ) ff ( vec{b} .(vec{a}+vec{c})=vec{b} cdot vec{b}+ )
( vec{a} cdot vec{c} ;|vec{b}-vec{a}|=3 ;|vec{c}-vec{b}|=4 ) then the
angle between the medians ( overline{A M} ) and
( B D ) is
A. ( pi-cos ^{-1}left(frac{1}{5 sqrt{13}}right) )
в. ( pi-cos ^{-1}left(frac{1}{13 sqrt{5}}right) )
c. ( cos ^{-1}left(frac{1}{5 sqrt{13}}right) )
D. ( cos ^{-1}left(frac{1}{13 sqrt{5}}right) )
12
837Find the angle between the vectors ( vec{a} ) and ( vec{b} ) with magnitude ( sqrt{3} ) and 2 respectively and ( overrightarrow{boldsymbol{a}} cdot overrightarrow{boldsymbol{b}}=sqrt{mathbf{6}} )12
838Find the distance between the points whose position vectors are given as follows ( -2 hat{i}+3 hat{j}+5 hat{k}, 7 hat{i}-hat{k} )
A. ( 3 sqrt{14} )
(i) 144
B. ( sqrt{54} )
c. ( 3 sqrt{19} )
D. ( sqrt{57} )
12
839Find the angle between ( 3 vec{i}+4 vec{j}, 2 vec{j}-5 vec{k} )
( ^{mathrm{A}} cdot cos ^{-1} frac{8}{5 sqrt{29}} )
B. ( sin ^{-1} frac{8}{5 sqrt{29}} )
c. ( cos ^{-1} frac{-1}{5 sqrt{29}} )
D. ( sin ^{-1} frac{-1}{5 sqrt{29}} )
12
840The unit vector orthogonal to ( -hat{mathbf{i}}+ ) ( hat{k}, 2 hat{j}-hat{k} ) and forming a right handed
system with them is
( mathbf{A} cdot 2 hat{i}+hat{j}+2 hat{k} )
B. ( frac{2 hat{i}+hat{j}+2 hat{k}}{3} )
c. ( -frac{2 hat{i}+hat{j}+2 hat{k}}{3} )
D. ( -frac{2 hat{i}+hat{j}+2 hat{k}}{9} )
12
841Given 3 vectors: ( vec{V}_{1}=a hat{i}+b hat{j}+c hat{k} )
( vec{V}_{2}=b hat{i}+c hat{j}+a hat{k} ) and ( vec{V}_{3}=c hat{i}+a hat{j}+ )
b ( hat{k} . ) In which of the following conditions
( vec{V}_{1}, vec{V}_{2} ) and ( vec{V}_{3} ) are linearly independent?
A ( cdot a+b+c=0 ) and ( a^{2}+b^{2}+c^{2} neq a b+b c+c a )
B . ( a+b+c=0 ) and ( a^{2}+b^{2}+c^{2}=a b+b c+c a )
C ( cdot a+b+c neq 0 ) and ( a^{2}+b^{2}+c^{2}=a b+b c+c a )
D . ( a+b+c neq 0 ) and ( a^{2}+b^{2}+c^{2} neq a b+b c+c a )
12
842If ( vec{a} cdot hat{i}=vec{a} cdot(hat{i}+hat{j})=vec{a}(hat{i}+hat{j}+hat{k}), ) thus
( vec{a}= )
( A )
B . ( hat{i}+j )
c. ( hat{k}-hat{j} )
D. ( hat{i}+hat{j}+hat{k} )
12
843If the position vectors ofthe vertices ( A ) B, C of a triangle ABC are ( 7 widehat{j}+ ) ( mathbf{1 0} widehat{boldsymbol{k}},-hat{mathbf{i}}+mathbf{6} widehat{mathbf{j}}+mathbf{6} widehat{boldsymbol{k}} ) and ( -mathbf{4} hat{mathbf{i}}+mathbf{9} hat{mathbf{j}}+mathbf{6} widehat{boldsymbol{k}} )
respectively, the triangle is
A . equilateral
B. isosceles
c. scalene
D. right angled and isoscele
12
844Find the unit vectors along ( vec{A} ) and ( vec{B} ) where ( vec{A}=3 hat{i}-2 hat{j}+4 hat{k} ) and ( vec{B}=4 hat{i}- )12
8456. If for two vector A and B , sum (A+B) is perpendicular
to the difference (A-B). The ratio of their magnitude is
(a) 1
(b) 2
(c) 3
(d) None of these
het is
12
846If ( overline{boldsymbol{a}}=mathbf{2} overline{mathbf{i}}-mathbf{3} overline{mathbf{j}}+overline{mathbf{k}}, overline{boldsymbol{b}}=_{-} overline{mathbf{i}}+overline{mathbf{k}}, overline{boldsymbol{c}}= )
( 2 bar{j}-bar{k}, ) then the area of the parallelogram is having diagonals ( bar{a}+bar{b} ) and ( bar{b}+bar{c}(text { in square units }) ) is
A ( cdot sqrt{21} )
B. ( frac{sqrt{21}}{2} )
c. ( sqrt{19} )
D. ( frac{sqrt{19}}{2} )
12
847For what value of ( lambda ), the vector ( i-lambda j+ )
( 2 k ) and ( 8 i+6 j-k ) are at right angles?
