8 Vector Algebra

10. If the position vectors of the vertices of a triangle 20. If a, b, c, d be the position vectors of the points A,
be 6i  4j  5k, 4i  5j  6k and 5i  6j  4k, B, C and D respectively referred to same origin O
then the triangle is such that no three of these points are collinear
(a) Right angled (b) Isosceles and a  c  b  d, then quadrilateral ABCD is a
(c) Equilateral (d) None of these (a) Square (b) Rhombus
11. The perimeter of the triangle whose vertices have (c) Rectangle (d) Parallelogram
the position vectors (i  j  k), (5i  3j  3k) 21. If the position vectors of A and B are i  3j  7k
and (2i  5j  9k), is given by and 5i  2j  4k, then the direction cosine of
[MP PET 1993] AB along y-axis is
(a) 15 (b) 15  [MNR 1989]
157 157
4 5
(c) 15  157 (d) 15  157 (a) (b) 
162 162
12. The position vectors of two points A and B are
(c) – 5 (d) 11
i j k and 2i  j  k respectively. Then
22. If the resultant of two forces is of magnitude P and
| AB |  [BIT Ranchi 1992] equal to one of them and perpendicular to it, then
(a) 2 (b) 3 the other force is
(c) 4 (d) 5 [MNR 1986]
13. The magnitudes of mutually perpendicular forces (a) P 2 (b) P
a, b and c are 2, 10 and 11 respectively. Then the (c) P 3 (d) None of these
magnitude of its resultant is 23. The direction cosines of vector a  3i  4j  5k in
[IIT 1984]
the direction of positive axis of x, is
(a) 12 (b) 15 [MP PET 1991]
(c) 9 (d) None 3 4
14. The system of vectors i, j, k is (a)  (b)
50 50
(a) Orthogonal (b) Coplanar
3 4
(c) Collinear (d) None of these (c) (d) 
15. The direction cosines of the resultant of the 50 50
vectors (i  j  k), (i  j  k), (i  j  k) and 24. The point having position vectors 2i  3j  4k,
(i  j  k), are 3i  4j  2k, 4i  2j  3k are the vertices of
[EAMCET 1988]
 1 1 1   1 1 1 
(a)  , , (b)  , ,  (a) Right angled triangle (b) Isosceles triangle
 2 3 6   6 6 6  (c) Equilateral triangle (d) Collinear
 25. Let  ,  ,  be distinct real numbers. The points
1 1 1   1 1 1 
(c)   , ,  (d)  , , with position vectors
 6 6 6  3 3 3  i  j  k, i  j  k, i  j  k
16. The position vectors of P and Q are 5i  4j  ak [IIT Screening 1994]
and i  2j  2k respectively. If the distance (a) Are collinear
between them is 7, then the value of a will be (b) Form an equilateral triangle
(a) – 5, 1 (b) 5, 1 (c) Form a scalene triangle
(c) 0, 5 (d) 1, 0 (d) Form a right angled triangle
17. A zero vector has 26. If | a|  3, | b |  4 and | a  b |  5, then
(a) Any direction (b) No direction | a  b| 
(c) Many directions (d) None of these [EAMCET 1994]
(a) 6 (b) 5
 (c) 4 (d) 3
18. A unit vector a makes an angle with z-axis. If
4 27. If OP = 8 and OP makes angles 45o and 60o
a  i  j is a unit vector, then a is equal to with OX-axis and OY-axis respectively, then
[IIT 1988]
OP 
i j k i j k
(a)   (b)   (a) 8 ( 2i  j  k) (b) 4 ( 2i  j  k)
2 2 2 2 2 2
1 1
i j k (c) ( 2i  j  k) (d) ( 2i  j  k)
(c)    (d) None of these 4 8
2 2 2 28. If a and b are two non-zero and non-collinear
19. A force is a vectors, then a + b and a – b are
[MP PET 1997]
(a) Unit vector (b) Localised vector
(a) Linearly dependent vectors
(c) Zero vector (d) Free vector
(b) Linearly independent vectors
(c) Linearly dependent and independent vectors
 9 Vector Algebra
(d) None of these 37. If the position vectors of the vertices A, B, C of a
29. If the vectors 6i  2j  3k, 2i  3j  6k and triangle ABC are 7j  10k, i  6j  6k and
3i  6j  2k form a triangle, then it is 4i  9j  6k respectively, the triangle is
[Karnataka CET 1999] [UPSEAT 2004]
(a) Right angled (b) Obtuse angled (a) Equilateral
(c) Equilteral (d) Isosceles (b) Isosceles
30. If the resultant of two forces of magnitudes P and (c) Scalene
Q acting at a point at an angle of 60o is 7Q, (d) Right angled and isosceles also
then P/Q is 38. The figure formed by the four points
[Roorkee 1999] i  j  k, 2i  3j, 3i  5j  2k and k  j is
3 [MP PET 2004]
(a) 1 (b) (a) Rectangle (b) Parallelogram
2
(c) Trapezium (d) None of these
(c) 2 (d) 4
39. ABC is an isosceles triangle right angled at A.
31. The direction cosines of the vector 3i  4j  5k
Forces of magnitude 2 2, 5 and 6 act along
are
[Karnataka CET 2000] BC, CA and AB respectively. The magnitude
of their resultant force is
3 4 1 3 4 1
(a) , , (b) , , [Roorkee 1999]
5 5 5 5 2 5 2 2 (a) 4 (b) 5
3 4 1 3 4 1 (c) 11 2 2 (d) 30
(c) , , (d) , ,
2 2 2 5 2 5 2 2 40. If ABCDEF is a regular hexagon and
32. The position vectors of A and B are 2i  9j  4k AB  AC  AD  AE  AF   AD, then  
[RPET 1985]
and 6i  3j  8k respectively, then the
(a) 2 (b) 3
magnitude of AB is
(c) 4 (d) 6
[MP PET 2000]
41. If P and Q be the middle points of the sides BC and
(a) 11 (b) 12
CD of the parallelogram ABCD, then AP  AQ 
(c) 13 (d) 14
33. If the position vectors of
P and Q are 1
(a) AC (b) AC
(i  3j  7k) and (5i  2j  4k), then | PQ | 2
is [MP PET 2001, 03] 2 3
(c) AC (d) AC
(a) 158 (b) 160 3 2
(c) 161 (d) 162 42. P is a point on the side BC of the  ABC and Q
34. If a is non zero vector of modulus a and m is a is a point such that PQ is the resultant of
