Vector Algebra

Topic : Addition of Vectors

1. P is the point of intersection of the diagonals of the parallelogram ABCD. If O is any point, then OA + OB + OC + OD =

(a) OP (b) 2OP (c) 3OP (d) 4 OP

2. If p = 7 i − 2 j + 3 k and q = 3 i + j + 5 k , then the magnitude of p – 2q is

(a) 29 (b) 4 (c) 62 − 2 35 (d) 66

3. If C is the middle point of AB and P is any point outside AB, then

(a) PA + PB = PC (b) PA + PB = 2 PC (c) PA + PB + PC = 0 (d) PA + PB + 2 PC = 0

4. If a = 2 i + 5 j and b = 2 i − j , then the unit vector along a + b will be

i−j i+j
(a) (b) i + j (c) 2 (i + j) (d)
2 2
5. What should be added in vector a = 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

6. If a = i + 2 j + 3 k , b = −i + 2 j + k and c = 3 i + j , then the unit vector along its resultant is

3i + 5 j + 4 k 3i + 5 j + 4 k
(a) 3 i + 5 j + 4 k (b) (c) (d) None of these
50 5 2

7. In the triangle ABC, AB = a, AC = c , BC = b , then

(a) a + b + c = 0 (b) a + b − c = 0 (c) a − b + c = 0 (d) −a + b + c = 0

8. If a has magnitude 5 and points north-east and vector b has magnitude 5 and points north-west, then | a − b | =

(a) 25 (b) 5 (c) 7 3 (d) 5 2

9. If | a | = 3 , | b | = 4 and | a + b | = 5 , then | a − b | =

(a) 6 (b) 5 (c) 4 (d) 3

10. If the sum of two unit vectors is a unit vector, then the angle between them is equal to
   2
(a) (b) (c) (d)
6 3 2 3

11. A, B, C, D, E are five coplanar points, then DA + DB + DC + AE + BE + CE is equal to

(a) DE (b) 3 DE (c) 2 DE (d) 4 ED

12. If a  0, b  0 and | a + b | =| a − b | , then the vectors a and b are

(a) Parallel to each other (b) Perpendicular to each other

(c) Inclined at an angle of 60o (d) Neither perpendicular nor parallel

13. If ABCDEF is a regular hexagon and AB + AC + AD + AE + AF =  AD ,then  =

(a) 2 (b) 3 (c) 4 (d) 6

14. If O be the circumcentre and O' be the orthocentre of a triangle ABC, then OA + OB + OC =

(a) 2OO ' (b) 2O' O (c) OO' (d) O' O

15. Let a = i be a vector which makes an angle of 120o with a unit vector b. Then the unit vector (a + b) is

PRATAP BHAWAN, BEHIND LEELA CINEMA, HAZRATGANJ, LUCKNOW.

PH.(0522)4026913, 9838162263. e-mail. id: inpsclasses@gmail.com. www.inpsmcalucknow.com
Vector Algebra

1 3 3 1 1 3 3 1
(a) − i+ j (b) − i+ j (c) i+ j (d) i− j
2 2 2 2 2 2 2 2

16. If  be the angle between the unit vectors a and b, then cos =
2
1 1 | a − b| | a + b|
(a) | a − b| (b) | a + b| (c) (d)
2 2 | a + b| | a − b|

17. If | a | = 3 , | b | = 4, | c | = 5 and a + b + c = 0 , then the angle between a and b is

  
(a) 0 (b) (c) (d)
6 3 2

18. If ABCD is a parallelogram, AB = 2i + 4 j − 5 k and AD = i + 2 j + 3 k , then the unit vector in the direction of BD is
1 1 1 1
(a) (i + 2 j − 8 k ) (b) (i + 2 j − 8 k ) (c) (−i − 2 j + 8 k) (d) (−i − 2 j + 8 k )
69 69 69 69

19. If a and b are unit vectors making an angle  with each other then | a − b | is

 
(a) 1 (b) 0 (c) cos (d) 2 sin
2 2
20. If the moduli of the vectors a, b, c are 3, 4, 5 respectively and a and b + c, b and c + a, c and a + b are mutually
perpendicular, then the modulus of a + b + c is