12
848Let ( a=2 i-j+k, b=i+2 j-k ) and
( c=i+j-2 k ) be three vectors. A vector
in the plane of ( b ) and ( c ) whose projection on ( a ) is of magnitude ( sqrt{frac{2}{3}} ) is This question has multiple correct options
A. ( 2 i+3 j-3 k )
B. ( 2 i+3 j+3 k )
c. ( 2 i-j+5 k )
( mathbf{D} cdot 2 i+j+5 k )
12
849The position vectors of ( boldsymbol{P} ) and ( boldsymbol{Q} ) are
respectively ( a ) and ( b ). If ( R ) is a point on
( P Q, P Q ) such that ( P R=5 P Q, ) then the
position vector of ( boldsymbol{R} ) is
A ( .5 b-4 a )
B. ( 5 b+4 a )
c. ( 4 b-5 a )
D. ( 4 b+5 a )
12
850Find a vector perpendicular to each of the vectors ( vec{a}+vec{b} ) and ( vec{a}-vec{b}, ) where ( vec{a}= ) ( hat{boldsymbol{i}}+hat{boldsymbol{j}}+hat{boldsymbol{k}}, overrightarrow{boldsymbol{b}}=hat{boldsymbol{i}}+boldsymbol{2} hat{boldsymbol{j}}+boldsymbol{3} hat{boldsymbol{k}} )
A ( cdot frac{hat{i}}{i}-hat{2} hat{j}+hat{2} k )
B.
c. ( -hat{i}-hat{3} j+hat{2 k} )
D. None of these
12
851What are coinitial vectors.?
A. Two or more vectors having the same magnitude are called coinitial vectors.
B. Two or more vectors are said to be coinitial if they are parallel to the same line, irrespective of their magnitudes and directions.
C. Two or more vectors having the same initial point are called coinitial vectors.
D. None of the above
12
852Define Rectangular components of
vector
12
853( A B C D ) a parallelogram, ( A_{1} ) and ( B_{1} ) are the midpoints of sides ( B C ) and ( C D ) respectively. If ( boldsymbol{A} boldsymbol{A}_{1}+boldsymbol{A} boldsymbol{B}_{1}=boldsymbol{lambda} boldsymbol{A} boldsymbol{C} )
then ( lambda ) is equal to
A ( cdot frac{1}{2} )
B. 1
( c cdot frac{3}{2} )
D. 2
12
854Assertion
Consider the points ( A, B ) and ( C ) The vector sum, ( overrightarrow{boldsymbol{A B}}+overrightarrow{boldsymbol{B C}}+overrightarrow{boldsymbol{C A}}=overrightarrow{mathbf{0}} )
Reason
( A, B ) and ( C ) form the vertices of a
triangle.
A. Assertion and Reason are correct and the Reason is the correct explanation for the Assertion
B. Assertion and Reason are correct but the Reason is not the correct explanation for the Assertion
c. Assertion is correct while the Reason is incorrect
D. Assertion is incorrect while the Reason is correct
12
855The angle between the vectors ( overrightarrow{boldsymbol{a}}=hat{boldsymbol{i}}+ ) ( hat{boldsymbol{j}}+hat{boldsymbol{k}} ) and ( overrightarrow{boldsymbol{b}}=hat{boldsymbol{i}}+boldsymbol{2} hat{boldsymbol{j}}+hat{boldsymbol{k}} ) is?
A ( cdot tan ^{-1} frac{1}{2 sqrt{2}} )
в. ( tan ^{-1} frac{1}{3} )
( ^{mathbf{c}} cdot tan ^{-1}left(frac{1}{2}right) )
D. ( 90^{circ} )
12
856A unit vector ( boldsymbol{d} ) is equally inclined at an angle ( alpha ) with the vectors ( a=cos theta . i+ )
( sin theta . j, b=-sin theta . i+cos =theta . j ) and
( c=k . ) Then ( alpha ) is equal to
A ( cdot cos ^{-1}left(frac{1}{sqrt{2}}right) )
B. ( cos ^{-1}left(frac{1}{sqrt{3}}right) )
c. ( cos ^{-1} frac{1}{3} )
D.