non-zero scalar, then ma is a unit vector if AP, PB, PC. Then ABQC is a
[MP PET 2002]
(a) Square (b) Rectangle
(a) m  1 (b) m  | a|
(c) Parallelogram (d) Trapezium
1 43. In the figure, a vector x satisfies the equation
(c) m  (d) m   2
| a| x  w  v . Then x = A
35. The position vectors of the points A, B, C are
(2i  j  k), (3i  2j  k) and (i  4j  3k) a
c
respectively. These points b
[Kurukshetra CEE
2002] B C
w D v
(a) Form an isosceles triangle
(b) Form a right-angled triangle (a) 2a  b  c (b) a  2b  c
(c) Are collinear
(c) a  b  2c (d) a b c
(d) Form a scalene triangle
44. A vector coplanar with the non-collinear vectors a
36. The vectors AB  3i  4k,and
and b is
AC  5i  2j  4k are the sides of a triangle
ABC. The length of the median through A is
(a) a b (b) a  b
[AIEEE 2003] (c) a. b (d) None of these
(a) 18 (b) 72
(c) 33 (d) 288
 Vector Algebra 10
45. If ABCD is a parallelogram, AB  2 i  4 j  5 k 53. In a trapezium, the vector BC   AD. We will
and AD  i  2 j  3 k, then the unit vector in then find that p  AC  BD is collinear with
the direction of BD is [Roorkee 1976] AD, If p   AD, then
1 1 (a)     1 (b)     1
(a) (i  2j  8k) (b) (i  2j  8 k)
69 69 (c)     1 (d)   2  
1 54. If a  2i  j  8k and b  i  3j  4k, then the
(c) (i  2j  8k) (d)
69 magnitude of a  b  [MP PET 1996]
1 13
(i  2j  8 k) (a) 13 (b)
69 3
46. If a, b and c be three non-zero vectors, no two of
3 4
which are collinear. If the vector a  2b is (c) (d)
collinear with c and b  3c is collinear with a, 13 13
then (  being some non-zero scalar) a  2b  6c 55. A, B, C, D, E are five coplanar points, then
is equal to [AIEEE 2004] DA  DB  DC  AE  BE  CE is equal to [RPET
1999]
(a) a (b) b
(a) DE (b) 3 DE
(c) c (d) 0
(c) 2 DE (d) 4 ED
47. If a  2i  5j and b  2i  j, then the unit
56. If a  3i  2j  k, b  2i  4j  3k and
vector along a  b will be
[RPET 1985, 95] c  i  2j  2k, then a  b  c is
[MP PET 2001]
ij
(a) (b) i  j (a) 3i  4j (b) 3i  4j
2
(c) 4i  4j (d) 4i  4j
ij
(c) 2 (i  j) (d) 57. Five points given by A, B, C, D, E are in a plane.
2 Three forces AC, AD and AE act at A and
48. What should be added in vector a  3i  4j  2k three forces CB, DB, EB act at B. Then their
to get its resultant a unit vector i resultant is [AMU 2001]
[Roorkee 1977]
(a) 2AC (b) 3AB
(a)  2i  4j  2k (b) 2i  4j  2k
(c) 3DB (d) 2BC
(c) 2i  4j  2k (d) None of these
58. The sum of two forces is 18 N and resultant whose
49. If a  i  2j  3k, b  i  2j  k and c  3i  j, direction is at right angles to the smaller force is
then the unit vector along its resultant is 12N. The magnitude of the two forces are
[Roorkee 1980] [AIEEE 2002]
3i  5j  4k (a) 13, 5 (b) 12, 6
(a) 3i  5j  4k (b)
50 (c) 14, 4 (d) 11, 7
59. The unit vector parallel to the resultant vector of
3i  5j  4k
(c) (d) None of these 2i  4j  5k and i  2j  3k is
5 2 [MP PET 2003]
50. In a regular hexagon ABCDEF, AE  [MNR 1984] 1 i jk
(a) (3i  6j  2k) (b)
(a) AC  AF  AB (b) AC  AF  AB 7 3
(c) AC  AB  AF (d) None of these i  j  2k 1
(c) (d) (i  j  8k)
51. 3 OD  DA  DB  DC  6 69
[IIT 1988] 60. If a, b, c are the position vectors of the vertices A,
(a) OA  OB  OC (b) OA  OB  BD B, C of the triangle ABC, then the centroid of
 ABC is
(c) OA  OB  OC (d) None of these
[MP PET 1987]
52. p  2a  3b, q  a  2b  c, r  3a  b  2c;
where a, b and c being non-zero, non-coplanar a b c 1 b c
(a) (b) a 
vectors, then the vector 2a  3b  c is equal to 3 2 2 
7q  r b c a b c
(a) p  4q (b) (c) a  (d)
5 2 2
(c) 2p  3q  r (d) 4p  2r
 11 Vector Algebra
61. If in the given figure OA  a, OB  b and 69. If the position vector of one end of the line
AP : PB  m : n, then OP  [RPET 1981; segment AB be 2i  3j  k and the position
MP PET 1988] vector of its middle point be 3(i  j  k), then
A P B the position vector of the other end is
(a) 4i  3j  5k (b) 4i  3j  7k
(c) 4i  3j  7k (d) 4i  3j  7k
70. If G and G' be the centroids of the triangles ABC
and A' B' C' respectively, then
ma  n b O n a  mb
(a) (b) AA' BB'  CC ' 
m n m n 2
(a) GG' (b) GG'
ma  n b 3
(c) ma  n b (d)
m n (c) 2GG' (d) 3GG'
62. If D, E, F be the middle points of the sides BC, CA 71. If O be the circumcentre and O' be the orthocentre
and AB of the triangle ABC, then AD  BE  CF of the triangle ABC, then O' A  O' B  O' C 
is (a) OO' (b) 2 O' O
(a) A zero vector (b) A unit vector
(c) 2OO' (d) 0
(c) 0 (d) None of these
63. If a and b are the position vectors of A and B 72. If the vectors represented by the sides AB and BC
respectively, then the position vector of a point C of the regular hexagon ABCDEF be a and b, then
on AB produced such that AC  3AB is the vector represented by AE will be
[MNR 1980; MP PET 1995, 99] (a) 2 b  a (b) b a
(a) 3a  b (b) 3b  a (c) 2 a  b (d) a b
(c) 3a  2b (d) 3b  2a 73. The position vector of a point C with respect to B is
64. The position vectors of A and B are i  j  2k and i  j and that of B with respect to A is i  j. The
3i  j  3k. The position vector of the middle position vector of C with respect to A is
