Vectors and their Applications

Type – 1
Choose the most appropriate option (a, b, c or d).
Q 1. ABCDEF is a regular hexagon where centre O is the origin. If the position vectors of A and B are
      
i  j  2 k and 2i  j  k respectively then BC is equal to
        
(a) i  j  2 k (b)  i  j  2k (c) 3 i  3 j  4 k (d) none of these
     
Q 2. The position vectors of two vertices and the centroid of a triangle are i  j,2 i  j  4 k and k

respectively. The position vector of the third vertex of the triangle is

     2
(a) 3 i  2 k (b) 3 i  2 k (c) i  k (d) none of these
3
        
Q 3. Let the position vectors of the points A, B, C be i  2 j, 3 k,  i  j  8 k and 4 i  4 j,6 k

respectively. Then the ABC is

(a) right angled (b) equilateral (c) isosceles (d) none of these
    
Q 4. a,b,c are three vectors of which every pair is noncollinear. If the vector a  b and are collinear
    
with c and a respectively then a  b  c is
  
(a) a unit vector (b) the null vector (c) equally inclined to a,b,c (d) none of these
                   
Q 5. If r  3 i  2 j  5 k,a  2 i  j  k,b  i  3 j  2k and s c  2 i  j  3 k such that r   a   b  v c

then

(a) , ,v arein AP (b) , , v are in AP (c) , , v are in HP (d) , , v are in GP
2
          
Q 6. The position vectors of three points are 2a  b  3 c,a  2b   c and  a  5 b where a,b,c are
noncoplanar vectors. The are collinear when
9 9 9
(a)   2  (b)    ,   2 (c)   ,   2 (d) none of these
4 4 4
            
Q 7. If a  i  j  k,b  4 i  3 j  4 k and c  i   j   k are linearly dependent vectors and | c |  3
then
(a)  = 1,  = – 1 (b)  = 1,  =  1 (c)  = – 1,  =  1 (d)  =  1,  = 1
   
Q 8. Let OA  a and OB  b . A vector along one of the bisectors of the angle AOB is
 
   
a b
(a) a  b (b) a  b (c) 
 
(d) none of these
|a| |b|
Q 9. A vector has components 2p and 1 with respect to a rectangular Cartesian system. The axes are
rotated through an angle  about the origin in the anticlockwise sense. If the vector has
components p + 1 and 1 with respect to the new system then
 1 1
(a) p = 1, – (b) p = 0 (c) p = – 1 , (d) p = 1, – 1
3 3
  
Q 10. If a and b are two vectors of magnitude inclined at an angle 60 then the angle between a and
 
a  b is
(a) 30 (b) 60 (c) 45 (d) none of these
     
Q 11. Let | a |  | b |  | a  b |  1 Then the angle between a and b is
   
(a) (b) (c) (d)
6 3 4 2
    
Q 12. A vector of magnitude 4 which is equally inclined to the vectors i  j, j  k k  i is
4    4    4   
(a) ( i  j  k) (b) ( i  j  k) (c) ( i  j  k) (d) none of these
3 3 3
        
Q 13. If a  b  2 i and 2a  b  i  j then cosine of the angle between a and b is
4 4 3
(a) sin 1 (b) cos 1 (c) cos1 (d) none of these
5 5 5
              
Q 14. Let | a |  1,| b | 2, | c | 3, and a  (b  c),b  (c  a) and c  (a  b) . Then | a  b  c | is
(a) 6 (b) 6 (c) 14 (d) none of
these
       
Q 15. (a. i ) i  (a. j ) j  (a.k) is equal to
    
(a) i  j  k (b) a (c) 3 a (d) none of these
Q 16. If a, b, are unit vectors such that a + b is also a unit vector then the angle between the vectors a
and b is
   2
(a) (b) (c) (d)
6 4 3 3
         