12
857If the vectors ( vec{a}, vec{b} ) and ( vec{c} ) are represented by the sides ( B C, C A ) and ( A B, ) of a triangle respectively, then
( mathbf{A} cdot vec{a} cdot vec{b}+vec{b} cdot vec{c}+vec{c} cdot vec{a}=0 )
В . ( vec{a} times vec{b}=vec{b} times vec{c}=vec{c} times vec{a} )
c. ( vec{a} cdot vec{b}=vec{b} cdot vec{c}=vec{c} cdot vec{a} )
D. ( vec{a} times vec{b}+vec{b} times vec{c}+vec{c} times vec{a}=0 )
12
858A vector ( vec{r} ) is equally inclined with the
coordinate axes. If the tip of ( vec{r} ) is in the
positive octant and ( |boldsymbol{r}|=mathbf{6}, ) then ( overrightarrow{boldsymbol{r}} ) is
A ( cdot 2 sqrt{3}(hat{i}-hat{j}+widehat{k}) )
B ( cdot 2 sqrt{3}(-hat{i}+widehat{j}+widehat{k}) )
c. ( 2 sqrt{3}(hat{i}+widehat{j}-widehat{k}) )
D ( cdot 2 sqrt{3}(hat{i}+hat{j}+widehat{k}) )
12
859If ( vec{a} ) and ( vec{b} ) are mutually perpendicular unit vectors, then ( (3 vec{a}+2 vec{b}) cdot(5 vec{a}- )
( mathbf{6} overrightarrow{boldsymbol{b}})=? )
( mathbf{A} cdot mathbf{5} )
B. 3
( c cdot 6 )
D. 12
12
860Given that ( vec{A} times vec{B}=vec{B} times vec{C}=overrightarrow{0} ) if ( overrightarrow{A B C} )
are not null vectors, Find the value of ( vec{A} times vec{C} )
A ( . vec{A} times vec{B} )
B. ( overrightarrow{0} )
c. ( vec{c} times vec{B} )
D. ( vec{C} times vec{A} )
12
861Calculate the scalar product of the following vectors. Find the vector c being given that it is perpendicular to the vectors ( boldsymbol{a}= )
( {2,3,-1}, b={1,-2,3} ) and
satisfies the condition
( boldsymbol{c} cdot{boldsymbol{2} boldsymbol{i}-boldsymbol{j}+boldsymbol{k}}=-boldsymbol{6} )
12
862The cosine of the angle ( A ) of the triangle with vertices
( boldsymbol{A}(mathbf{1},-mathbf{1}, mathbf{2}), boldsymbol{B}(mathbf{6}, mathbf{1 1}, mathbf{2}), boldsymbol{C}(mathbf{1}, mathbf{2}, boldsymbol{6}) ) is
( ^{text {A }} cdot frac{63}{65} )
в. ( frac{36}{65} )
c. ( frac{16}{65} )
D. ( frac{13}{64} )
12
863If ( overrightarrow{boldsymbol{a}}+overrightarrow{boldsymbol{b}}+overrightarrow{boldsymbol{c}}=mathbf{0},|overrightarrow{boldsymbol{a}}|=mathbf{3},|overrightarrow{boldsymbol{b}}|=mathbf{5},|overrightarrow{boldsymbol{c}}|=mathbf{7} )
then the angle between ( vec{a} ) and ( vec{b} ) is
A ( . pi / 6 )
в. ( 2 pi / 3 )
c. ( 5 pi / 3 )
D . ( pi / 3 )
12
864A vector ( vec{B} ) which has magnitude 8 is added to a vector ( vec{A} ) which lies along the
x-axis. the sum of their two vectors is a
third vector which lies along the y-axis and has a magnitude that is twice the magnitude of ( vec{A} ). Find the magnitude of ( boldsymbol{A} )
12
865Find the position vector of a point ( boldsymbol{R} )
which divides the line joining two
points ( P ) and ( Q ) whose position vectors ( operatorname{are} hat{boldsymbol{i}}+boldsymbol{2} hat{boldsymbol{j}}-hat{boldsymbol{k}} ) and ( -hat{boldsymbol{i}}+hat{boldsymbol{j}}+hat{boldsymbol{k}} )
respectively, in the ratio 2: 1
(i) internally
(ii) externally
12
866Let ( P, Q, R ) and ( S ) be the points on the plane with position vectors ( (-2 i-j), 4 i ) ( (3 i+3 j) ) and ( (-3 i+2 j) ) respectively. The quadilateral PQRS must be a
A. Parallelogram, which is neither a rhombus nor a rectangle
B. Square
c. Rectangle, but not a square
D. Rhombus, but not a square
12
867A unit radial vector ( hat{r} ) makes angles of
( boldsymbol{alpha}=mathbf{3 0}^{o} ) relative to the ( boldsymbol{x} ) -axis, ( boldsymbol{beta}=mathbf{6 0}^{circ} )
relative to the ( y ) -axis, and ( gamma=90^{circ} )
relative to the ( z- ) axis. The vector ( hat{r} ) can
be written as:
A ( cdot frac{1}{2} hat{i}+frac{sqrt{3}}{2} hat{j} )
B. ( frac{sqrt{3}}{2} hat{i}+frac{1}{2} hat{j} )
c. ( frac{sqrt{2}}{3} hat{i}+frac{1}{sqrt{3}} hat{j} )
D. none of these
12
868Find the sine of the angle between the vectors ( hat{i}+2 hat{j}+2 hat{k} ) and ( 3 hat{i}+2 hat{j}+6 hat{k} )12
869A vector ( vec{r} ) is inclined at equal angles to
( boldsymbol{O} boldsymbol{X}, boldsymbol{O} boldsymbol{Y} ) and ( boldsymbol{O} boldsymbol{Z} . ) If the magnitude of ( boldsymbol{r} )
is 6 units, then ( vec{r} ) is equal to This question has multiple correct options
A ( cdot sqrt{3}(hat{i}+widehat{j}+widehat{k}) )
B. ( -sqrt{3}(hat{i}+hat{j}+widehat{k}) )
c ( cdot 2 sqrt{3}(hat{i}+hat{j}+widehat{k}) )
D. ( -2 sqrt{3}(hat{i}+hat{j}+widehat{k}) )
12
870Find the angle between the vectors ( hat{i}- ) ( 2 hat{j}+3 hat{k} ) and ( 3 hat{i}-2 hat{j}+hat{k} )12
871Find the sin angle between the vectors ( A=3 i-4 j+5 k ) and ( B=i-j+k ? )12
872If ( vec{A}=vec{B}+vec{C} ) and the magnitude of
( A, B ) and ( C ) are 5,4 and 3 minutes
respectively,then the angle between ( boldsymbol{A} ) and ( C ) is:
A ( cdot cos ^{-1}(4 / 5) )
B ( cdot cos ^{-1}(3 / 5) )
c. ( tan ^{-1}(3 / 4) )
D ( cdot sin ^{-1}(3 / 5) )
12
873( |vec{a}+vec{b}|^{2}+|vec{a}-vec{b}|^{2} ) is equal to ( 2left(|vec{a}|^{2}right. )
( left.|vec{b}|^{2}rightrangle )
If true enter 1 else enter 0
12
874Write the associative law for addition of
vectors
12
875Find out the resultant of the two vectors
one of magnitude 3 directed towards the east and another one of magnitude
4 directed towards the north.
