Physics
Scalars and Vectors
1)
Given vector A  2ˆi  3 ˆj, the angle between A and y-axis is
(a)
tan 1 3 / 2
(b)
tan 1 2 / 3
(c)
sin 1 2 / 3
(d) cos 1 2 / 3
2)
Which of the following is a scalar quantity
(a) Displacement
(b) Electric field
3)
If A  4ˆi  3ˆj and B  6ˆi  8 ˆj then magnitude and direction of A  B will be
(d) Work
(b) 5 5 , tan 1 (1 / 2) (c) 10 , tan 1 (5 ) (d)
(a) 5, tan 1 (3 / 4 )
4)
(c) Acceleration
25 , tan 1 (3 / 4 )
Two vectors A and B lie in a plane, another vector C lies outside this plane, then the resultant of these three vectors i.e., A  B  C
(a) Can be zero
(b) Cannot be zero

(c) Lies in the plane containing A  B
(d) Lies in the plane containing C
5)
The resultant of two vectors P and Q is R. If Q is doubled, the new resultant is perpendicular to P. Then R equals
(a) P
(b) (P+Q)
(c) Q
(d) (P–Q)
6)
Two forces F1  5ˆi  10 ˆj  20 kˆ and F2  10ˆi  5ˆj  15 kˆ act on a single point. The angle between F1 and F2 is nearly
(a) 30°
(b) 45°
(c) 60°
(d) 90°
ˆ
ˆ
ˆ
ˆ
ˆ
If two vectors 2i  3 j  k and  4 i  6 j  kˆ are parallel to each other then value of  be
(a) 0
(b) 2
(c) 3
(d) 4
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
7)
8)
9)




If A  B  C, then which of the following statements is wrong
(a) C  A
(b) C  B
(c)
(d) C  ( A  B)
C  ( A  B)
10) Consider two vectors F1  2ˆi  5 kˆ and F 2  3ˆj  4 kˆ . The magnitude of the scalar product of these vectors is
(a) 20
(b) 23
(c)
(d) 26
5 33
11) If P.Q  PQ, then angle between P and Q is
(a) 0°
(b) 30°
(c) 45°
(d) 60°
12) A vector F 1 is along the positive X-axis. If its vector product with another vector F 2 is zero then F 2 could be
(a) 4 ˆj
(b)  (ˆi  ˆj)
(c) (ˆj  kˆ )
(d) (4ˆi )
13) What is the angle between ( P  Q) and (P  Q )
(a) 0
(b)

2
(c)

4
(d) 
14) The resultant of the two vectors having magnitude 2 and 3 is 1. What is their cross product
(a) 6
(b) 3
(c) 1
(d) 0
15) The angle between two vectors given by 6i  6 j  3k and 7 i  4 j  4 k is
 1 


 3
(a) cos 1 
 5 


 3
(b) cos 1 
(c)
 2 

sin 1 

 3
 5

 3 


(d) sin 1 
16) Three vectors a, b and c satisfy the relation a . b  0 and a . c  0. The vector a is parallel to
(a) b
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(b) c
(c)
b .c
(d) b  c
17) The value of ( A  B) ( A  B) is
(a) 0
(b) A 2  B 2
(c) B  A (d) 2(B  A)
18) Can the resultant of 2 vectors be zero
(a) Yes, when the 2 vectors are same in magnitude and direction
(b) No
(c) Yes, when the 2 vectors are same in magnitude but opposite in sense
2
with each other
3
19) The vectors from origin to the points A and B are A  3ˆi  6 ˆj  2kˆ and B  2ˆi  ˆj  2kˆ respectively. The area of the
(d) Yes, when the 2 vectors are same in magnitude making an angle of
triangle OAB be
5
17 sq.unit
2
(a)
2
17 sq.unit
5
(b)
(c)
3
17 sq.unit
5
(d)
5
17 sq.unit
3
20) If the resultant of n forces of different magnitudes acting at a point is zero, then the minimum value of n is
(a) 1
(b) 2
(c) 3
(d) 4
21) If for two vectors A and B, A  B  0, the vectors
(a) Are perpendicular to each other
(b) Are parallel to each other
(c) Act at an angle of 60°
(d) Act at an angle of 30°
22) Consider a vector F  4ˆi  3ˆj. Another vector that is perpendicular to F is
(a) 4ˆi  3 ˆj
(c) 7kˆ
(b) 6 î
(d) 3ˆi  4 ˆj
23) Given that A  B  C and that C is  to A . Further if | A | | C |, then what is the angle between A and B


(c)
3
radian
4
(c)
7 
tan 1  
5
(d)  radian
  



 
24) 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
(a)
4

(a)
(b)
radian
2
radian
(b) cos 1 (0.6)
2
(d)

4
25) Maximum and minimum magnitudes of the resultant of two vectors of magnitudes P and Q are in the ratio 3 : 1. Which of
the following relations is true
(a) P  2 Q
(b) P  Q
(c) PQ  1
(d) None of these
26) The vector that must be added to the vector ˆi  3ˆj  2kˆ and 3ˆi  6 ˆj  7kˆ so that the resultant vector is a unit vector along
the y-axis is
(a) 4ˆi  2ˆj  5kˆ
(b)  4ˆi  2ˆj  5kˆ
(c) 3ˆi  4 ˆj  5kˆ
(d) Null vector
27) Any 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