Q 17. If a. i  a.( i  j )  a.( i  j  k)  1 then a is
    
(a) i  j (b) i (c) j (d) k
     
Q 18. (a. i )2  (a. j )2  (a  k)2 is equal to
     
(a) a (b) 3 (c) | a.( i  j  k) |2 (d) k
   
Q 19. | a  b |2  | a b |2 is equal to
   
(a) 4 a.b (b) 0 (c) 4 | a.b | (d) none of these
Q 20. If a,b,c are the pth, qth, rth terms of an HP and
  
  i j k   
then   (q  r) i  (r  p) j  (p  q) k, v  
a b c
   
(a)  . v are parallel vectors (b)  . v are orthogonal vectors
      
(c)  .v  1 (d)  x v  i  j  k
    
Q 21. If a  b  a and | b | 2 | a | then
            
(a) (2 a  b) || b (b) (2 a  b)  b (c) (2 a  b)  b (d) (2 a  b)  a
           
Q 22. Let a  2 i  j  k,b  i  2 j  k and a unit vector c be coplanar. If c is perpendicular to a then c
=
1   1    1   1   
(a) (  j  k) (b) (  i  j  k) (c) ( i 2 j) (d) ( i  j  k)
2 3 5 3
           
Q 23. Let   a x(b  c),   b x(c  a) and v  c x(a  b) . Then
        
(a)    v (b) , ,v are coplanar (c)  v  2  (d) none of these
          
Q 24. Let a,b,c be three unit vectors such that | a  b  c |  1 a  b. If c makes angles ,  with a,b
respectively then cos  + cos  is equal to
3
(a) (b) 1 (c) – 1 (d) none of these
2
 

Q 25. If a,b c are three vectors of equal magnitude and the angle between each pair of vectors is
3
   
such that | a b  c |  6 then | a | is equal to
1
(a) 2 (b) – 1 (c) 1 (d) 6
3
   
Q 26. If | a |  5,| a  b |  8 and | a  b | = 10 then | b | is
(a) 1 (b) 57 (c) 3 (d) none of these
  
Q 27. If a and b are unit vectors and  is the angle between them then cos is
2
 
1   1  
(a) | a b | (b) | ab | (c) | a  b | (d) none of these
2 2
   
Q 28. If | a  b |  | a  b | then
     
(a) a|| b (b) a  b (c) | a |  | b | (d) none of these
 
  i  i 
Q 29. Two vectors a  i  and b   j are
3 3
(a) perpendicular to each other (b) parallel to each other
 
(c) inclined to each other at an angle (d) inclined to each other at an angle
3 6
             
Q 30. Let a  2 i  j  k,b  i  3 j  k and c  i  j  2k . Avector in the plane if b and c whose
 
projection on a has the magnitude a is
           
(a) 2 i  3 j  3 k (b) 2 i  3 j  3 k (c) 2 i  j  5 k (d) 2 i  j  5 k
     
Q 31. ABC is an equilateral triangle of side a. The value of AB.BC  BC.CA  CA.AB is equal to
3a 2 3a2
(a) (b) 3a2 (c)  (d) none of these
2 2

             
r
Q 32. If a  i  j,b  2 j  k and r x a  b x a, r x b  a x b then 
is equal to
|r |
 1    1    1   
(a) ( i  3 j  k) (b) ( i  3 j  k) (c) ( i  j  k) (d) none of these
11 11 11
   
Q 33. (a x b)2  (a.b)2 is equal to
   
(a) 0 (b) | a |2 | b |2 (c) (| a |  | b |)2 (d) 1
 
Q 34. If p,q are two noncollinear and nonzero vector such that
   
(b  c)p x q  (c  a)p  (a  b) q  0,
where a,b,c are the lengths of the sides of a triangle, then the triangle is
(a) right angled (b) obtuse angled (c) equilateral (d) isosceles
           
Q 35. If a,b,c are any three vectors such that (a  b).c  (a  b).c  0 then (a x b) x c is
  
(a) 0 (b) a (c) b (d) none of these
       
Q 36. Then nit vector perpendicular to both the vectors a  i  j  k and b  2 i  j  3 k and making an

acute angle with the vector k is
1    1    1   
(a)  (4 i  j  3 k) (b) (4 i  j  3 k) (c) (4 i  j  3 k) (d) none of these
26 26 26
            