A. 5 northeast
B. 7 northeast
c. 2 southwest
D. 1 northeast
E. 12 northeast.
12
876If ( 2 vec{a}+3 vec{b}+4 vec{c}=0 Rightarrow vec{a} times vec{b}+vec{b} times )
( vec{c}+vec{c} times vec{a}= )
( A )
B ( cdot 3 vec{a} times vec{b} )
c. ( 3 vec{b} times vec{c} )
D. ( 3 vec{c} times vec{a} )
12
877The vectors ( overline{A B}=overline{3 i}-overline{2 j}+overline{2 k} ) and
( B C=-bar{i}-overline{2 k} ) are the adjacent sides
of a parallelogram. The angle between its diagonals is
A ( cdot frac{pi}{2} )
В ( cdot frac{pi}{3} ) or ( frac{2 pi}{3} )
c. ( frac{3 pi}{4} ) or ( frac{pi}{4} )
D. ( frac{5 pi}{6} ) or ( frac{pi}{6} )
12
878If ( I ) is the center of a circle inscribed in
a triangle ( A B C, ) then ( |B C| I A+ ) ( |boldsymbol{C A}| boldsymbol{I B}+|boldsymbol{A B}| boldsymbol{I C} )
( mathbf{A} cdot mathbf{0} )
B. ( I A+I B+I C )
c. ( frac{I A+I B+I C}{3} )
D. ( frac{I A+I B+I C}{2} )
12
8795. The angle between the vectors A and B is e. The value
of the triple product Ā (B x A) is
(a) A2B
(b) Zero
(c) A2B sin e
(d) A2B cos e
12
880If ( vec{a}=x hat{i}-y hat{j} ) and ( vec{b}=y hat{i}+x hat{j}, mid vec{a} times )
( vec{b} mid=5 ) then locus of ( (x, y) ) is
A. Hyperbola
B. Parabola
c. Ellipse
D. Circle
12
881Match the following:
( begin{array}{ll}text { List – I } & text { List – II } \ text { a) Angular velocity } & text { e) null vector } \ text { b) Power } & text { f) Scalar } \ text { c) If } overrightarrow{mathrm{A}}+overrightarrow{mathrm{B}}=overrightarrow{mathrm{A}}-overrightarrow{mathrm{B}} text { then } overrightarrow{mathrm{B}} text { is } & text { g) unit vector } \ mathrm{d}) frac{vec{A}}{|vec{A}|} & text { h) axial vector }end{array} )
( mathbf{A} cdot mathbf{a} rightarrow mathbf{f} ; mathbf{b} rightarrow mathbf{h} ; mathbf{c} rightarrow mathbf{e} ; mathrm{d} rightarrow mathbf{f} )
B ( cdot a rightarrow h ; b rightarrow f ; c rightarrow g ; d rightarrow e )
( mathbf{C} cdot mathbf{a} rightarrow mathbf{g} ; mathbf{b} rightarrow mathbf{e} ; mathbf{c} rightarrow mathbf{h} ; mathbf{d} rightarrow mathbf{f} )
( mathbf{D} cdot mathbf{a} rightarrow mathbf{h} ; mathbf{b} rightarrow mathbf{f} ; mathbf{c} rightarrow mathbf{e} ; mathbf{d} rightarrow mathbf{g} )
12
8829. Two vector A and B have equal magnitudes. Then the vector
A + B is perpendicular to
(a) AXB
(b) A – B
(c) 3A – 3B
(d) All of these
12
883OABC is a tetrahedron express the vectors ( overrightarrow{B C}, overrightarrow{C A}, ) and ( overrightarrow{A B} ) in terms of the vectors ( overrightarrow{boldsymbol{O A}}, overrightarrow{boldsymbol{O B}} ) and ( overrightarrow{boldsymbol{O C}} )
A ( cdot overrightarrow{B C}=overrightarrow{O C}+overrightarrow{O B} quad ; overrightarrow{C A}=overrightarrow{O A}+overrightarrow{O C} quad ; overrightarrow{A B}=overrightarrow{O B}+ )
( overrightarrow{O A} )
в . ( overrightarrow{B C}=overrightarrow{O C}-overrightarrow{O B} quad ; overrightarrow{C A}=overrightarrow{O A}-overrightarrow{O C} quad ; overrightarrow{A B}=overrightarrow{O B} )
( overrightarrow{O A} )
c. ( overrightarrow{B C}=overrightarrow{O C}-overrightarrow{B O} quad ; overrightarrow{C A}=overrightarrow{O A}-overrightarrow{C O} quad ; overrightarrow{A B}=overrightarrow{O B} )
( overrightarrow{A O} )
D. none of these
12
884If two unit vectors ( bar{a} ) and ( bar{b} ) are inclined
at an angle ( 2 theta(0<theta<pi) ) and ( mid bar{a}- )
( vec{b} mid leq 1, ) then ( theta in )
A ( cdotleft[0, frac{pi}{6}right) )
в. ( left[0, frac{pi}{3}right] cupleft[frac{2 pi}{3}, piright] )
( ^{mathrm{c}} cdotleft[0, frac{pi}{6}right] cupleft[frac{5 pi}{6}, piright] )
D.