[MP PET 1989]
point of the line AB is [MP PET 1988]
(a) 2 i (b) 2 j
1 1 5 (c) – 2 j (d) – 2 i
(a) i jk (b) 2i  j  k
2 2 2 74. A and B are two points. The position vector of A is
3 1 3 6b  2a. A point P divides the line AB in the ratio
(c) i j k (d) None of these 1 : 2. If a  b is the position vector of P, then the
2 2 2
position vector of B is given by
65. If ABCD is a parallelogram and the position vectors [MP PET 1993]
of A, B, C are i  3j  5k, i  j  k and
(a) 7a  15b (b) 7a  15b
7i  7j  7k, then the position vector of D will be
(c) 15a  7b (d) 15a  7b
(a) 7i  5j  3k (b) 7i  9j  11k
75. If the position vectors of the points A and B are
(c) 9i  11j  13k (d) 8i  8j  8k i  3j  k and 3i  j  3k, then what will
66. P is the point of intersection of the diagonals of be the position vector of the mid point of AB
the parallelogram ABCD. If O is any point, then [MP PET 1992]
OA  OB  OC  OD  [RPET 1989; J & K (a) i  2j  k (b) 2i  j  2k
2005]
(c) 2i  j  k (d) i  j  2k
(a) OP (b) 2 OP
76. If C is the middle point of AB and P is any point
(c) 3 OP (d) 4 OP outside AB, then
67. If the position vectors of the point A, B, C be i, j, k [MNR 1991; UPSEAT 2000; AIEEE 2005]
respectively and P be a point such that (a) PA  PB  PC (b) PA  PB  2 PC
AB  CP , then the position vector of P is
(c) PA  PB  PC  0 (d) PA  PB  2 PC  0
(a) i  j  k (b) i  j  k
77. If in a triangle AB  a, AC  b and D, E are the
(c) i  j  k (d) None of these
mid-points of AB and AC respectively, then DE is
68. If the position vectors of the points A, B, C, D be equal to
2i  3j  5k, i  2j  3k,  5i  4j  2k and [RPET 1986]
i  10j  10k respectively, then
a b a b
[MNR 1982] (a)  (b) 
(a) AB  CD (b) AB | | CD
4 4 2 2
(c) AB  CD (d) None of these
 Vector Algebra 12
b a b a (a)  i  j  k (b) i  j  k
(c)  (d) 
4 4 2 2 (c) i  j  k (d) i  j  k
78. In the triangle ABC, AB  a, AC  c, BC  b , 86. If a and b are P.V. of two points A and B and C
then divides AB in ratio 2 : 1, then P.V. of C is
[RPET 1996]
[RPET 1984]
(a) a b c  0 (b) a b c  0 a  2b 2a  b
(a) (b)
(c) a  b  c  0 (d)  a  b  c  0 3 3
79. ABCDE is a pentagon. Forces a 2 a b
(c) (d)
AB, AE, DC, ED act at a point. Which force 3 2
should be added to this system to make the 87. If A, B, C are the vertices of a triangle whose
position vectors are a, b, c and G is the centroid of
resultant 2 AC [MNR 1984]
the ABC, then GA  GB  GC is
[Karnataka CET 2000]
(a) AC (b) AD
(a) 0 (b) A  B  C
(c) BC (d) BD
80. Let A and B be points with position vectors a and a b c a b c
(c) (d)
b with respect to the origin O. If the point C on OA 3 3
is such that 2AC  CO, CD is parallel to OB 88. If O is origin and C is the mid point of A(2,  1)
and | CD |  3| OB |, then AD is equal to
and B(4, 3) . Then value of OC is
a a [RPET 2001]
(a) 3b  (b) 3b 
2 2 (a) i + j (b) i – j
(c) – i + j (d) – i – j
a a
(c) 3b  (d) 3b  89. If ABCDEF is regular hexagon, then
3 3
AD  EB  FC 
81. In a
triangle ABC, if 2AC  3CB, then
[Karnataka CET
2OA  3OB equals 2002]
[IIT 1988; Pb. CET 2003] (a) 0 (b) 2AB
(c) 3AB (d) 4 AB
(a) 5OC (b)  OC 90. If position vectors of a point A is a + 2b and a
(c) OC (d) None of these divides AB in the ratio 2 : 3 , then the position
vector of B is [MP PET 2002]
82. If AO  OB  BO  OC, then A, B, C form [IIT
1983] (a) 2a – b (b) b – 2a
(a) Equilateral triangle (b) Right angled triangle (c) a – 3b (d) b
(c) Isosceles triangle (d) Line 91. If D, E, F are respectively the mid points of
83. The sum of the three vectors determined by the AB, AC and BC in ABC , then BE
medians of a triangle directed from the vertices is  AF  [EAMCET 2003]
[MP PET 1997]
(a) 0 (b) 1 1
(a) DC (b) BF
1 2
(c) – 1 (d)
3 3
(c) 2BF (d) BF
84. The position vector of the points which divides 2
internally in the ratio 2 : 3 the join of the points 92. If 4i  7j  8k, 2i  3j  4k and 2i  5j  7k
2a  3b and 3a  2b, is are the position vectors of the vertices A, B and C
[AI CBSE 1985] respectively of triangle ABC. The position vector of
12 13 12 13 the point where the bisector of angle A meets BC
(a) a b (b) a b is
5 5 5 5
[Pb. CET 2004]
3 2 1 2
(c) a b (d) None of these (a) (6i  13j  18k) (b) (6i  12j  8k)
5 5 3 3
85. If position vector of points A, B, C are respectively
1
i, j, k and AB  CX , then position vector of (c) (6i  8j  9k) (d)
point X is
3
[MP PET 1994] 2
(6i  12j  8k)
3
 13 Vector Algebra
93. If a  i  j and b  i  k , then a unit vector 103. Three points whose position vectors are
coplanar with a and b and perpendicular to a is a  b, a  b and a  kb will be collinear, if the
(a) i (b) j value of k is [IIT 1984]
(c) k (d) None of these (a) Zero
94. If the position vectors of the points A, B, C be (b) Only negative real number
i  j, i  j and a i  b j  c k respectively, then (c) Only positive real number
the points A, B, C are collinear if (d) Every real number
(a) a  b  c  1 104. If the position vectors of A, B, C, D are 2 i  j,
(b) a  1, b and c are arbitrary scalars i  3 j, 3 i  2 j and i  j respectively and

