Vector | Physics Optional Notes for UPSC PDF Download

 Page 1


 
VECTOR 
Vector (Extra) 
Example: A man walks 10 m east and then 15 m30
°
N of E and then 25 m40
°
E of S? 
Solution: 
 
Example: 
 
 
 
Example: A man moves 4 m in the north direction and then 4 m in the east direction. 
Find resultant displacement vectors resultant displacement is 4v2 in the N- E direction. 
Page 2


 
VECTOR 
Vector (Extra) 
Example: A man walks 10 m east and then 15 m30
°
N of E and then 25 m40
°
E of S? 
Solution: 
 
Example: 
 
 
 
Example: A man moves 4 m in the north direction and then 4 m in the east direction. 
Find resultant displacement vectors resultant displacement is 4v2 in the N- E direction. 
 
Example: Two forces of magnitude 5 N are acting on a particle such that the angle 
between them is 120
°
 . Find the resultant force acting on the particle. 
 
 
?  Ans. 5 N 
 
A
? ? 
+ B
? ? 
= C
? 
 
|A
? ? 
| + |B
? ? 
| ? |C
? 
| 
Key Point: 
(i) Electric current does not follow the triangle rule of vector addition and it is not a 
vector quantity. 
Example: ?
?? 
?? 
?? ? 
 ?? 
+ ?? ? 
= ?? 
 |?? 
| + |?? ? 
| = |?? 
| 
 
 
Page 3


 
VECTOR 
Vector (Extra) 
Example: A man walks 10 m east and then 15 m30
°
N of E and then 25 m40
°
E of S? 
Solution: 
 
Example: 
 
 
 
Example: A man moves 4 m in the north direction and then 4 m in the east direction. 
Find resultant displacement vectors resultant displacement is 4v2 in the N- E direction. 
 
Example: Two forces of magnitude 5 N are acting on a particle such that the angle 
between them is 120
°
 . Find the resultant force acting on the particle. 
 
 
?  Ans. 5 N 
 
A
? ? 
+ B
? ? 
= C
? 
 
|A
? ? 
| + |B
? ? 
| ? |C
? 
| 
Key Point: 
(i) Electric current does not follow the triangle rule of vector addition and it is not a 
vector quantity. 
Example: ?
?? 
?? 
?? ? 
 ?? 
+ ?? ? 
= ?? 
 |?? 
| + |?? ? 
| = |?? 
| 
 
 
Polygon Rule of Vector Addition: 
 
If given vectors represent ( ?? - 1 ) sides of the n-sided polygon in the same sense then 
their resultant vector will be represented by n
th 
 side in opp. Sense 
Example: ?
?
?
5 N
?
3 N
  Resultant =? 
We can also see that 5 and 5 cancel each other and we get 3 N ? 
Parallelogram (|gm) Rule of Vector Addition : 
When you place two vectors side by side with their tails connected, representing two 
sides of a parallelogram, the diagonal starting from their shared tail indicates the 
combined result or the total effect of those two vectors. 
 
Resultant magnitude and direction 
|A
? ? 
| = A |B
? ? 
| = B 
? between A
? ? 
 and B
? ? 
 is ?? 
?QRS;cos ?? =
QR
B
QR = Bcos ?? RS = Bsin ?? 
Page 4


 
VECTOR 
Vector (Extra) 
Example: A man walks 10 m east and then 15 m30
°
N of E and then 25 m40
°
E of S? 
Solution: 
 
Example: 
 
 
 
Example: A man moves 4 m in the north direction and then 4 m in the east direction. 
Find resultant displacement vectors resultant displacement is 4v2 in the N- E direction. 
 
Example: Two forces of magnitude 5 N are acting on a particle such that the angle 
between them is 120
°
 . Find the resultant force acting on the particle. 
 
 
?  Ans. 5 N 
 
A
? ? 
+ B
? ? 
= C
? 
 
|A
? ? 
| + |B
? ? 
| ? |C
? 
| 
Key Point: 
(i) Electric current does not follow the triangle rule of vector addition and it is not a 
vector quantity. 
Example: ?
?? 
?? 
?? ? 
 ?? 
+ ?? ? 
= ?? 
 |?? 
| + |?? ? 
| = |?? 
| 
 
 
Polygon Rule of Vector Addition: 
 
If given vectors represent ( ?? - 1 ) sides of the n-sided polygon in the same sense then 
their resultant vector will be represented by n
th 
 side in opp. Sense 
Example: ?
?
?
5 N
?
3 N
  Resultant =? 
We can also see that 5 and 5 cancel each other and we get 3 N ? 
Parallelogram (|gm) Rule of Vector Addition : 
When you place two vectors side by side with their tails connected, representing two 
sides of a parallelogram, the diagonal starting from their shared tail indicates the 
combined result or the total effect of those two vectors. 
 
Resultant magnitude and direction 
|A
? ? 
| = A |B
? ? 
| = B 
? between A
? ? 
 and B
? ? 
 is ?? 
?QRS;cos ?? =
QR
B
QR = Bcos ?? RS = Bsin ?? 
 