12
885If ( hat{a}, hat{b} ) and ( hat{c} ) are unit vectors satisfying ( |hat{boldsymbol{a}}-hat{boldsymbol{b}}|^{2}+|hat{boldsymbol{b}}-hat{boldsymbol{c}}|^{2}+|hat{boldsymbol{c}}-hat{boldsymbol{a}}|^{2}=mathbf{9}, ) then
( |2 hat{a}+5 hat{b}+5 hat{c}| ) is
( A cdot 3 )
B. 4
c. 5
D. 6
12
886If ( boldsymbol{O P}=2 hat{boldsymbol{i}}+boldsymbol{3} hat{boldsymbol{j}}-hat{boldsymbol{k}} ) and ( boldsymbol{O} boldsymbol{Q}=boldsymbol{3} hat{boldsymbol{i}}- )
( 4 hat{j}-2 hat{k} ) then the modulus ( bar{P} Q ) is
A ( cdot sqrt{13} )
B. ( sqrt{51} )
c. ( sqrt{39} )
D. ( sqrt{67} )
12
8874. The angle between the two vectors Ā=3i +4ị+ 5k and
B = 3 +4j – 5k will be
(a) 90° (b) 0° (c) 60° (d) 45°
12
88828.
If a, 6, 7 are three non-zero, non-coplanar vectors and
b. a



b. a

>
a, b
= b + bia →
la 12
lap

→ boc →
at
br. c by,
→ → coa
C2 = C-
lak
lop
16 2
C4 =
C
a
=-
c. a → b.c→
C3 = C- at hi,
Ich ich
then the set of orthogonal vectors is
Ich
(2005)
(a) (a, b, c3)
(b) (a, ,c2)
(c) (a,b,c)
(a) (a, b, cz)
12
889Two forces act at the vertex ( mathbf{A} ) of
quadrilateral ABCD represented by ( A B, overline{A D} ) and two at ( C ) represented by ( C D ) and ( overline{C B} . ) If ( mathrm{E}, mathrm{F} ) are mid points of ( A C ) and ( overline{B D} ) respectively, then their resultant is
A. ( overline{E F} )
в. ( 2 overline{E F} )
c. ( frac{3}{2} overline{E F} )
D. ( 4 overline{E F} )
12
890If ( theta ) is the angle between the lines whose vector equations are ( overrightarrow{boldsymbol{r}}=boldsymbol{3} overrightarrow{boldsymbol{i}}+boldsymbol{2} overrightarrow{boldsymbol{j}}+ )
( 4 vec{k}+lambda(vec{i}+2 vec{j}+2 vec{k}) ) and ( vec{r}=5 vec{i} )
( 2 vec{k}+mu(3 vec{i}+2 vec{j}+6 vec{k}) ; lambda ) and ( mu ) being
parameters, then
This question has multiple correct options
( ^{mathbf{A}} cdot cos theta=frac{19}{21} )
B. ( sin theta=frac{19}{21} )
( ^{mathbf{C}} sin theta=frac{4 sqrt{5}}{21} )
D. ( cos theta=frac{4 sqrt{5}}{21} )
12
891L

LU
U
U
13. Let a= 2i +j-2k and b=i+j. Ifc is a vector such that a .
c=c,|c-a)=2 12 and the angle between (a + b) and cis
30°, then (ax b) c=
(1999 – 2 Marks)
(a) 2/3 (b) 3/2 (c) 2
(d) 3
111
12
892If ( a=3, b=4, c=5 ) each one of ( vec{a}, vec{b} )
and ( vec{c} ) is perpendicular to the sum of the remaining then ( |overrightarrow{boldsymbol{a}}+overrightarrow{boldsymbol{b}}+overrightarrow{boldsymbol{c}}| )
A ( cdot frac{5}{sqrt{2}} )
B. ( frac{2}{sqrt{5}} )
( c cdot 5 sqrt{2} )
D. ( sqrt{5} )
12
893If ( vec{a}, vec{b}, vec{c} ) are unit vectors such that ( vec{a}+vec{b}+vec{c}=overrightarrow{0}, ) then the value of
( vec{a} cdot vec{b}+vec{b} cdot vec{c}+vec{c} cdot vec{a} ) is equal to
( A )
B. ( frac{3}{2} )
( c cdot 3 )
D. ( -frac{3}{2} )
12
894A vector ( a ) can be written as
( mathbf{A} cdot(a cdot i) i+(a cdot j) j+(a cdot k) k )
B. ( (a cdot j) i+(a cdot k) j+(a cdot i) k )
( mathbf{c} cdot(a cdot k) j+(a cdot i) j+(a cdot j) k )
( mathbf{D} cdot(a cdot a) i+(i+j+k) k )
12
895If three mutually perpendicular lines have direction cosines
( left(ell_{1}, mathbf{m}_{1}, mathbf{n}_{1}right),left(ell_{2}, mathbf{m}_{2}, mathbf{n}_{2}right) ) and
( left(ell_{3}, boldsymbol{m}_{3}, boldsymbol{n}_{3}right), ) then the line having
direction cosines ( ell_{1}+ell_{2}+ell_{3}, boldsymbol{m}_{1}+ )
( boldsymbol{m}_{2}+boldsymbol{m}_{3} ) and ( boldsymbol{n}_{1}+boldsymbol{n}_{2}+boldsymbol{n}_{3} ) make an
angle of ( ldots ldots ) with each other.
