Exercise - I

Definition, Types of Vector & Angle

6. The vector joining the points A (1, 1, –1)
Between the Vectors and B (2, –3, 4) & pointing from A to B is-
1. Which one of the following statement (1) – î + 4 ˆj – 5 k̂ (2) î + 4 ˆj + 5 k̂
is false :
(3) î – 4 ˆj + 5 k̂ (4) – î – 4 ˆj – 5 k̂
(1) Mass, speed and energy are scalars
(2) Momentum, force and torque are

vectors 7. If A = 3 î + 4 ˆj then find Â
(3) Distance is a scalar while
displacement is a vector 3iˆ + 4ˆj 3iˆ – 4ˆj
(1) (2)
(4) A vector has only magnitude where 5 5
as a scalar has both magnitude and
4iˆ + 3ˆj 4iˆ – 3ˆj
direction (3) (4)
5 5
2. If n̂ is a unit vector in the direction of
 Addition & Subtraction of Vectors
the vector A , then :-
  
A   8. Given : A = 2 î + 3 ˆj and B = 5 î – 6 ˆj .
(1) n̂ =  (2) n̂ = A | A |  
|A| The magnitude of A + B is

|A|  (1) 4 units (2) 10 units
(3) n̂ =  ˆ= n
(4) n ˆ×A
A (3) 58 units (4) 61 units

3. The forces, which meet at one point  

but their lines of action do not lie in 9. Given : A = 2 î – ˆj +2 k̂ and B = – î – ˆj + k̂ .
 
one plane, are called: The unit vector of A – B is
(1) non-coplanar and non-concurrent
3iˆ + kˆ 3iˆ
forces (1) (2)
(2) coplanar and non-concurrent forces 10 10
(3) non-coplanar and concurrent forces k̂ –3iˆ – kˆ
(4) coplanar and concurrent forces (3) (4)
10 10
4. A vector is not changed if  
(1) it is displaced parallel to itself 10. Two vectors A and B lie in a plane,

(2) it is rotated through an arbitrary another vector C lies outside this
angle plane, then the resultant of these three
(3) it is cross-multiplied by a unit   
vectors i.e. A + B + C :
vector
(4) it is multiplied by an arbitrary (1) Can be zero
scalar. (2) Cannot be zero
 
(3) Lies in the plane containing A & B
 
5. The forces, each numerically equal to 5 (4) Lies in the plane containing B & C
N, are acting as shown in the Figure.
Find the angle between forces:    
11. Given that P + Q =
P – Q . This can be
true when:
5N  
60º (1) P = Q
 
5N (2) Q = 0
 
(1) 60° (2) 110° (3) Neither P nor Q is a null vector
(3) 115° (4) 120°  
(4) P is perpendicular to Q
  
12. The resultant of A and B makes an 19. Force 3 N, 4 N and 12 N act at a point
  in mutually perpendicular directions.
angle α with A and β with B , then :
The magnitude of the resultant force is:
(1) α < β
(1) 19 N (2) 13 N
(2) α < β if A < B
(3) 11 N (4) 5 N
(3) α < β if A > B
(4) α < β if A = B   
20. If vectors P , Q and R have magnitudes
  
13. The minimum number of vectors of 5, 12 and 13 units and P + Q = R , the
equal magnitude required to produce a  
angle between Q and R is :
zero resultant is :
–1 5 –1 5
(1) 2 (2) 3 (1) cos   (2) cos  
(3) 4 (4) more than 4  12   13 
–1  12  –1 2
(3) cos   (4) cos  
14. How many minimum number of  13   13 
coplanar vectors having different
magnitudes can be added to give zero 21. In vector diagram shown in figure
resultant:- 
where ( R ) is the resultant of vectors
(1) 2 (2) 3
  B
(3) 4 (4) 5 ( A ) and ( B ). If R = , the value of
2

15. How many minimum number of vectors angle θ is :

in different planes can be added to give
zero resultant:-
(1) 2 (2) 3
(3) 4 (4) 5
(1) 30° (2) 45°
16. Which of the following pair of forces (3) 60° (4) 75°
will never give resultant force of 2 N :
 