Q 37. Let a  i  2 j  3 k,b  2 i  3 j  k and c   i  j (2  1)k . If c is parallel to the plane of the
 
vectors a and b then  is
(a) 1 (b) 0 (c) – 1 (d) 2
   
Q 38. Let a be a unit vector perpendicular to unit vectors b and c and if the angle between b and be 
 
then b x c is
  
(a) cos  a (b) cosec  a (c) sin  a (d) none of these
    
Q 39. If a.b  0 and a x b  0 then
       
(a) a || b (b) a  b (c) a  b or  b  0 (d) none of these
     
Q 40. The area of the parallelogram whose diagonals represent the vectors 3 i  j  2 k and i  3 j  4 k
is
(a) 10 3 (b) 5 3 (c) 8 (d) 4
           
Q 41. ( r . i )( r x i )  ( r . j )( r x j )  ( r .k)( r x k) is equal to
 
(a) 3 r (b) r (c) 8 (d) none of these
             
Q 42. Let a  i  j  k,c  j  k . If b is a vector satisfying a x b  c and a.b  3 then b is
  
1    1   
(a) (5 i  2 j  2 k ) (b) (5 i  2 j  2 k) (c) 3 i  j  k (d) none of these
3 3
Q 43. A unit vector perpendicular to the plane passing through the points whose position vector are
      
i  j  2k  2 i  k and 2 i  k is
   1    1   
(a) 2 i  j  k (b) (2 i  j  k) (c) ( i  2 j  k) (d) none of these
6 6
      
Q 44. Let r x a  b x a and v, where a.b  0 . Then r is equal to
  
   b.c 
(a) b  t a where t is a scalar (b) b    a
a.c
 
(c) a  c (d) none of these
  
Q 45. For the vectors u,v,w which of the following expressions is not equal to any one of the remaining
three option?
           
(a) u.(v x w) (b) (v x w).u (c) v .(u x w) (d) (u x v).w
  
Q 46. For three noncoplanar vectors a,b,c the relation hold
     
| a x b.c |  | a | | b | | c |
holds if and only if
                 
(a) b.c  c .a  0 (b) a.b  b.c  0 (c) a.b  b.c  c .a  0 (d) c .a  a.b  0
     
Q 47. [a  b b  c c  a] is equal to
        
(a) 2[a b c] (b) 3[a b c] (c) [a b c] (d) 0
     
Q 48. a  b b  c c  a is equal to
     
(a) 2[a b c] (b) [a b c] (c) 0 (d) none of these
         
Q 49. Let a,b,c be three unit vectors and a.b  a.c  0 . If the angle between b and c is then
3

| [a b c] | is equal to
3 1
(a) (b) (c) 1 (d) none of these
2 2
  
Q 50. Let a,b,c, be three distinct positive real numbers. If p,q, r lie in a plane, where
          
p  a i  a j  b k,q  i  k and r  c i  c j  b k , then b us
(a) then AM of a,c (b) then GM if a,c (c) sthem HM of a,c (d) equal to 0
  
Q 51. Which of the following is not equal to s [a b c] ?
           
(a) a.b x c (b) c x a.b (c) b.a x c (d) c .a x b
        
   a.b x c b.c x a c .a x b
Q 52. If [a b c] = 1 then   
        is equal to
c x a.b a x b.c b x c .a
(a) 3 (b) 1 (c) 0 (d) none of these
     
Q 53. a,b,c are noncoplanar vectors and p,q, r are defined as
     
 bxc  c xa  axb
p 
,q  
,r  
[b c a] [c a b] [a b c]
        
(a  b).p  (b  c).q (c  a). r is equal to
(a) 0 (b) 1 (c) 2 (d) 3
   
Q 54. If a,b,c are three noncoplanar vectors represented by concurrent edges of a parallelepiped of
volume 4 then
          
(a  b).(b x c)  (b xc).(c x a)  (c  a).(a x b)
is equal to
(a) 12 (b) 4 (c)  12 (d) 0
  
Q 55. If a,b,c are three noncoplanar nonzero vectors then
           
(a.a)b x c  (a.b)c x a  (a.c)a x b
Is equal to
           
(a) [b c a]a (b) [c a b]b (c) [a b c] c (d) none of these
             
Q 56. Let r be a vector perpendicular to a  b  c , where [a b c] = 2. If r  l( b x c)  m(c x a)  n(a x b)
then l + m + n is
(a) 2 (b) 1 (c) 0 (d) none of these
         
Q 57. If a,b,c are any three vectors in space then (c  b)x(c  a).(c  b  a) is equal to
     
