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Addition of Vectors - Lesson 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 â†’   +   BC â†’   =   AC â†’

Reverse the direction of vector AC to get vector CA.

AB â†’   +   BC â†’   +   CAâ†’   =   AA â†’

AB â†’   +   BC â†’   +   CA â†’   =   AA â†’   =   0 â†’

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

Here   BC â†’   = -   BC â†’   =   -b â†’

AC' â†’ = AB' â†’ + BC' â†’

â‡’ AC' â†’ = AB â†’ + (-BC â†’ )

â‡’ AC' â†’ = a â†’ +  (- b â†’ ) = a â†’ -  b â†’

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 â†’ + OB â†’ = OC â†’ ....parallelogram law

AC â†’ = OB â†’ = b â†’

OA â†’ + AC â†’ = OC â†’ ....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 â†’   and  b â†’      :

Let   AB â†’     = a â†’ and  BC â†’   = b â†’

By triangle law of vector addition:

AB â†’ + BC â†’ = AC â†’ = a â†’    + b â†’ ...(1)

In parallelogram ABCD:

AD â†’ = BC â†’ = b â†’

DC â†’ = AB â†’ = a â†’

By triangle law of vector addition:

AD â†’ + DC â†’ = AC â†’ = b â†’    + a â†’ ...(2)

From (1) and (2):

a â†’   +  b â†’     = b â†’ + a â†’

AB â†’   + 0 â†’   = 0 â†’   + AB â†’   = AB â†’

Associative Property of Vector Addition

Given three vectors  a â†’ , b â†’     and c â†’     :

(  a â†’   +  b â†’     ) + c â†’ = a â†’ + ( b â†’    +  c â†’ )

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

In âˆ† ABC:

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

In âˆ† ACD:

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

In âˆ† BCD:

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

In âˆ† ABD:

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

From (1) and (2):

( a â†’   + b â†’ ) +  c â†’ =  a â†’ + ( b â†’   + c â†’ )