(c) a  b  c  0 AB|| CD , then  will be

[RPET 1988]
(d) c  0, a  1 and b is arbitrary scalars
(a) – 8 (b) – 6
95. If the points a  b, a  b and a  k b be (c) 8 (d) 6
collinear, then k = 105. If the vectors 3 i  2 j  k and 6 i  4xj  yk are
(a) 0 (b) 2 parallel, then the value of x and y will be
(c) – 2 (d) Any real number [RPET 1985, 86]
96. If the position vectors of the points A, B, C be (a) – 1, – 2 (b) 1, – 2
a, b , 3a  2b respectively, then the points A, (c) – 1, 2 (d) 1, 2
B, C are 106. If (x, y, z)  (0, 0, 0) and
[MP PET 1989] (i  j  3 k) x  (3i  3j  k)y
(a) Collinear (b) Non-collinear (4i  5j) z   (xi  yj  zk), then the value
(c) Form a right angled triangle(d) None of these
of  will be [IIT 1982; RPET 1984]
97. If a, b, c are non-collinear vectors such that for
(a) – 2, 0 (b) 0, – 2
some scalars x, y, z, xa  yb  zc  0, then (c) – 1, 0 (d) 0, – 1
[RPET 2002]
107. The vectors a, b and a + b are
(a) x  0, y  0, z  0 (b) x  0, y  0, z  0 (a) Collinear (b) Coplanar
(c) x  0, y  0, z  0 (d) x  0, y  0, z  0 (c) Non-coplanar (d) None of these
98. The vectors 3 i  j  5 k and a i  b j  15k are 108. If a, b, c are the position vectors of three collinear
collinear, if points, then the existence of x, y, z is such that
[RPET 1986; MP PET 1988] (a) xa  yb  zc  0, x  y  z  0
(a) a  3, b  1 (b) a  9, b  1 (b) xa  yb  zc  0, x  y  z  0
(c) a  3, b  3 (d) a  9, b  3 (c) xa  yb  zc  0, x  y  z  0
99. The points position vectors 60i  3 j ,
with (d) xa  yb  zc  0, x  y  z  0
40i  8j, , a i  52j are collinear, if a  109. If a  (2, 5) and b  (1, 4), then the vector
[RPET 1991; IIT 1983; MP PET 2002]
parallel to (a  b) is
(a) – 40 (b) 40
(a) (3, 5) (b) (1, 1)
(c) 20 (d) None of these
(c) (1, 3) (d) (8, 5)
100. If O be the origin and the position vector of A be
4 i  5 j, then a unit vector parallel to OA is 110. The vectors a and b are non-collinear. The value of
x for which the vectors c  (x  2)a  b and
4 5
(a) i (b) i d  (2x  1)a  b are collinear, is
41 41
1
1 1 (a) 1 (b)
(c) (4 i  5 j) (d) (4 i  5 j) 2
41 41
1
101. If the position vectors of the points A and B be (c) (d) None of these
2 i  3 j  k and 2 i  3 j  4 k, then the line 3
AB is parallel to 111. The vectors i  2j  3k, i  4j  7k,
(a) xy-plane (b) yz-plane 3i  2j  5k are collinear, if  equals
(c) zx-plane (d) None of these [Kurukshetra CEE 1996]
(a) 3 (b) 4
102. The points with position vectors
10i  3 j, 12i  5 j a i  11j (c) 5 (d) 6
and are
112. The position vectors of four points P, Q, R, S are
collinear, if a  2a  4c, 5a  3 3 b  4c,  2 3b  c and
[MNR 1992; Kurukshetra CEE 2002]
(a) – 8 (b) 4
2a  c respectively, then
[MP PET 1997]
(c) 8 (d) 12
(a) PQ is parallel to RS
 Vector Algebra 14
(b) PQ is not parallel to RS (c) 0 (d) None of these
(c) PQ is equal to RS 2. If r. i  r. j  r. k and | r|  3, then r 
(d) PQ is parallel and equal to RS 1
(a)  3(i  j  k) (b)  (i  j  k)
113. If a  (1,  1) and b  ( 2, m) are two 3
collinear vectors, then m = 1
[MP PET 1998] (c)  (i  j  k) (d)  3 (i  j  k)
3
(a) 4 (b) 3
3. If a, b, c are non-zero vectors such that
(c) 2 (d) 0
a . b  a . c, then which statement is true
114. If three points A, B, C are collinear, whose position
[RPET 2001]
vectors are i  2j  8k, 5i  2k and
(a) b = c (b) a  (b  c)
11i  3 j  7k respectively, then the ratio in
which B divides AC is [RPET 1999] (c) b  c or a  (b  c) (d) None of these
(a) 1 : 2 (b) 2 : 3 4. If a and b be unlike vectors, then a . b =
(c) 2 : 1 (d) 1 : 1 (a) | a | | b | (b) – | a | | b |
(c) 0 (d) None of these
115. If a and b are two non-collinear vectors and
xa yb  0 5. If a, b, c are unit vectors such that a  b  c  0,
[RPET 2001] then a . b  b . c  c . a 
[MP PET 1988; Karnataka CET 2000; UPSEAT 2003, 04]
(a) x  0 , but y is not necessarily zero
(a) 1 (b) 3
(b) y  0 , but x is not necessarily zero (c) – 3/2 (d) 3/2
(c) x  0, y  0 6. If a, b, c are mutually perpendicular vectors of
(d) None of these equal magnitudes, then the angle between the
116. If three points A, B and C have position vectors vectors a and a  b  c is
(1, x, 3), (3, 4,7) and (y,  2,  5)  
(a) (b)
respectively and if they are collinear, then 3 6
(x, y)  [EAMCET 2002]
1 1 
(a) (2, – 3) (b) (– 2, 3) (c) cos (d)
3 2
(c) (2, 3) (d) (– 2, – 3)
7. If a, b, c are mutually perpendicular unit vectors,
117. a and b are two non-collinear vectors, then
then | a  b  c |  [Karnataka CET 2002, 05;
xa  yb (where x and y are scalars) represents a
J & K 2005]
vector which is
[MP PET 2003] (a) 3 (b) 3
(a) Parallel to b (b) Parallel to a (c) 1 (d) 0
(c) Coplanar with a and b (d) None of these 8. If | a|  | b | | c | and a  b  c, then the angle
118. If a, b, c are three non-coplanar vectors such that between a and b is
a b c   d and b  c  d   a, then 
(a) (b) 
a  b  c  d is equal to 2
(a) 0 (b) a (c) 0 (d) None of these
(c)  b (d) (   )c 9. If a has magnitude 5 and points north-east and
vector b has magnitude 5 and points north-west,
119. The value of k for which the vectors a  i  j and then | a  b | 
b  2i  k j are collinear is [MNR 1984]
[Pb. CET 2004] (a) 25 (b) 5
1 (c) 7 3 (d) 5 2
(a) 2 (b)
2 10. If  be the angle between the unit vectors a and b,
1 
(c) (d) 3 then cos  [MP PET 1998;
3 2
Pb. CET 2002]