(1) 2 N and 2 N (2) 1 N and 1 N 22. The resultantof A and B is
(3) 1 N and 3 N (4) 1 N and 4 N 
perpendicular to A . What is the angle
   
17. If A + B is a unit vector along x-axis and between A and B :
  A  A
= ˆ , then what is B :
A ˆi – ˆj + k (1) cos–1   (2) cos–1  – 
B  B
ˆ
(1) ˆj + k ˆ
(2) ˆj – k
A  A
ˆ
(3) ˆi + ˆj + k (4) ˆi + ˆj – kˆ (3) sin–1   (4) sin–1  – 
B  B

18. What happens, when we multiply a  

23. When two vector a and b are added,
vector by (– 2):
(1) direction reverses and unit changes the magnitude of the resultant vector is
(2) direction reverses and magnitude is always
doubled (1) greater than (a + b)
(3) direction remains unchanged and (2) less than or equal to (a + b)
unit changes (3) less than (a + b)
(4) none of these (4) equal to (a + b)
24. Rain is falling vertically downwards 30. The sum and difference of two
with a speed 5 m/s. If unit vector along perpendicular vectors of equal lengths
upward is defined as ˆj , represent are
(1) of equal lengths and have an acute
velocity of rain in vector form.
angle between them
(1) 5ˆj (2) – 5ˆj (2) of equal length and have an obtuse
(3) 8ˆj (4) – 8ˆj angle between them
(3) also perpendicular to each other
  and are of different lengths
25. Two vectors a and b inclined at an
(4) also perpendicular to each other
angle θ w.r.t. each other have a
 and are of equal lengths
resultant c which makes an angle β
   
with a . If the directions of a and b are 31. What is the angle between A and the
interchanged, then the resultant will  
have the same (
ˆ and A – B
resultant of A + B )
ˆ : ( )
(1) magnitude –1 A
(1) 0° (2) tan  
(2) direction B
(3) magnitude as well as direction
–1 B –1  A –B 
(4) neither magnitude nor direction (3) tan   (4) tan  
A  A +B
26. A set of vectors taken in a given order
gives a closed polygon. Then the Dot Product and Cross Product
resultant of these vectors is a

(1) scalar quantity (2) pseudo vector 32. = 2iˆ + 3ˆj
The angle that the vector A
(3) unit vector (4) null vector makes with y-axis is :
-1 -1
27. The vector sum of two force P and Q is (1) tan (3/2) (2) tan (2/3)
-1 -1
minimum when the angle θ between (3) sin (2/3) (4) cos (3/2)
their positive directions, is

(1)
π
(2)
π 33. A vector perpendicular to 4iˆ – 3ˆj may ( )
4 3
be :
π
(3) (4) π (1) 4iˆ + 3ˆj ˆ
(2) 7k
2
 (3) 6iˆ (4) 3iˆ – 4ˆj
28. The vector sum of two vectors A and

B is maximum, then the angle θ
between two vectors is - 34. ( )
A force 3iˆ + 2ˆj N displaces an object
(1) 0° (2) 30°
(3) 45° (4) 60°
(
through a distance 2iˆ – 3ˆj m. The work )
   done is :
29. Given : C = A + B . Also, the magnitude (1) zero (2) 12 J
  
of A , B and C are 12, 5 and 13 units (3) 5 J (4) 13 J

respectively. The angle between A and 
 35. = 5iˆ + 2ˆj – Skˆ is perpendicular
The vector B
B is

π to the vector A = 3iˆ + ˆj + 2kˆ if S =
(1) 0° (2)
4 (1) 1 (2) 4.7
π (3) 6.3 (4) 8.5
(3) (4) π
2
36. The angle between vectors (iˆ + ˆj) and 43. If î , ˆj and k̂ are unit vectors along X,
ˆ is :
(ˆj + k) Y & Z axis respectively, then tick the
wrong statement:
(1) 90° (2) 180°
(1) ˆi.iˆ = 1 (2) ˆi × ˆj =ˆ
k
(3) 0° (4) 60°
(3) ˆi.ˆj = 0 ˆ=
(4) ˆi × k –iˆ
37. The angle between two vectors given
ˆ and (7iˆ + 4ˆj + 4k)
by (6iˆ + 6ˆj – 3k) ˆ is :  
44. Two vectors P and Q are inclined to
 1  1 each other at angle θ. Which of the
(1) cos–1   (2) cos–1  
2 3 following is the unit vector perpendicular
 
