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Addition of Vectors

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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 →   +   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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