Scalar or Dot product of two vectors and its 1 1

(a) | a  b| (b) | a  b|
applications 2 2
| a  b| | a  b|
(a. i)i  (a. j)j  (a. k)k  (c) (d)
1. | a  b| | a  b|
[Karnataka CET 2004]
(a) a (b) 2 a
 15 Vector Algebra
11. If | a|  3, | b|  4, | c |  5 and a  b  c  0, scalar product of F1  F2  F3 and AB will be
then the angle between a and b is [MP PET 1989; [Roorkee 1980]
Bihar CEE 1994] (a) 3 (b) 6
 (c) 9 (d) 12
(a) 0 (b)
6 20. If the moduli of a and b are equal and angle
between them is 120o and a. b   8, then | a
 
(c) (d) | is equal to [RPET 1986]
3 2 (a) – 5 (b) – 4
12. If | a  b|  | a  b|, then the angle between a (c) 4 (d) 5
and b is 21. If | a|  3, | b |  4 and the angle between a and
(a) Acute (b) Obtuse
b be 120o , then | 4a  3b| 

(c) (d)  (a) 25 (b) 12
2 (c) 13 (d) 7
13. If a, b, c are three vectors such that a  b  c and 22. A vector whose modulus is 51 and makes the
the angle between b and c is  / 2, then i  2j  2k  4i  3k
same angle with a  , b
[EAMCET 2003] 3 5
(a) a2  b2  c 2 (b) b2  c 2  a2 and c  j, will be
(c) c 2  a2  b2 (d) 2a2  b2  c 2 [Roorkee 1987]
(a) 5i  5j  k (b) 5i  j  5k
(Note : Here a  | a|, b | b|, c | c |)
(c) 5i  j  5k (d)  (5i  j  5k)
14. If the angle between the vectors a and b be  and 23. If a, b, c are coplanar vectors, then [IIT 1989]
a. b  cos , then the true statement is
a b c
(a) a and b
are equal vectors (a) b c a  0 (b)
(b) a and b
are like vectors c a b
(c) a and b
are unlike vectors
a b c
(d) a and b
are unit vectors
a. a a. b a. c  0
15. If the vector i  j  k makes angles  ,  ,  with
b. a b. b b. c
vectors i, j, k respectively, then
a b c
(a)      (b)     
(c) c.a c. b c.c  0 (d)
(c)      (d)     
b. a b. c b. b
16. (r . i)2  (r . j)2  (r . k)2 
a b c
(a) 3r 2 (b) r 2 a. b a. a a. c  0
(c) 0 (d) None of these c.a c.c c.b
17. The value of b such that scalar product of the 
24. If  is a unit vector perpendicular to plane of
vectors (i  j  k) with the unit vector parallel to vector a and b and angle between them is , then
the sum of the vectors (2i  4j  5k) and a . b will be
[RPET 1985]
(bi  2j  3k) is 1, is  
(a) | a | | b | sin  (b) | a| | b | cos 
[MNR 1992; Roorkee 1985, 95; Kurukshetra CEE 1998;
UPSEAT 2000] (c) | a| | b| cos (d) | a| | b| sin
(a) – 2 (b) – 1 25. If p  i  2j  3k and q  3i  j  2k, then a
(c) 0 (d) 1 vector along r which is linear combination of p
18. If a unit vector lies in yz–plane and makes angles and q and also perpendicular to q is
[MNR 1986]
of 30o and 60o with the positive y-axis and z-
(a) i  5j  4k (b) i  5j  4k
axis respectively, then its components along the
co-ordinate axes will be 1
(c)  (i  5j  4k) (d) None of these
3 1 3 1 2
(a) , ,0 (b) 0, ,
2 2 2 2 26. If d   (a  b)   (b  c)   (c  a) and