 1  2 to P and Q :
(3) cos–1   (4) cos–1    
 3 3 P×Q ˆ
P̂ × Q
(1) (2)
P·Q sin θ
   
38. If P.Q = PQ, then angle between P and ˆ
P̂ × Q P̂ × Q
 (3) (4)
Q is : PQ sin θ PQ sin θ
(1) 0° (2) 30°
(3) 45° (4) 60° 45. The magnitude of the vector product of
 
two vectors A and B may not be :
39. For a body, angular velocity (1) Greater than AB (2) Less than AB
 ˆi − 2ˆj + 3kˆ
(ω) = and radius vector (3) Equal to AB (4) Equal to zero
 ˆ , then its velocity is
(r ) = ˆi + ˆj + k   
   46. If P × Q = R , then which of the following
(v = ω × r ): statements is not true :
(1) –5 î – 2 ˆj + 3 k̂    
(1) R ⊥ P (2) R ⊥ Q
     
(2) –5 î + 2 ˆj – 3 k̂ (3) R ⊥ (P + Q) (4) R ⊥ (P × Q)
(3) –5 î + 2 ˆj + 3 k̂
(4) –5 î – 2 ˆj – 3 k̂ 47. (
If the vectors ˆi + ˆj + k )
ˆ and 3iˆ form two

sides of a triangle, then area of the

40. Area of a parallelogram, whose
triangle is :
diagonals are 3iˆ + ˆj – 2kˆ and ˆi – 3ˆj + 4kˆ
(1) 3 unit (2) 2 3 unit
will be :
3
(1) 95 (2) 75 (3) unit (4) 3 2 unit
2
(3) 105 (4) 100
   
41.

A vector A points vertically downward

48. (
What is the value of A + B · A × B : ) ( )
& B points towards east, then the (1) 0
2
(2) A – B
2
  2 2
vector product A × B is (3) A + B + 2AB (4) none of these
(1) along west (2) along east
(3) zero (4) along south 49. If n̂ aiˆ + bjˆ is perpendicular to the
=

42. A vector F1 is along the positive X-axis. ( )
vector, ˆi + ˆj , then the value of a and b
If its
vector product with another may be :
 
vector F2 is zero then F2 may be : (1) 1, 0 (2) –2, 0
(1) 4 ˆj (2) – (iˆ + ˆj) 1 1
(3) 3, 0 (4) , −
ˆ 2 2
(3) (iˆ + k) ˆ
(4) (–4i)
Resolution of Vector, Projection of Vector, 53. What is the maximum number of
Miscellaneous rectangular components into which a
vector can be split in space:
50. The x and y components of a force are (1) 2 (2) 3
2 N and – 3 N. The force is (3) 4 (4) ∞
(1) 2iˆ – 3ˆj (2) 2iˆ + 3ˆj
54. The direction cosines of a vector
(3) –2iˆ – 3ˆj (4) 3iˆ + 2ˆj ˆ are:
ˆi + ˆj + 2 k

51. What is the maximum number of 1 1 1 1 1

(1) , ,1 (2) , ,
components into which a vector can be 2 2 2 2 2
split: 1 1 1 1 1 1
(3) , , (4) , ,
(1) 2 (2) 3 2 2 2 2 2 2
(3) 4 (4) ∞
55. One of the rectangular components of
52. What is the maximum number of –1 –1
a velocity of 60 km h is 30 km h .
rectangular components into which a
Find other rectangular component:
vector can be split in its own plane: –1 –1
(1) 2 (2) 3 (1) 20 3 km h (2) 30 2 km h
–1 –1
(3) 4 (4) ∞ (3) 20 2 km h (4) 30 3 km h

ANSWER KEY
Que. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Ans. 4 1 3 1 4 3 1 3 1 2 2 3 1 2 3 4 2 2 2 3 2 2 2 2 1
Que. 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
Ans. 4 4 1 3 4 1 2 2 1 4 4 4 1 3 2 4 4 4 2 1 4 3 1 4 1
Que. 51 52 53 54 55
Ans. 4 1 2 3 4