Addition of vectors , Magnitude and Direction of the Resultant of Two Vectors with special cases of parallel , Anti-parallel and Perpendicular vectors

1. Physics Helpline
L K Satapathy
Theory of Vectors 2

2. Physics Helpline
L K Satapathy Theory of Vectors - 2
Addition of vectors :
Vectors are added geometrically and not by the rules of algebra.
+ =
+ =
+ =
( I )
( II )
( III )
A = 4m East B = 3m East
A = 4m East B = 3m west
A = 4m East
B = 3m North
 A + B  = 7m
 A + B  = 1m
 A + B  = 5m

3. Physics Helpline
L K Satapathy Theory of Vectors - 2
Triangle law of vector addition
If two vectors acting simultaneously at a point are represented in
magnitude and direction by the two sides of a triangle, taken in the
same order, then their resultant is represented in magnitude and
direction by the third side of the triangle taken in opposite order.
O P
Q
A
B
A
R
B
 A + B = R
OP PQ OQ 

4. Physics Helpline
L K Satapathy Theory of Vectors - 2
Parallelogram law of vector addition
If two vectors acting simultaneously at a point are represented in
magnitude and direction by the two sides of a parallelogram drawn
from a point, then their resultant is represented in magnitude and
direction by the diagonal of the parallelogram passing through that
point.
O P
Q S
OQ PS OP OQ OP PS OS     
 A + B = R
A
RB
A
B

5. Physics Helpline
L K Satapathy Theory of Vectors - 2
Polygon law of vector addition
If a number of vectors acting simultaneously at a point are
represented in magnitude and direction by the sides of an open
polygon taken in the same order, then their resultant is
represented in magnitude and direction by the closing side of the
polygon taken in the opposite order.
O P
Q
S
OP PQ QS OQ QS OS    
 A + B + C = R
A
R
B
C
C
A
B

6. Physics Helpline
L K Satapathy Theory of Vectors - 2
Magnitude of the Resultant of Two vectors
2 2 2
OS OT ST 
O P
Q S
T

In  OST
2 2
( )OP PT ST  
2 2 2
2 .OP PT OP PT ST   
2 2
2 .OP PS OP PT  
2 2
2 . cosOP PS OP PS   
2 2 2
2 cosR A B AB    
2 2
2 cos . . . (1)R A B AB    
A
RB
2 2 2
[ ]PT ST PS 
[cos ]PT PS 

7. Physics Helpline
L K Satapathy Theory of Vectors - 2
Direction of the Resultant of Two vectors
tan ST ST
OT OP PT
  
 O P
Q S
T

In  OST
sin
cos
PS
OP PS




sintan
cos
B
A B




 1 sintan . . . (2)
cos
B
A B





A
RB
( Angle between R and A )
cos , sinPT ST
PS PS
   
  

8. Physics Helpline
L K Satapathy Theory of Vectors - 2
Special Cases
0tan 0 0
1
B
A B
    
 
Case 1 : Parallel Vectors
2 2
2R A B AB A B    
0 cos 1 , sin 0o
     
When two vectors are pointing in the same direction, the
magnitude of their resultant = the sum of their
magnitudes and it points in the same direction
A
R
B
( Angle between R and A )

9. Physics Helpline
L K Satapathy Theory of Vectors - 2
Special Cases
0tan 0 0
( 1)
B
A B
    
  
Case 2 : Anti-Parallel Vectors
2 2
2R A B AB A B    
180 cos 1 , sin 0o
      
When two vectors are pointing in opposite directions,
magnitude of their resultant = the difference of their
magnitudes and it points in the same direction as the
larger of the two vectors
A
R
B
( Angle between R and A )

10. Physics Helpline
L K Satapathy Theory of Vectors - 2
Special Cases
11tan tan
0
B B B
A B A A
     
 
Case 3 : Perpendicular Vectors
2 2 2 2
2 0R A B AB A B     
90 cos 0 , sin 1o
     
When two vectors are perpendicular to each other, they
represent the adjacent sides of a rectangle, whose
diagonal gives the magnitude and direction of their
resultant
A
R
B
( Angle between R and A )

11. Physics Helpline
L K Satapathy
For More details:
www.physics-helpline.com
Subscribe our channel:
youtube.com/physics-helpline
Follow us on Facebook and Twitter:
facebook.com/physics-helpline
twitter.com/physics-helpline