A . 0
B. 30
( c cdot 60^{circ} )
D. ( 90^{circ} )
12
896Prove that:
( |vec{a}+vec{b}| leq|vec{a}|+|vec{b}| )
12
897If vector ( overrightarrow{A B}=3 hat{i}-3 hat{k}, overrightarrow{A C}=hat{i}- )
( 2 hat{j}+hat{k} ) represents the sides of any triangle ( A B C ) then the length of median AM is-
A ( cdot sqrt{6} )
B. ( sqrt{3} )
( c cdot 2 sqrt{3} )
D. ( 3 sqrt{2} )
12
8981D – Daru KAO
11.
Determine the value of ‘c so that for all real x, the vector
cxi-6-3k and xi +2 +2cxk make an obtuse angle
with each other.
(1991 – 4 Marks)
cond AC
12
899Prove that for any three vectors ( a, b, c[b ) ( +c c+a a+b]=2[a b c] )12
900Find the area of the triangle whose two adjecent sides are represented by the vectors ( bar{a}=3 bar{i}+4 bar{j} ) and ( bar{b}=-5 bar{i}+7 bar{j} )12
901If ( overline{D A}=bar{a} ; overline{A B}=bar{b} ) and ( overline{C B}=k bar{a} )
where ( k>0 ) and ( x, y ) are the midpoints
of ( D B ) and ( A C ) respectively such that ( |overline{boldsymbol{a}}|=mathbf{1 7} ) and ( |overline{boldsymbol{X}} boldsymbol{Y}|=mathbf{4}, ) then ( mathbf{k}= )
A. ( frac{8}{17} )
в. ( frac{9}{17} )
c. ( frac{11}{17} )
D. ( frac{4}{17} )
12
902The ( x ) and ( y ) components of vector ( vec{A} ) are
( 4 m ) and ( 6 m ) respectively. The ( x ) and ( y ) components of vector ( overrightarrow{boldsymbol{A}}+overrightarrow{boldsymbol{B}} ) are ( mathbf{1 0} boldsymbol{m} )
and 9 m respectively. For the vector ( vec{B} ) calculate the following
(i) ( x ) and ( y ) components
(ii) length and
(iii) the angle it makes with ( x ) -axis
A. (i) ( 6 m, 3 m )
(ii) ( sqrt{45} m )
(iii) ( tan ^{-1}left(frac{1}{2}right) )
B.
(i) ( 4 m, 2 m ) (ii) ( sqrt{90} m ) (iii) ( tan ^{-1}left(frac{1}{2}right) )
c. ( left(text { i) } 3 m, 6 m text { (ii) } sqrt{45} m text { (iii) } tan ^{-1}left(frac{3}{2}right)right. )
( D )
(i) ( 4 m, 3 m ) (ii) ( sqrt{60} m ) (iii) ( tan ^{-1}left(frac{5}{2}right) )
12
903Let L be the line of intersection of the planes
2x+3y+z=1 andx + 3y + 2z=2. IfL makes an angle a with
the positive x-axis, then cos a equals
[2007]
(d) 1
12
904The position vectors of two points ( A ) and ( mathrm{B} ) are ( i+j-k ) and ( 2 i-j+k )
respectively. Then ( |boldsymbol{A} boldsymbol{B}|= )
( A cdot 2 )
B. 4
( c .3 )
D. 5
12
90511. A particle moves in the x-y plane under the action of a
force F such that the value of its linear momentum (P) at
anytime t is P = 2 cos t, p, = 2 sin t. The angle between
F and P at a given time t will be
(a) 0 = 0°
(b) 0= 30°
(c) 0= 90°
(d) 0= 180°
12
906veen V
7. The angle between vectors (AXB) and (B x A) is
(a) Zero (b) T (c) 10/4 (d) 1/2
12
907Find the co ordinates of the point where the line through (3,4,1) and (5,1,6) ( operatorname{crosses} X Y- ) plane12
908If a unit vector ( vec{a} ) makes an angle ( frac{pi}{3} ) with ( hat{i}, frac{pi}{4} ) with ( hat{j} ) and an accute angle ( theta ) with ( hat{k}, ) then find ( theta ) and hence, the
components of ( overrightarrow{boldsymbol{a}} ).
A ( cdot frac{pi}{3} ; vec{a}=frac{1}{2} hat{i}-frac{1}{sqrt{2}} hat{j}+frac{1}{2} hat{k} )
B ( cdot frac{pi}{3} ; vec{a}=frac{-1}{2} hat{i}+frac{1}{sqrt{2}} hat{j}+frac{1}{2} hat{k} )
C ( cdot frac{pi}{3} ; vec{a}=frac{1}{2} hat{i}+frac{1}{sqrt{2}} hat{j}+frac{1}{2} hat{k} )
D ( cdot frac{pi}{3} ; vec{a}=frac{1}{2} hat{i}+frac{1}{sqrt{2}} hat{j}-frac{1}{2} hat{k} )
12
909Under what condition will the direction
of sum and difference of two vectors be
same?
12
910Show that the points ( boldsymbol{A}(mathbf{2}, mathbf{1},-mathbf{1}) )
( B(0,-1,0), C(4,0,4) ) and ( D(2,0,1) ) are
coplanar.
12
911The angle that the vector ( vec{A}=2 hat{i}+3 hat{j} )
makes with ( x ) -axis is
( A cdot tan ^{-1}(3 / 2) )
B cdot ( tan ^{-1}(2 / 3) )
( c cdot sin ^{-1}(2 / 3) )
D ( cdot cos ^{-1}(3 / 2) )
12
91224.
If u, v, w, are three non-coplanar unit vectors and a, B.
are the angles between ū and V and w, w and
respectively and x, y, z are unit vectors along the bisectors
of the angles a, b, y respectively. Prove that
2
B
se
[xx] jxi Exă] = lu v wjsee? sec? see?
