Notes On Addition of Vectors - CBSE Class 12 Maths

Triangle Law of Vector Addition:

The sum of two vectors representing two sides of a triangle taken in the same order is

given by the vector representing the third side of the triangle taken in the opposite order.

Ex:

Let vectors A and B be represented by the vectors AB and BC, respectively. By joining A

and C, we get triangle ABC.

    AB   +   BC   =   AC      

Reverse the direction of vector AC to get vector CA.

  AB   +   BC   +   CA   =   AA  

  AB   +   BC   +   CA   =   AA   =   0  

Thus, the sum of the vectors representing the three sides of a triangle in order is a zero vector.

Here   BC   = -   BC   =   -b  

AC' = AB' + BC'

AC' = AB + (-BC )

AC' = a +  (- b ) = a b


Parallelogram Law of Vector Addition:

The sum of two vectors representing adjacent sides of a parallelogram is given by the diagonal of the parallelogram passing through the common point of the two adjacent sides.

OA + OB = OC ....parallelogram law

AC = OB = b

OA + AC = OC ....traingle law

Thus, we see that the triangle and parallelogram laws of vector addition are equivalent

to each other.

Commutative Property of Vector Addition

Given two vectors  a   and  b      :

Let   AB     = a and  BC   = b             


By triangle law of vector addition:

AB + BC = AC = a    + b ...(1)

In parallelogram ABCD:

AD = BC = b

DC = AB = a


By triangle law of vector addition:

AD + DC = AC = b    + a ...(2)

From (1) and (2):

 a   +  b     = b + a       


Additive Identity:

AB   + 0   = 0   + AB   = AB

Associative Property of Vector Addition

Given three vectors  a , b     and c     :

(  a   +  b     ) + c = a + ( b    +  c )

Consider the vectors   AB    =  a    ,  BC   =   b   and   CD   =  c    

In ∆ ABC:

AB + BC = AC =  a   +  b    [Triangle law of vector addition]

In ∆ ACD:

AC + CD = AD = ( a    +   b ) + c ...(1) [Triangle law of vector addition]

In ∆ BCD:

BC + CD = BD = b   + c       [Triangle law of vector addition]

In ∆ ABD:

AB + BD = AD = a + ( b   + c ) ...(2) [Triangle law of vector addition]

From (1) and (2):

( a   + b ) +  c =  a + ( b   + c )

Summary

Triangle Law of Vector Addition:

The sum of two vectors representing two sides of a triangle taken in the same order is

given by the vector representing the third side of the triangle taken in the opposite order.

Ex:

Let vectors A and B be represented by the vectors AB and BC, respectively. By joining A

and C, we get triangle ABC.

    AB   +   BC   =   AC      

Reverse the direction of vector AC to get vector CA.

  AB   +   BC   +   CA   =   AA  

  AB   +   BC   +   CA   =   AA   =   0  

Thus, the sum of the vectors representing the three sides of a triangle in order is a zero vector.

Here   BC   = -   BC   =   -b  

AC' = AB' + BC'

AC' = AB + (-BC )

AC' = a +  (- b ) = a b


Parallelogram Law of Vector Addition:

The sum of two vectors representing adjacent sides of a parallelogram is given by the diagonal of the parallelogram passing through the common point of the two adjacent sides.

OA + OB = OC ....parallelogram law

AC = OB = b

OA + AC = OC ....traingle law

Thus, we see that the triangle and parallelogram laws of vector addition are equivalent

to each other.

Commutative Property of Vector Addition

Given two vectors  a   and  b      :

Let   AB     = a and  BC   = b             


By triangle law of vector addition:

AB + BC = AC = a    + b ...(1)

In parallelogram ABCD:

AD = BC = b

DC = AB = a


By triangle law of vector addition:

AD + DC = AC = b    + a ...(2)

From (1) and (2):

 a   +  b     = b + a       


Additive Identity:

AB   + 0   = 0   + AB   = AB

Associative Property of Vector Addition

Given three vectors  a , b     and c     :

(  a   +  b     ) + c = a + ( b    +  c )

Consider the vectors   AB    =  a    ,  BC   =   b   and   CD   =  c    

In ∆ ABC:

AB + BC = AC =  a   +  b    [Triangle law of vector addition]

In ∆ ACD:

AC + CD = AD = ( a    +   b ) + c ...(1) [Triangle law of vector addition]

In ∆ BCD:

BC + CD = BD = b   + c       [Triangle law of vector addition]

In ∆ ABD:

AB + BD = AD = a + ( b   + c ) ...(2) [Triangle law of vector addition]

From (1) and (2):

( a   + b ) +  c =  a + ( b   + c )

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