28) If P  Q then which of the following is NOT correct
(a) Pˆ  Qˆ
29) Surface area is
(a) Scalar


(b) | P| | Q|
(b) Vector
(c)
ˆ  QP
ˆ
PQ
(d)
(c) Neither scalar nor vector
 
ˆ
ˆ Q
PQ  P
(d) Both scalar and vector
30) If a particle moves from point P (2,3,5) to point Q (3,4,5). Its displacement vector be
(a) ˆi  ˆj  10 kˆ
(b) ˆi  ˆj  5kˆ
(c) ˆi  ˆj
(d) 2ˆi  4 ˆj  6kˆ
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Answers:
1) (b)
2) (d) Displacement, electrical and acceleration are vector quantities
3)
(b)
A  B  4ˆi  3ˆj  6ˆi  8 ˆj  10ˆi  5 ˆj
| A  B |  (10 )2  (5)2  5 5
tan  

5
1
1
   tan 1  

10
2
2
4)
5)
(b) If C lies outside the plane then resultant force can not be zero.
(c)
6)
(b) cos  
F1 .F2
| F1 || F2 |
(5ˆi  10 ˆj  20 kˆ ).(10ˆi  5 ˆj  15 kˆ )

25  100  400 100  25  225
1
 cos  

50  50  300
525 350
   45 
2
7)
(b) Let A  2ˆi  3 ˆj  kˆ and B  4ˆi  6 ˆj  kˆ
A and B are parallel to each other
a1 a 2 a 3
2
3
1


i.e.


   2.
4 6

b1 b 2 b 3
8)
(a) ( A  B) is perpendicular to ( A  B) . Thus
( A  B) . ( A  B) = 0
or A 2  B . A  A . B  B 2  0
Because of commutative property of dot product A.B  B. A
 A 2  B 2  0 or A  B
Thus the ratio of magnitudes A/B = 1
9)
(d) From the property of vector product, we notice that C must be perpendicular to the plane formed by vector A and B . Thus
C is perpendicular to both A and B and ( A  B) vector also, must lie in the plane formed by vector A and B . Thus C must
be perpendicular to ( A  B) also but the cross product ( A  B) gives a vector C which can not be perpendicular to itself. Thus
the last statement is wrong.
 
10) (d) F1 .F2  (2ˆj  5kˆ )(3ˆj  4 kˆ )
 6  20  20  6  26
 
P.Q
11) (a) cos  
 1    0
PQ
12) (d)




13) (b)Vector (P  Q) lies in a plane and vector (P  Q ) is perpendicular to this plane i.e. the angle between given vectors is
2 2  3 2  2  2  3  cos   1
 
By solving we get   180   A  B  0

56
AB
42  24  12

15) (d) cos  

AB
36  36  9 49  16  16
9 71
14) (d)
cos  
16) (d)
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56
 sin  
 5
5

or   sin 1 
 3 
3


9 71


 
a . b  0 i.e. a and b will be perpendicular to each other


 
a . c  0 i.e. a and c will be perpendicular to each other

2
.
 


b  c will be a vector perpendicular to both b and c
 

So a is parallel to b  c












17) (d) ( A  B)  ( A  B)  A  A  A  B  B  A  B  B
   
   
 
 0  A  B  B  A  0  B  A  B  A  2(B  A)
18) (c)
19) (a) Given OA  a  3ˆi  6 ˆj  2kˆ and OB  b  2ˆi  ˆj  2kˆ
ˆi
ˆj
kˆ
 (a  b)  3
6
2
2
1
2
 (12  2)ˆi  (4  6)ˆj  (3  12 )kˆ
 10 ˆi  10 ˆj  15 kˆ  | a  b |  10 2  10 2  15 2
 425  5 17
20) (c) If vectors are of equal magnitude then two vectors can give zero resultant, if they works in opposite direction. But if the
vectors are of different magnitudes then minimum three vectors are required to give zero resultant.


21) (b) A  B  0  sin   0    0
Two vectors will be parallel to each other.
22) (c)Force F lie in the x-y plane so a vector along z-axis will be perpendicular to F.
23) (c)
24) (a) C  A 2  B 2
= 32  4 2  5
 Angle between A and B is


C

B
2
25) (a) According to problem P  Q  3 and P  Q 1
A
P
By solving we get P  2 and Q  1 
 2  P  2Q
Q
26) (b) Unit vector along y axis  ˆj so the required vector  ˆj  [(ˆi  3 ˆj  2kˆ )  (3ˆi  6 ˆj  7 kˆ )]   4ˆi  2ˆj  5kˆ
27) (c)


28) (d) P  Q  PPˆ  Q Qˆ
29) (a)   B. A . In this formula A is a area vector.
30) (c) Displacement vector r  xˆi  yˆj  zkˆ
 (3  2)ˆi  (4  3)ˆj  (5  5)kˆ  ˆi  ˆj
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