3 1 1 3 1
(c) , 0, (d) 0, , [a b c]  , then      is equal to
2 2 2 2 8
19. If F1  i  j  k, F2  i  2j  k, F3  j  k, (a) 8d.(a  b  c) (b) 8d (a  b  c)
 
A  4i  3j  2k and B  6i  j  3k, then the
 Vector Algebra 16
d d (a) a  (a . i) i  (a . j) j  (a . k) k
(c) .(a  b  c) (d)  (a  b  c)
8 8 (b) a  (a  i)  (a  j)  (a  k)
27. The horizontal force and the force inclined at an (c) a  j (a . i)  k (a . j)  i (a . k)
angle 60o with the vertical, whose resultant is in
(d) a  (a  i) i  (a  j) j  (a  k) k
vertical direction of P kg, are
[IIT 1983] 36. If vectors a, b, c satisfy the condition
(a) P, 2P (b) P, P 3  a b
| a  c || b  c | , then (b  a).  c   is
(c) 2P , P 3 (d) None of these  2 
28. If a and b are mutually perpendicular vectors, equal to [AMU 1999]
then (a  b)2  [MP PET 1994; Pb. CET 2002] (a) 0 (b) –1
(c) 1 (d) 2
(a) a b (b) a b
37. (a .b) c and (a.c) b are [RPET 2000]
(c) a2  b2 (d) (a  b)2 (a) Two like vectors
29. a. b  0, then [RPET 1995] (b) Two equal vectors
(a) a  b (c) Two vectors in direction of a
(d) None of these
(b) a || b
38. If a  (1,  1, 2), b  (2, 3, 5) ,
(c) Angle between a and b is 60o
c  (2,  2, 4) and i is the unit vector in the x-
(d) None of these
direction, then (a  2b  3c). i 
30. If | a|  3, | b |  1, | c |  4 and a  b  c  0,
[Karnataka CET 2001]
then a. b  b . c  c . a  [MP PET 1995; RPET
(a) 11 (b) 15
2000]
(a) – 13 (b) – 10 (c) 18 (d) 36
(c) 13 (d) 10 39. For any three non-zero vectors r1, r2 and r3 ,
31. If ABCDEF is regular hexagon, the length of whose r1 . r1 r1 . r2 r1 . r3
1 2
side is a, then AB . AF  BC  r2 . r1 r2 . r2 r2 . r3  0 . Then which of
2
r3 . r1 r3 . r2 r3 . r3
(a) a (b) a2
the following is false [AMU 2000]
(c) 2a2 (d) 0 (a) All the three vectors are parallel to one and
32. If in a right angled triangle ABC, the hypotenuse the same plane
AB  p, then AB . AC  BC . BA  CA . CB (b) All the three vectors are linearly dependent
is equal to (c) This system of equation has a non-trivial
solution
p2
(a) 2p2 (b) (d) All the three vectors are perpendicular to each
2 other
(c) p2 (d) None of these 40. Let a, b and c be vectors with magnitudes 3, 4
and 5 respectively and a + b + c = 0, then the
33. A, B, C, D are any four points, then
values of a.b + b.c + c.a is
AB . CD  BC . AD  CA . BD  [MNR [IIT 1995; DCE 2001; AIEEE 2002; UPSEAT 2002;
1986] Kerala (Engg.) 2005]
(a) 2 AB . BC . CD (b) AB  BC  CD (a) 47 (b) 25
(c) 5 3 (d) 0 (c) 50 (d) – 25
41. If a and b are adjacent sides of a rhombus, then
34. The vector a coplanar with the vectors i and j, [RPET 2001]
perpendicular to the vector b  4i  3j  5k such (a) a.b = 0 (b) a × b = 0
that | a| | b | is (c) a.a = b.b (d) None of these
(a) 2 (3i  4j) or  2 (3i  4j) 42. If x and y are two unit vectors and  is the angle
(b) 2 (4i  3j) or  2 (4i  3j) 1
between them, then | x  y| is equal to
2
(c) 3 (4i  5j) or  3 (4i  5j)
[UPSEAT 2001]
(d) 3 (5i  4j) or  3 (5i  4j) (a) 0 (b)  / 2
35. If a is any vector in space, then (c) 1 (d)  / 4
[MP PET 1997]
 17 Vector Algebra
43. If a. i  a.(i  j)  a.(i  j  k) , then a = then the angle between the vectors AB and
[EAMCET 2002] CD is
(a) i (b) k  
(c) j (d) i + j + k (a) (b)
4 3
44. If i, j, k are unit vectors, then

[MP PET 2001]
(c) (d) 
(a) i . j 1 (b) i . i 1 2
(c) i  j  1 (d) i  (j  k)  1 52. If  be the angle between the unit vectors a and b,
then a  2 b will be a unit vector if  
45. If | a| | b|, then (a  b) . (a  b) is [MP PET
2002]
 