(a) 3[a b c] (b) 0 (c) [a b c] (d) none of these
         
Q 58. If a,b,c are three noncoplanar vectors then [a b  c a  c a  b] is equal to
        
(a) 0 (b) [a b c] (c) 3[a b c] (d) 2[a b c]
     
Q 59. [a b  c a  b  c] is equal to
     
(a) 0 (b) 2[a b c] (c) [a b c] (d) none of these
            
Q 60. If a,b are nonezero and noncolinear vectors then [a,b i ] i  [ a b j ] j  [a b k]k is equal to
       
(a) a  b (b) a x b (c) a  b (d) b x a
     
Q 61. The three concurrent edges of a parallelpiped respesent the vectors a,b,c such that [a b c] =.
Then the volume of the parallelpiped whose three concurrent edges are the three concurrent
diagonals of three faces of the given parallelpiped is
(a) 2 (b) 3 (c)  (d) none of these
        
Q 62. i x( a x i )  j x( a x j )  k x(a x k) is equal to
  
(a) 2 a (b) 3 a (c) 0 (d) none of these
       
Q 63. Let a,b and c be three vectors having magnitudes 1,1 and 2 respectively. If a x(a x c)  b  0 ,
 
the acute angle between a and c is
  
(a) (b) (c) (d) none of these
3 4 6
      
Q 64. If b is a unit vector then (a.b)b  b x(a x b) is equal to
2     
(a) a b (b) (a.b)a (c) a (d) none of these
  
      b c    
Q 65. Let a,b,c be three unit vectors such that a x(b x c)  and the angles between a,c and a,b
2
be  and  respectively then
3   7  3
(a)   ,  (b)   ,  (c)   ,  (d) none of these
4 4 4 4 4 4
   
Q 66. Let p,q, r be three mutually perpendicular vectors of the same magnitude. If a vector x satisfies
the equation
            
p x{(x  q)x p}  q x{(x  r ) x q}  r {(x  p)x r }  0
this x is given by
1    1    1    1   
(a) (p  q 2 r ) (b) (p  q r ) (c) (p  q r ) (d) (2 p  q  r )
2 2 3 3
Q 67. If . and x represent dot product and cross product respectively then which of the following is
meaningless?
               
(a) (a x b).(c x d) (b) (a x b)x(c x d) (c) (a.b)(c x d) (d) (a.b)x(c x d)
     
Q 68. (a x i )2  (a x j )2  (a x k)2 is equal to
2 2 2
(a) a (b) 3 a (c) 2 a (d) none of these
     
Q 69. If || a b x c then (a x b).(a x c) is equal to
2   2   2  
(a) a (b.c) (b) b (a.c) (c) c (a.b) (d) none of these
  
Q 70. If a,b,c are noncoplanar nonzero vectors then
           
(a x b)x(a x c)  (b x c)x(b x a)  (c x a)x(c x b)
is equal to
            
(a) [a b c]2 (a  b  c) (b) [a b c](a  b  c) (c) 0 (d) none of these
   
Q 71. If a,b,c are three noncoplanar nonzero vectors and r is any vector in space then
           
(a x b)x( r x c)  (b x c)x( r x a)  (c x a)x( r x b) is equal to
           
(a) 2[a b c] r (b) 3[a b c] r (c) [a b c] r (d) none of these
      
Q 72. Let a,b,c be three unit vectors of which b and c are nonparallel. Let the angle between a and b
       1
be  and that between a and b be  and that between a and c be . If a x( b x c)  b then
2
     
(a)   , (b)   , (c)   ,  (d) none of these
3 2 2 3 6 3
            
Q 73. Let a  2 i  j  2k and b  i  j . If c is a vector such that a.c | c |,| c  a | 2 2 and the angle
     
between a x b and c is 30 then | (a x b)x c | is equal to
2 3
(a) (b) (c) 2 (d) 3
3 2
          
Q 74. Let a , and b be two noncollinear unit vectors. If u  a  (a.b)b and v  a x b then | v | is
           
(a) | u | (b) | u |  | u.a | (c) | u |  | u.b | (d) | u |  u.(a  b)
           
Q 75. If a,b,c be three vectors such that [a b c] = 4 then [a x b b x c c x a] is equal to
(a) 8 (b) 16 (c) 64 (d) none of these
       