[xy ý xz Zxx]=
2002 AM
12
913If the scalar projection of the vector ( x hat{i}-hat{j}+hat{k} ) on the vector ( 2 hat{i}-hat{j}+5 hat{k} ) is
( frac{1}{sqrt{30}}, ) then value of ( x ) is equal to
A ( cdot frac{-5}{2} ) units
B. 6 units
c. -6 units
D. 3 units
12
9142.
The number of vectors of unit length perpendicular to
vectors ā = (1, 1, 0) and ✓ = (0,1,1) is (1987 – 2 Marks)
(a) one (b) two (c) three (d) infinite
(e) None of these.
12
915If ( |vec{a}|=3,|vec{b}|=4, ) then a value of ( lambda ) for which ( vec{a}+lambda vec{b} ) is perpendicular to ( vec{a}- ) ( lambda vec{b} ) is
A ( cdot frac{9}{16} )
B. ( frac{3}{4} )
( c cdot frac{3}{2} )
D. ( frac{4}{3} )
12
916Show that vectors ( 2 hat{i}-hat{k}, hat{i}-3 hat{j}-5 hat{k} )
and ( 3 hat{i}-4 hat{j}-4 hat{k} ) form the vertices of
the triangle.
12
91718. Let APQR be a triangle. Let a = QR, b = RP and c = POJE
lā = 12, 151 = 413, 6.c = 24, then which of the following
is (are) true?
(JEE Adv. 2015)
(a) let -|āl=12 (6) le +lāt = 30
(c) |āx5+ēxă = 48/3 (d) a. b =-72
12
9188. The unit vector parallel to the resultant of the vecto
A=4+3j+6k and B=-i +3j – 8k is
(a) +(3+63–2) (b) (3 +63 + zł)
(c) . Gi +63-2) ( ) (59 – 69 +2h)
12
9196. The resultant of two vectors A and B is perpendicular
to the vector A and its magnitude is equal to half of the
magnitude of vector B Fig. 3.77. The angle between A
and B is
в
в
Ph 90°
a. 120°
Fig. 3.77
b. 150°
d. None of these
c. 135°
12
920Find unit vector in the direction of
vector ( boldsymbol{a}=mathbf{2} boldsymbol{i}+boldsymbol{3} boldsymbol{j}+boldsymbol{k} )
12
921Form the greatest and the smallest ( 4- ) digit numbers using any four different digits with the condition that digit 5 is always at the ten’s place.12
922If ( vec{a}=hat{i}+hat{j}+hat{k}, vec{b}=4 hat{i}+3 hat{j}+4 hat{k} ) and
( vec{c}=hat{i}+alpha hat{j}+beta hat{k} ) are linearly dependent
vectors and ( |vec{c}|=sqrt{3} ) then the value of ( alpha )
may be.
( mathbf{A} cdot mathbf{1} )
B . 2
c. -3
D. 4
12
923If ( boldsymbol{a} cdot boldsymbol{i}=boldsymbol{a} cdot(boldsymbol{j}+boldsymbol{i})=boldsymbol{a} cdot(boldsymbol{i}+boldsymbol{j}+boldsymbol{k}) )
then ( a ) is equal to
( mathbf{A} cdot i )
B. ( k )
( c cdot j )
D ( cdot(i+j+k) )
12
92414. The component of vector A = 2î +3j along the vector
i ti is
(b) 1072 (C) 52 (d) 5
(a)
12
925Suppose ( vec{V}_{1}=hat{i}+hat{j}-2 hat{k}, vec{V}_{2}=hat{i}- )
( mathbf{2} hat{boldsymbol{j}}+hat{boldsymbol{k}}, overrightarrow{boldsymbol{V}}_{3}=-2 hat{boldsymbol{i}}+boldsymbol{2} hat{boldsymbol{j}}+hat{boldsymbol{k}} ) are three
vectors. Let ( vec{V} ) be a vector such that it
can be expressed as a linear combination of ( vec{V}_{1} ) and ( vec{V}_{2} . ) Also ( vec{V} vec{V}_{3}= )
0 and the projection of the vector ( vec{V} ) on ( hat{boldsymbol{i}}-hat{boldsymbol{j}}+hat{boldsymbol{k}} ) is ( boldsymbol{6} sqrt{boldsymbol{3}} . ) If ( overrightarrow{boldsymbol{V}}=boldsymbol{lambda}(hat{boldsymbol{i}}+boldsymbol{3} hat{boldsymbol{j}}- )
4 ( hat{k} ) ) then find the absolute value of ( lambda )
12
926If ( theta ) be the angle between the lines
whose direction cosines’s are given by ( mathbf{3} l+boldsymbol{m}+mathbf{5} boldsymbol{n}=mathbf{0}, boldsymbol{6} boldsymbol{m} boldsymbol{n}-boldsymbol{2} boldsymbol{n} l+mathbf{5} boldsymbol{l} boldsymbol{m}= )
0. Find value of ( 6 cos theta )
12
927The position vectors of points ( overrightarrow{boldsymbol{A}}, overrightarrow{boldsymbol{B}}, overrightarrow{boldsymbol{C}} )
are respectively ( vec{a}, vec{b}, vec{c} ). If ( P ) divides ( overrightarrow{A B} ) in the ratio 3: 4 and ( Q ) divides ( overrightarrow{B C} ) in
the ratio 2: 1 both externally then ( overrightarrow{P Q} )
is
A. ( vec{b}+vec{c}-overrightarrow{2 a} )
and
В . ( 2(vec{b}+vec{c}-overrightarrow{2 a}) )
c. ( 4 vec{a}-vec{b}-vec{c} )
D. ( frac{-2 vec{a}-vec{b}-vec{c}}{2} )
12
92832. Let ā, b, c be unit vectors such that a +b + c = 0. Which
one of the following is correct?