(a) 12 (b) 12 (c) 5 2 (d) 50

21. If a and b are unit vectors and a – b is also a unit vector, then the angle between a and b is
   2
(a) (b) (c) (d)
4 3 2 3

22. If in a triangle AB = a , AC = b and D, E are the mid-points of AB and AC respectively, then DE is equal to
a b a b b a b a
(a) − (b) − (c) − (d) −
4 4 2 2 4 4 2 2

23. ABCDE is a pentagon. Forces AB , AE , DC, ED act at a point. Which force should be added to this system to make the

resultant 2 AC

(a) AC (b) AD (c) BC (d) BD

24. In a regular hexagon ABCDEF, AE =

(a) AC + AF + AB (b) AC + AF − AB (c) AC + AB − AF (d) None of these

25. 3OD + DA + DB + DC =

(a) OA + OB − OC (b) OA + OB − BD (c) OA + OB + OC (d) None of these

26. In a triangle ABC, if 2 AC = 3CB , then 2OA + 3 OB equals

(a) 5OC (b) − OC (c) OC (d) None of these

27. If | AO + OB | =| BO + OC | , then A, B, C form

(a) Equilateral triangle (b) Right angled triangle (c) Isosceles triangle (d) Line

28. Three forces of magnitudes 1, 2, 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 these

PRATAP BHAWAN, BEHIND LEELA CINEMA, HAZRATGANJ, LUCKNOW.

PH.(0522)4026913, 9838162263. e-mail. id: inpsclasses@gmail.com. www.inpsmcalucknow.com
Vector Algebra

29. If p + q + r = 0, | p | = 3, | q | = 5,| r | = 7 . Then angle between p and q is

 2  
(a) (b) (c) (d)
16 3 6 3

30. If A, B, C are the vertices of a triangle whose position vectors are a, b, c and G is the centroid of the ABC , then
a +b+c a −b −c
GA + GB + GC is (a)0 (b) A + B + C (c) (d)
3 3

31. If a = 3 i − 2 j + k , b = 2i − 4 j − 3 k and c = −i + 2 j + 2 k , then a + b + c is

(a) 3 i − 4 j (b) 3 i + 4 j (c) 4 i − 4 j (d) 4 i + 4 j

1
32. If x and y are two unit vectors and  is the angle between them, then | x − y | is equal to
2

(a) 0 (b)  / 2 (c) 1 (d)  / 4

33. If D, E, F are respectively the mid points of AB, AC and BC in ABC , then BE + AF =

1 3
(a) DC (b) BF (c) 2 BF (d) BF
2 2

34. If ABCD is a rhombus whose diagonals cut at the origin O, then OA + OB + OC + OD equals

(a) AB + AC (b) O (c) 2( AB + BC ) (d) AC + BD

35. ABCD is a parallelogram with AC and BD as diagonals. Then AC − BD =

(a) 4 AB (b) 3 AB (c) 2 AB (d) AB

36. The vectors b and c are in the direction of north-east and north-west respectively and | b | =| c | = 4 . The magnitude and
direction of the vector d = c − b , are

(a) 4 2 , towards north (b) 4 2 , towards west (c) 4, towards east (d) 4, towards south

37. Let a and b be two unit vectors inclined at an angle  , then sin( / 2) is equal to

1 1
(a) | a − b| (b) | a + b| (c) | a − b | (d) | a + b |
2 2

38. If a, b, c are three vectors such that a = b + c and the angle between b and c is  / 2 , then

(a) a 2 = b 2 + c 2 (b) b 2 = c 2 + a 2 (c) c 2 = a 2 + b 2 (d) 2a 2 − b 2 = c 2

(Note : Here a =| a |, b =| b |, c =| c | )

39. If a, b, c are three vectors of equal magnitude and the angle between each pair of vectors is such that | a + b + c | = 6
3
then | a | is equal to
1
(a) 2 (b) –1 (c) 1 (d) 6
3
40. Let a, b, c be three unit vectors such that | a + b + c | = 1 and 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
41. A vector of magnitude 2 along a bisector of the angle between the two vectors 2 i − 2 j + k and i + 2 j − 2 k is
2 1 2
(a) (3 i − k ) (b) (i − 4 j + 3 k) (c) (i − 4 j + 3 k) (d) None of these
10 26 26

PRATAP BHAWAN, BEHIND LEELA CINEMA, HAZRATGANJ, LUCKNOW.