? PRS; 
(???? )
2
 +(RS)
2
= (PS)
2
(PS)
2
 = (A+ Bcos ?? )
2
+ (Bsin ?? )
2
 = A
2
+ B
2
cos
2
 ?? + 2ABcos ?? + B
2
sin
2
 ?? 
|?? ? 
| = v?? 2
+ ?? 2
+ 2???? cos ?? 
tan ?? =
RS
PR
 
tan ?? =
Bsin ?? A+ Bcos ?? 
?? is ? between ?? ? 
 and ?? ¨
 
Example: Two forces are acting on the block as shown. Find the resultant force and its 
? with 10 N force. 
Solution: 
R = v5
2
+ 10
2
+ 2 × 5 × 10cos 60 = v175
tan ?? =
5 ×
v3
2
10+
5
2
=
v3
5
 ? ?? = tan
-1
 (
v3
5
)
 
Example: Two forces of 5 N are acting on the block. One is towards the east and the 
other 30
°
W of N. Find the resultant force. 
Solution: 
Page 5


 
VECTOR 
Vector (Extra) 
Example: A man walks 10 m east and then 15 m30
°
N of E and then 25 m40
°
E of S? 
Solution: 
 
Example: 
 
 
 
Example: A man moves 4 m in the north direction and then 4 m in the east direction. 
Find resultant displacement vectors resultant displacement is 4v2 in the N- E direction. 
 
Example: Two forces of magnitude 5 N are acting on a particle such that the angle 
between them is 120
°
 . Find the resultant force acting on the particle. 
 
 
?  Ans. 5 N 
 
A
? ? 
+ B
? ? 
= C
? 
 
|A
? ? 
| + |B
? ? 
| ? |C
? 
| 
Key Point: 
(i) Electric current does not follow the triangle rule of vector addition and it is not a 
vector quantity. 
Example: ?
?? 
?? 
?? ? 
 ?? 
+ ?? ? 
= ?? 
 |?? 
| + |?? ? 
| = |?? 
| 
 
 
Polygon Rule of Vector Addition: 
 
If given vectors represent ( ?? - 1 ) sides of the n-sided polygon in the same sense then 
their resultant vector will be represented by n
th 
 side in opp. Sense 
Example: ?
?
?
5 N
?
3 N
  Resultant =? 
We can also see that 5 and 5 cancel each other and we get 3 N ? 
Parallelogram (|gm) Rule of Vector Addition : 
When you place two vectors side by side with their tails connected, representing two 
sides of a parallelogram, the diagonal starting from their shared tail indicates the 
combined result or the total effect of those two vectors. 
 
Resultant magnitude and direction 
|A
? ? 
| = A |B
? ? 
| = B 
? between A
? ? 
 and B
? ? 
 is ?? 
?QRS;cos ?? =
QR
B
QR = Bcos ?? RS = Bsin ?? 
 
? PRS; 
(???? )
2
 +(RS)
2
= (PS)
2
(PS)
2
 = (A+ Bcos ?? )
2
+ (Bsin ?? )
2
 = A
2
+ B
2
cos
2
 ?? + 2ABcos ?? + B
2
sin
2
 ?? 
|?? ? 
| = v?? 2
+ ?? 2
+ 2???? cos ?? 
tan ?? =
RS
PR
 
tan ?? =
Bsin ?? A+ Bcos ?? 
?? is ? between ?? ? 
 and ?? ¨
 
Example: Two forces are acting on the block as shown. Find the resultant force and its 
? with 10 N force. 
Solution: 
R = v5
2
+ 10
2
+ 2 × 5 × 10cos 60 = v175
tan ?? =
5 ×
v3
2
10+
5
2
=
v3
5
 ? ?? = tan
-1
 (
v3
5
)
 
Example: Two forces of 5 N are acting on the block. One is towards the east and the 
other 30
°
W of N. Find the resultant force. 
Solution: 
 
?? = v5
2
+ 5
2
+ 2 × 5 × 5cos 120
°
= 5 
tan ?? =
5sin 120
°
5 + 5cos 120
°
= v3 
? ?? = 60
°
 
Example: Two vectors of magnitude ?? and ?? are at 60
°
 and their resultant is of 
magnitude v13?? ? 
Solution: v13 B = vA
2
+ B
2
+ 2ABcos 60
°
 
v13 B = vA
2
+ B
2
+ AB
13 B
2
= A
2
+ B
2
+ AB
A
2
- 12 B
2
+ AB = 0
 A
2
+ AB- 12 B
2
= 0
 divide by B
2
 
(
?? ?? )
2
+
?? ?? - 12 = 0 
Let 
?? ?? = ?? 
 
?? 2
+ ?? - 12 = 0
?? 2
+ 4?? - 3?? - 12 = 0
 (?? + 4)(?? - 3)= 0
?? = -4,?? = 3