(a) (b)
(a) Positive (b) Negative 6 4
(c) Zero (d) None of these  2
(c) (d)
46. a,b,c are three vectors, such that a  b  c  0 , 3
3
| a|  1,| b|  2,| c|  3 , then a.b b.c  c.a is 53. If the angle between a and b be 30o , then the
equal to angle between 3 a and – 4 b will be
[AIEEE 2003]
(a) 0 (b) – 7 (a) 150o (b) 90o
(c) 7 (d) 1 (c) 120o (d) 30o
47. A unit vector which is coplanar to vector
54. The angle between the vectors i j k and
i  j  2k and i  2j  k and perpendicular to
i  2j  k is
i  j  k, is
[BIT Ranchi 1991]
[IIT 1992; Kurukshetra CEE 2002]
ij  j k 1  1  1  4 
(b)    (a) cos  (b) cos 
15  15 
(a) 
2  2   
ki i j k 1  4  
(c) (d) (c) cos   (d)
2 3  15  2
48. If | a |  3, | b |  4 then a value of  for which 55. The position vector of vertices of a triangle ABC
are 4i  2j, i  4j  3k and i  5j  k
a  b is perpendicular to a  b is
[Karnataka CET 2004] respectively, then ABC 
(a)
9
(b) [RPET 1988, 97]
16

3 (a)  / 6 (b)  / 4
4 (c)  / 3 (d)  / 2
3 4 56. The value of x for which the angle between the
(c) (d) vectors a  xi  3j  k, b  2xi  xj  k is acute
2 3
and the angle between the vectors b and the axis
49. a, b and c are three vectors with magnitude of ordinate is obtuse, are
| a|  4, | b|  4, | c |  2 and such that a is (a) 1, 2 (b) – 2, – 3
perpendicular to (b  c), b is perpendicular to (c) x > 0 (d) None of these
(c  a) and c is perpendicular to (a  b). It 57. If a and b are unit vectors and a  b is also a unit
follows that | a  b  c | is equal to vector, then the angle between a and b is
[UPSEAT 2004] [RPET 1991; MP PET 1995; Pb. CET 2001]
(a) 9 (b) 6  
(c) 5 (d) 4 (a) (b)
4 3
50. The angle between the vectors 3 i  j  2 k and
 2
2 i  2 j  4 k is [MP PET 1990] (c) (d)
2 3
1 2 1 2
(a) cos (b) sin 58. If  be the angle between two vectors a and b,
7 7 then a.b  0 if
1 2 1 2 [MP PET 1995]
(c) cos (d) sin
5 5 
(a) 0     (b)  
51. If the position vectors of the points A, B, C, D be 2
i  j  k, 2 i  5 j, 3 i  2 j  3k and i  6 j  k,

(c) 0    (d) None of these
2
 Vector Algebra 18
59. If a  i  2j  3k and b  3i  j  2k, then the 67. If  be the angle between the vectors
angle between the vectors a  b and a  b is a  2i  2j  k and b  6i  3j  2k , then
[Karnataka CET 1994; Orissa JEE 2005] [MP PET 2001, 03]