PH.(0522)4026913, 9838162263. e-mail. id: inpsclasses@gmail.com. www.inpsmcalucknow.com
Vector Algebra

42. The vector i + x j + 3 k is rotated through an angle  and doubled in magnitude, then it becomes 4 i + (4 x − 2)j + 2 k . The value
of x is
2 1 2
(a) − (b) (c) (d) 2
3 3 3

43. If I is the centre of a circle inscribed in a triangle ABC, then | BC | IA+ | CA | IB + | AB | IC is

IA + IB + IC
(a) 0 (b) IA + IB + IC (c) (d) None of these
3
44. If the vector −i + j − k bisects the angle between the vector e and the vector 3 i + 4 j, then the unit vector in the direction of
e is
1 1 1 1
(a) (11 i + 10 j + 2 k ) (b) − (11 i − 10 j + 2 k ) (c) − (11 i + 10 j − 2 k ) (d) − (11 i + 10 j + 2 k )
15 15 15 15
45. The sides of a parallelogram are 2i + 4 j − 5 k, i + 2 j + 3k ,then the unit vector parallel to one of the diagonals
1 1 1 1
(a) (3 i + 6 j − 2 k ) (b) (3 i − 6 j − 2 k ) (c) (−3 i + 6 j − 2 k ) (d) (3 i + 6 j + 2 k )
7 7 7 7

46. A point O is the centre of a cricle circumscribed about a triangle ABC. Then OA sin 2 A + OB sin 2 B + OC sin 2C is equal to

(a) (OA + OB + OC) sin 2 A (b) 3. OG , where G is the centroid of triangle ABC

(c) O (d) None of these

47. If a + b + c = d, b + c + d =  a and a, b, c are non-coplanar, then the sum of a + b + c + d =

(a) 0 (b) ( − 1)d + ( − 1)a (c) ( − 1)d − ( − 1)a (d) ( − 1)d + ( − 1)a

48. Let a and b be two non-parallel unit vectors in a plane. If the vectors (a + b ) bisects the internal angle between a and b,
then  is
1
(a) (b) 1 (c) 2 (d) 4
2
49. The horizontal force and the force inclined at an angle 60o with the vertical, whose resultant is in vertical direction of P kg, are

(a)P, 2P (b) P, P 3 (c) 2 P, P 3 (dNone of these

50. If the resultant of two forces is of magnitude P and equal to one of them and perpendicular to it, then the other force is

(a) P 2 (b) P (c) P 3 (d) None of these

51. ABC is an isosceles triangle right angled at A. Forces of magnitude 2 2 , 5 and 6 act along BC, CA and AB respectively. The
magnitude of their resultant force is

(a) 4 (b) 5 (c) 11 + 2 2 (d) 30

52. If the resultant of two forces of magnitudes P and Q acting at a point at an angle of 60o is 7 Q , then P / Q is

(a) 1 (b) 3/2 (c) 2 (d) 4

53. Five points given by A, B, C, D, E are in a plane. Three forces AC, AD and AE act at A and three forces CB , DB, EB act at B.
Then their resultant is

(a) 2 AC (b) 3 AB (c) 3 DB (d) 2 BC

PRATAP BHAWAN, BEHIND LEELA CINEMA, HAZRATGANJ, LUCKNOW.

PH.(0522)4026913, 9838162263. e-mail. id: inpsclasses@gmail.com. www.inpsmcalucknow.com
Vector Algebra

Answer Key
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

d d b d a c b d b d b b b c c b d c d c

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

b d c c c a c c d a c c a c c b a a c c

41 42 43 44 45 46 47 48 49 50 51 52 53

a,c a,d a d a c a b c a b c b

PRATAP BHAWAN, BEHIND LEELA CINEMA, HAZRATGANJ, LUCKNOW.

PH.(0522)4026913, 9838162263. e-mail. id: inpsclasses@gmail.com. www.inpsmcalucknow.com