class 11 Vector (all concepts)

Turtle Tutorials Vectors



ADDITION OF VECTORS
Scalars can be added or subtracted following the simple rules of algebra or arithmetic.
But vectors do not follow the same simple rules, because while adding or subtracting
vectors, their direction also has to be considered. For example, when a mass of 5 kg is
added to another mass of 5 kg, the result is exactly 10kg. But when a vector of magnitude
5 units is added to another vector of magnitude 5 units (of course of same physical
quantity), the result may have a magnitude from zero to a maximum of 10 units,
depending on relative orientations of the two vectors. The vector sum is also called
resultant.
ADDITION OF TWO VECTORS IN SAME DIRECTION
If two vectors are in the same direction, their resultant (sum) is obtained by adding
their vector lengths as shown in the figure (a). The direction of resultant is same as
the individual vectors. Fig (a)- Addition of vectors in same direction

A



B

 
A B

  
C  A B

figure (a)
ADDITION OF TWO VECTORS IN OPPOSITE DIRECTION
If the vectors are mutually opposite, their resultant is obtained by subtracting the
length of smaller vector
Addition of vectors in opposite from that of larger vector as shown in figure (b). The
direction of resultant is same as that of larger vector. (Triangle law)

A


B



  
C  A  ( B)

figure (b)
ADDITION OF TWO VECTORS INCLINED MUTUALLY
If two vectors are mutually inclined, the following procedure is adopted to find their
sum. A and B are the given vectors. B is slides parallel to itself, such that its ‘tail’
coincides with the head of A as shown in figure (c). Then the directed line segment
drawn from the tail of A to the head of B represents the addition of A and B.


B

A

  
C  AB




A B
Turtle Tutorials, Marris Road, Aligarh 202001. Ph. 7617733444, 7617733555. Web: www.turtletutorails.com

figure (c)

, Turtle Tutorials Vectors

TRIANGLE LAW OF VECTORS : If two vectors are represented in magnitude and
direction by the two sides of a triangle taken in order, the third side of the triangle
taken in reverse order represents their resultant in magnitude and direction.
PARALLELOGRAM LAW OF VECTORS :
Two vector quantities (say, velocity, acceleration, force, etc) can be added using
parallelogram law. This law is useful to find both magnitude and direction of resultant.
Statement: If two vectors are represented in magnitude and direction by the adjacent
sides of a parallelogram drawn from a point, the diagonal passing through that point
represents their resultant both in magnitude and direction.

D D C
  
Q Q R


 
 
A P B A P B E

figure (d)
   
Explanation: P and Q are two vectors represented by AB and AD . Both vectors act
at the common point A and mutually inclined at angle ‘  ’ as shown in fig (d). If the
parallelogram ABCD is completed taking AB and AD as adjacent sides, then the diagonal
 
 
AC represents their resultant R both in magnitude and direction.

Magnitude of the resultant: The line of action of P is extended. The perpendicular
drawn from ‘C’ meets the extension of AB at E.
  
From the figure, it is obvious that BC  AD  Q and CBE  
 
 Length of AB = magnitude of P = P Length of BC = magnitude of Q = Q

Length of AC = magnitude of R = R

BE
From triangle CBE and , BE = BC cos   BE = Q cos  ........... [1]
BC
EC
From the triangle AEC and = sin  and EC = Q sin  ........... [2]
BC
(AC)2 = (AE)2 + (EC)2
 (AC)2 = (AB + BE)2 + (EC)2  (AC)2 = (AB)2 + (BE)2 + 2AB . BE + (EC)2
 R2 = P2 + Q2 cos2  + 2PQ cos  + Q2 sin2   R2 = P2 + Q2 + 2PQ cos 

2 2
 R  P  Q  2PQcos   (3)

Direction of the resultant : The resultant makes angle ‘  ’ with A (say)

EC EC Qsin 
From triangle CAE : tan     tan  
AE AB  BE P  Q cos 

 Qsin  
   tan 1    (4)
 P  Q cos  


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