Q 76. If d is a unit vector such that d   b x c   c x a  v a x b then
           
| (d.a)(b x c)  (d.b)(c x a)  (d.c)(a x b)
is equal to
     
(a) | [a b c] | (b) 1 (c) 3 | [a b c] | (d) none of these
        
Q 77. a x( b x c),b x(c x a) and c x( a x b) are
(a) linearly (b) dependent (c) equal vectors (d) none of these
     
Q 78. [b c b x c]  (b.c)2 is equal to
     
(a) | b|2 |c |2 (b) (b  c)2 (c) | b |2  | c |2 (d) none of these
       
Q 79. If the vector a,b,c and d are coplanar then (a x b) x ( c x d) is equal to
        
(a) a  b  c  d (b) 0 (c) a  b  c  d (d) none of these
          
Q 80. If a x(a x b)  b x(b x c) and a.b  0 then [a b c] is equal to
(a) 0 (b) 1 (c) 2 (d) none of these
      
Q 81. Let OA  a,OB  10 a  2b and OC  b, where O, A and C noncollinear points. Let p denote the
area of the quadrilateral OAB, and q denote the area of the parallelogram with OA and OC as
adjacent sides. Then p/q is equal to
 
1 | a b |
(a) 4 (b) 6 (c) 
(d) none of these
2 |a |
     
Q 82. The position vectors of the vertices A, B, C of a triangle are i  j  3 k,2 i  j  2k and
  
5 i  2 j  6 k respectively. The length of the bisector AD of the angle BAC where D is on the line
segment BC, is
15 1 11
(a) (b) (c) (d) none of these
2 4 2

Q 83. P is a point on the line through the point A whose position vector is a and the line is parallel to the

vector b . If PA = 6, the position vector of P is
   6     6 
(a) a  6 b (b) a  
b (c) a  6 b (d) b  
a
|b| |a|
Q 84. The coplanar points A,B,C,D are (2 – x, 2, 2), (2, 2 – y, 2), (2, 2, 2, 2  z) and (1, 1, 1)
respectively. Then
1 1 1 1 1 1
(a)    1 (b) x + y + z = 1 (c)   (d) none of these
x y z 1 x 1 y 1 z
       
Q 85. Let AB  3 i  j  k and AC  i  j  3 k . If the point P on the line segment BC is equidistant from

AB and AC then AP is
      
(a) 2 i  k (b) i  2 k (c) 2 i  k (d) none of these
Q 86. The cosine of the angle between two diagonals of a cube is
1 2 2 1
(a) (b) (c) (d) none of these
3 3 2
   
Q 87. If AB  b and AC  c then the length of the perpendicular from A to the line BC is
     
| bxc | | bxc | 1| bxc |
(a)  
(b)  
(c)  
(d) none of these
| b c | | b c | 2|b c|
Q 88. The distance of the point (1, 1, 1) from the plane passing through the points (2, 1, 1), (1, 2, 1) and
(1, 1, 2) is
1
(a) (b) 1 (c) 3 (d) none of these
3
  
Q 89. The projection of the vector i  j  k one the line whose vector equation is
   
r  (3  t) i  (2t  1) j  3t k , t being the scalar parameter, is
1 6
(a) (b) 6 (c) (d) none of these
14 14
      
Q 90. If the vertices of a tetrahedron have the position vectors 0, i  j,2 j  k and i  k then the volume
of the volume if of the tetrahedron is
1
(a) (b) 1 (c) 2 (d) none of these
6

Type 2
Choose the correct options. One or more option may be correct.
     
