PROPERTIES OF VECTOR ADDITION

PROPERTIES OF VECTOR ADDITION

Commutative law of vector addition
Associative law of vector addition
Multiplication of a vector by a number
Division of a vector by a number

COMMUTATIVE LAW OF VECTOR ADDITION

Vector addition is commutative in nature,i.e.

                    `\Rightarrow` `\vecA+\vecB=\vecB+\vecA`

If `\vecA` and `\vecB` are two vectors represent the two adjacent sides of a parallelogram `OPOS`, then the diagonal `\overline{OP}` represents the resultant vector `\vecR`, then from the diagram:

                    `\Rightarrow` `\vecR=\vecA+\vecB`         `\rightarrow (i)`

                    `\Rightarrow` `\vecR=\vecB+\vecA`         `\rightarrow (ii)`

Also,

comparing equations (i) and (ii)

                    `\Rightarrow` `\vecA+\vecB=\vecB+\vecA`

This is commutative property of vector addition.

ASSOCIATIVE LAW OF VECTOR ADDITION 

Vector addition is associative in nature, i.e.

                    `\Rightarrow` `(\vecA+\vecB)+\vecC``=``\vecA+(\vecB+\vecC)`

If `\vecA`,`\vecB` and `\vecC` are three vectors then using head to tail rule the resultants of two vectors as `(\vecA+\vecB)` and `(\vecB+\vecC)` are obtained. If the head to tail rule is again used then.

                    `\Rightarrow` `\vecR=(\vecA+\vecB)+\vecC`        `\rightarrow (i)`

                    `\Rightarrow` `\vecR=\vecA+(\vecB+\vecC)`        `\rightarrow (ii)`

Also,

comparing equations (i) and (ii),

                    `\Rightarrow` `(\vecA+\vecB)+\vecC=\vecA+(\vecB+\vecC)`

This is the associative law of vector. It is proved in the diagram.

MULTIPLICATION OF A VECTOR BY A NUMBER

When a vector `\vecA` is multiplied be a number "`m`", we get a vector say `\vecB. The magnitude of vector `\vecB` is "`m`" times the magnitude of `\vecA`. If `m` is positive the direction of `\vecB` is same as that of the vector `\vecA` while the direction of `\vecB` is opposite to `\vecA` if `m` is negative.

RULES FOR MULTIPLICATION

`m\vecA` `=` `\vecAm`                                       Commutative Law

`m\left(n\vecA)` `=` `\left(mn\right)\vecA`                          Associative Law

`\left(m+n\right)\vecA=m\vecA+n\vecA`                  Distributive Law

`m\left(\vecA+\vecB\right)=m\vecA+m\vecB`               Distributive Law

DIVISION OF A VECTOR BY A NUMBER

Suppose `n` is a non-zero number. When a vector `\vecA` is divided by the number `n`, we get a `\vecB` whose magnitude is `1/n` times the magnitude of `\vecA`. Direction of `\vecB` is same as that of `\vecA` if `n` is positive. Direction of `\vecB` is opposite to that of `\vecA` if `n` is negative.