(a) 30 o (b) 60 o 4 3
(a) cos  (b) cos 
21 19
(c) 90o (d) 0o
2 5
60. The value of x for which the angle between the (c) cos  (d) cos 
vectors a   3i  xj  k and b  xi  2xj  k is 19 21
acute and the angle between b and x-axis lies 68. If a and b are two unit vectors such that a  2 b
between  / 2 and  satisfy and 5a  4b are perpendicular to each other,
[Kurukshetra CEE then the angle between a and b is
1996] [IIT Screening 2002]
(a) x0 (b) x0 (a) 45 o (b) 60o
(c) x  1 only (d) x  1 only
1  1 1  2
61. The angle between the vectors (2i  6j  3k) (c) cos   (d) cos  
 3  7
and (12i  4j  3k) is [MP PET 1996]
69. Let a and b be two unit vectors inclined at an
1  1  1  9  angle  , then sin( / 2) is equal to
(a) cos   (b) cos  
 10   11 [BIT Ranchi 1991; Karnataka CET 2000, 01;
UPSEAT 2002]
1  9  1  1
(c) cos   (d) cos   1 1
 91  9 (a) | a  b| (b) | a  b|
2 2
62. If the angle between two vectors i  k and (c) | a  b| (d) | a  b|
i  j  ak is  / 3, then the value of a 
70. The angle between the vectors a + b and a – b,
[MP PET 1997]
when a  (1,1, 4) and b  (1,  1, 4) is
(a) 2 (b) 4
[Karnataka CET 2003]
(c) – 2 (d) 0
(a) 90o (b) 45o
63. If three vectors a, b, c satisfy a  b  c  0 and
| a|  3, | b|  5, | c |  7, then the angle (c) 30o (d) 15o
between a and b is 71. A vector of length 3 perpendicular to each of the
[Kurukshetra CEE 1998; UPSEAT 2001; vectors 3 i  j  4 k and 6 i  5 j  2 k is
AIEEE 2002; MP PET 2002]
(a) 2 i  2 j  k (b)  2 i  2 j  k
(a) 30o (b) 45o
(c) 2 i  2 j  k (d) None of these
(c) 60o (d) 90o
72. If a  0, b  0 and | a  b| | a  b|,then the
64. If a, b and c are unit vectors such that vectors a and b are [Roorkee 1986; MNR 1988; IIT
a  b  c  0, then the angle between a and b is Screening 1989;
[Roorkee Qualifying 1998; MP PET 1999; MP PET 1990, 97; RPET 1984, 90, 96, 99; KCET 1999]
UPSEAT 2000; RPET 2002] (a) Parallel to each other
(a)  / 6 (b)  / 3 (b) Perpendicular to each other
(c)  / 2 (d) 2 / 3 (c) Inclined at an angle of 60o
65. If the sum of two unit vectors is a unit vector, then (d) Neither perpendicular nor parallel
the magnitude of their difference is
73. The vector 2 i  a j  k is perpendicular to the
[Kurukshetra CEE 1996; RPET 1996]
vector 2 i  j  k, if a 
(a) 2 (b) 3 [MP PET 1987]
1 (a) 5 (b) – 5
(c) (d) 1
3 (c) – 3 (d) 3
74. If a  2 i  2 j  3 k, b  i  2 j  k and
66. The angle between the vector 2i  3j  k and
c  3 i  j, then a  t b is perpendicular to c if
2i  j  k is
[MNR 1990; UPSEAT 2000] t
[MNR 1979; MP PET 2002]
(a)  / 2 (b)  / 4
(a) 2 (b) 4
(c)  / 3 (d) 0
(c) 6 (d) 8
 19 Vector Algebra
75. The vector 2i  j  k is perpendicular to 1
84. The vector (2i  2j  k) is
i  4j  k, if   3
[MNR 1983; MP PET 1988] [IIT Screening 1994]
(a) 0 (b) – 1 (a) A unit vector
(c) – 2 (d) – 3 
(b) Makes an angle with the vector
76. The vectors 2 i  3 j  4 k and a i  b j  c k 3
are perpendicular, when 2i  4j  3k
[MNR 1982; MP PET 1988; MP PET 2002]
1
(a) a  2, b  3, c  4 (b) (c) Parallel to the vector  i  j  k
2
a  4, b  4, c  5
(d) Perpendicular to the vector 3i  2j  2k
(c) a  4, b  4, c   5 (d) None of these
85. If the vectors ai  2j  3k and i  5j  ak are
77. A unit vector in the xy  plane which is perpendicular to each other, then a  [MP PET
perpendicular to 4i  3j  k is 1996]
[RPET 1991] (a) 6 (b) – 6
ij 1 (c) 5 (d) – 5
(a) (b) (3i  4j) 86. Which of the following is a true statement
2 5 [Kurukshetra CEE 1996]
1 (a) (a  b)  c is coplanar with c
(c) (3i  4j) (d) None of these
5 (b) (a  b)  c is perpendicular to a
78. If l a  mb  n c  0, where l, m, n are scalars (c) (a  b)  c is perpendicular to b
and a, b, c are mutually perpendicular vectors, (d) (a  b)  c is perpendicular to c
then
87. If a  i  2j and b  2i  j are parallel, then 
(a) l  m n  1 (b) l  m n  1 is
(c) l  m  n  0 (d) l  0, m  0, n  0 [RPET 1996]
(a) 4 (b) 2
79. The unit normal vector to the line joining i  j
(c) – 2 (d) – 4
and 2 i  3 j and pointing towards the origin is
88. If ai  6j  k and 7i  3j  17k are
[MP PET 1989]
perpendicular vectors, then the value of a is
4i  j 4 i  j [Karnataka CET 2001]
(a) (b) (a) 5 (b) – 5
17 17
1
2i  3 j  2i  3 j (c) 7 (d)
(c) (d) 7
13 13 89. If 4i  j  k and 3i  mj  2k are at right angle,
80. If the vectors a i  2j  3k and 3i  6j  5k are then m 
perpendicular to each other, then a is given by [MP [Karnataka CET
PET 1993] 2002]
(a) 9 (b) 16 (a) – 6 (b) – 8
(c) – 10 (d) – 12
(c) 25 (d) 36
90. If the vectors 3i   j  k and 2i  j  8k are
81. The value of  for which the vectors 2i  j  k
perpendicular, then  is
and 2j  k are perpendicular, is [Kerala (Engg.) 2002]
[MP PET 1992] (a) – 14 (b) 7
(a) None (b) – 1 (c) 14 (d) 1/7
(c) 1 (d) Any value 91. If a and b are two non-zero vectors, then the
component of b along a is
82. If the vectors ai  bj  ck and pi  qj  rk are [MP PET 1991]
perpendicular, then [RPET 1989] (a. b) a (a . b) b
(a) (a  b  c)(p  q  r)  0 (b) (a) (b)
b. b a. a
(a  b  c)(p  q  r)  1 (a . b) b (a. b) a
(c) (d)
(c) ap  bq  cr  0 (d) ap  bq  cr  1 a. b a. a
83. If a  2i  4j  2k and b  8i  3j  k and 92. A vector of magnitude 14 lies in the xy-plane and
a  b, then value of  will be makes an angle of 60o with x-axis. The
[RPET 1995] components of the vector in the direction of x-axis
(a) 2 (b) – 1 and y-axis are
(c) – 2 (d) 1 (a) 7, 7 3 (b) 7 3, 7
 Vector Algebra 20
(c) 14 3, 14 / 3 (d) 14 / 3, 14 3 (c) 5 (d) 6
93. If a  4i  6j and b  3 j  4 k, then the 100. The projection of the vector i  2j  k on the
component of a along b is vector 4i  4j  7k is [RPET 1990; MNR
[IIT Screening 1989; 1980; MP PET 2002;
MNR 1983, 87; UPSEAT 2000] UPSEAT 2002; Pb. CET 2004]
18 18 5 6 19
(a) (3j  4k) (b) (3j  4k) (a) (b)
10 3 25 10 9
18 9 6
(c) (3j  4k) (d) (3j  4k) (c) (d)
3 19 19
94. Let b  3j  4k, a  i  j and let b1 and b2 be
component vectors of b parallel and perpendicular
3 3
to a. If b1  i  j , then b2 
2 2
[MP PET 1989]
3 3 3 3
(a) i  j  4k (b)  i  j  4k
2 2 2 2
3 3
(c)  i  j (d) None of these
2 2
95. The component of i  j along j  k will be
ij j k
(a) (b)
2 2
ki
(c) (d) None of these
2
96. The projection of vector 2i  3j  2k on the
vector i  2j  3k will be
[RPET 1984, 90, 97, 99; Karnataka CET 2004]
1 2
(a) (b)
14 14
3
(c) (d) 14
14
97. If vector a  2i  3j  6k and vector
b  2i  2j  k, then
Projectionof vector
a on vectorb

Projectionof vector
b on vectora
[MP PET 1994, 99; Pb. CET 2000]
3 7
(a) (b)
7 3
(c) 3 (d) 7
98. The projection of a along b is
[RPET 1995]
a. b a b
(a) (b)
| a| | a|
a. b a b
(c) (d)
| b| | b|
99. If a  2i  j  2k and b  5i  3j  k, then the
projection of b on a is [Karnataka CET
2002]
(a) 3 (b) 4