Q 91. A line passes through the point whose position vectors are i  j  2 k and i  3 j  k . The position
vector of a point on it at a unit distance from the first point is
  
1    1   
(a) (6 i  j  7 k ) (b) (4 i  9 j  13 k) (c) i  4 j  3 k) (d) none of these
5 5
  
Q 92. A vector of magnitude 2 along bisector of the angle between the two vectors 2 i  2 j  k and
  
i  2 j  2 k is
2   1    2   
(a) ( i  k) (b) ( i  4 j  3 k) (c) ( i  4 j  3 k) (d) none of these
10 26 26
        
Q 93. A unit vector coplanar with i  j  2 k and i  2 j  k , and perpendicular to i  j  k , is
1   1   1   1  
(a) (  j  k) (b) (k  i ) (c) ( i  k) (d) ( j  k)
2 2 2 2
    
2 i  j  2 k  4 j  3 k

Q 94. A unit vector which is equally inclined to the vectors i ,  k ) and is
3 5
1    1    1    1   
(a) ( i  5 j  5 k) (b) ( i  5 j  5 k) (c) ( i  5 j  5 k) (d) ( i  5 j  5 k)
51 51 51 51
       
Q 95. If | a |  4,| b |  2 and the angle between a and b is then (a x b)2 equal to
6
2
(a) 48 (b) 16 (c) a (d) none of these
  
Q 96. Three points whose position vectors are a,b,c will be collinear if
            
(a)  a  b  (  )c (b) a x b  b x c  c x a  0 (c) [a b c] (d) none of these
    
Q 97. Let b  4 i  3 j . Let c be a vector perpendicular to b and it lies in the x–y plane. A vector in the
 
xy plane having projections 1 and 2 along b and c is
   
1  
(a) 2 i  j (b) i  2 j (c) (  i  11 j ) (d) none of these
5
   
Q 98. If a,b,c are noncoplanar nonzero vectors and r is any vector in space then
           
[b c r ]a  [c a r ]b  [a b r ] c is equal to
           
(a) 3[a b c] r (b) [a b c] r (c) [b c a] r (d) none of these
           
Q 99. If a,b,c are nonecoplanar vectors such that b x c  a,a x b  c and c x a  b then
    
(a) | a |  1 (b) | b |  1 (c) | a |  | b |  | c |  3 (d) none of these
     
     
bxc c xa axb
Q 100. Let a,b,c be noncoplanar vectors and p  
,q  
,r 
then
[a b c] [b c a] [c a b]
             
(a) p.a  1 (b) p.a  q.b  r .c  3 (c) p.a  q.b  r .c  0 (d) none of these
     
Q 101. If a,b,c are any three vectors then (a x b)x c is a vector
   
(a) perpendicular to a x b (b) coplanar with a and b
  
(c) parallel to c (D) parallel to either a or b
     
Q 102. If c  a x b and b  c x a then
  2   2     
(a) a.b  c (b) c.a  b (c) a  b (d) a ||b c
      
Q 103. If x b  c x b and x  a then x is equal to

        
b (a c) (b x c)x a (a x(c x b)
(a)  
(b)  
(C)  
(d) none of these
b.c b.a a.b
    
Q 104. The resolved part of the vector a along the vector b is  and that perpendicular to b is  . Then
              
 (a.b) a  (a.b)b  (b.b)a  (a.b)b  b x(a x b)
(a)   (b)   (c)   (d)  
2 2 2 2
a b b b
   
Q 105. (a x b.(c x d) is equal to
                   
(a) a.{b x(c x d)} (b) (a.c)(b.d)  (a.d)(b.c) (c) {(a x b) x c}.d (d) (d x c).(b x a)
       
Q 106. If a,b,c,d are any four then (a x b)x(c x d) is a vactor
    
(a) perpendicular to a,b,c,d
   
(b) along the line of intersection of two planes, one containing a,b and the other containing c,d
   
(c) equally inclined to both a x b and c x d
(d) none of these

Answers
1b 2a 3b 4b 5a 6c 7d 8c 9a 10a
11b 12c 13b 14a 15b 16d 17b 18a 19a 20b
21b 22a 23b 24c 25c 26b 27a 28b 29d 30c
31c 32a 33b 34c 35a 36a 37b 38c 39c 40b
41c 42a 43b 44b 45c 46c 47a 48c 49a 50c
51c 52a 53d 54c 55a 56c 57c 58c 59a 60b
61a 62a 63c 64c 65c 66b 67d 68c 69a 70b
71a 72b 73b 74a 75b 76a 77a 78a 79b 80a
81b 82a 83b 74a 85c 86a 87b 88a 89c 90a
91a,b 92a,c 93a,d 94a,d 95b,c 96a,b 97a,c 98b,c 99a,b,c 100a,b
101a,b 102c,d 103b,c 104b,c,d 105a,b,c,d 106b,c