(2007-3 marks)
(a) ã xb = b xễ = xã = 0
(b) ã xb = b xễ =ở xã + 0
(c) 4 xb = b xễ = 4 x + 0
(d) & x6,6 xễ,xã are muturally perpendicular
12
929Let ( vec{a}, vec{b}, vec{c} ) be vectors of length 3,4 and 5 respectively. Let ( vec{a} ) be perpendicular to ( (vec{b}+vec{c}), vec{b} ) to ( (vec{c}+vec{a}) ) and ( vec{c} ) to ( (vec{a}+vec{b}) )
Find the length of the vector ( vec{a}+vec{b}+vec{c} ).
12
930If ( vec{a}+vec{b} perp vec{a} ) and ( |vec{b}|=sqrt{2}|vec{a}| ), then?
( mathbf{A} cdot(2 vec{a}+vec{b} | vec{b} vec{b} )
B . ( (2 vec{a}+vec{b}) perp vec{b} )
c. ( (2 vec{a}-vec{b}) perp vec{b} )
D. ( (2 vec{a}+vec{b}) perp vec{a} )
12
9313. Let ā= 2î – +k, b = i +2j – Ã and c = i+j – 2k – 2k
be three vectors. A vector in the plane of 5 and ĉ, whose
projection on ā is of magnitude 7273, is: (1993 – 2 Marks)
(a) 2î +3j –
(b) 2ỉ +39 +3ť
(c) -21 – 9 +5h (d) zi tŷ + 5h
12
932The co-ordinates of head and tail of a
vector are ( (mathbf{2}, mathbf{1}, mathbf{0}) ) and ( (-mathbf{4}, mathbf{2},-mathbf{3}) )
respectively. The magnitude of the vector is:
A. ( sqrt{23} ) units
B. ( sqrt{46} ) units
c. ( sqrt{84} ) units
D. ( sqrt{12} ) units
12
933Vectors ( overrightarrow{boldsymbol{a}}=-mathbf{4} hat{mathbf{i}}+mathbf{3} hat{boldsymbol{k}} ; overrightarrow{boldsymbol{b}}=mathbf{1 4} hat{mathbf{i}}+mathbf{2} hat{mathbf{j}}- )
( mathbf{5} hat{k} ) are laid from one point. Vector ( overrightarrow{boldsymbol{d}} )
which is being laid off from the same point dividing the angle between vectors ( vec{a} ) and ( vec{b} ) in equal halves and having the magnitude ( sqrt{mathbf{6}} ) is
( mathbf{A} cdot hat{i}+hat{j}+2 hat{k} )
B ( cdot hat{i}-hat{j}+2 hat{k} )
c. ( hat{i}+hat{j}-2 hat{k} )
D . ( 2 hat{i}-hat{j}-2 hat{k} )
12
934Let ( A, B, C ) be unit vectors. Suppose
( A cdot B=A cdot C=0 ) and the angle between ( B ) and ( C ) is ( frac{pi}{6} . ) Then ( A ) equals
( mathbf{A} cdot pm 2(B times C) )
B . ( -2(B times C) )
( mathbf{c} cdot 2(B times C) )
D. ( (B times C) )
12
935Statement-1: the two vectors ( a=1 i+ )
( boldsymbol{P j}+mathbf{2 k} ) and ( boldsymbol{b}=boldsymbol{3} boldsymbol{i}+boldsymbol{3} boldsymbol{j}+boldsymbol{Q} boldsymbol{k} ) are
parallel only if ( boldsymbol{P}=mathbf{1} ) and ( boldsymbol{Q}+boldsymbol{6} )
Statement-2: if two vectors ( boldsymbol{a}=boldsymbol{a}_{1} boldsymbol{i}+ )
( a_{2} j+a_{3} k ) and ( b=b_{1} i+b_{2} j+b_{3} k ) are
parallel then ( frac{boldsymbol{a}_{1}}{boldsymbol{a}_{2}}=frac{boldsymbol{b}_{1}}{boldsymbol{b}_{2}} )
A. Both statements are true and Statement-2 is correct explanation of Statement- –
B. Both statements are true and Statement-2 is not correct explanation of Statement-
c. statement-2 is false but other is true
D. Statement-1 is false but other is true
12
936Find the equation of the line parallel to ( 2 bar{i}-bar{j}+2 bar{k} ) and which passes through
point ( ^{prime} A^{prime}(3 bar{i}+bar{j}-bar{k}) . ) if ( P ) is a point on
the line such that ( A P=15 . ) Find the
position vectors of ( boldsymbol{P} )
12
9373. Which of the following is the unit vector perpendicular to
A and B ?
2x
Å x ß
AB sin 0
AB cos e
Āx B
AB sin e
(d)
Ax B
AB cos e
12
93824. A line with direction cosines proportional to 2, 1,2 meets
each of the lines x = y + a = z and x+a = 2y = 2z. The
co-ordinates of each of the points of intersection are given
[2004]
(a) (2a, 3a, 3a),(2a,a,a) (b) (3a, 2a, 3a),(a, a, a)
(c) (3a, 2a, 3a),(a, a, 2a) (d) (3a, 3a, 3a),(a, a, a)
by
12

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