ADDITION AND SUBTRACTION OF VECTORS

Head to Tail Rule

To add non-zero vectors $\vec{p}$

and $\vec{q}$

join the tail of the second vector $\vec{q}$

with the head of the first vector $\vec{p}$

. Now the vector obtained by joining the tail of the first vector to the head of the second vector is the vector $\vec{p} + \vec{q}$

+ $q$

called the resultant vector of $\vec{p}$

and $\vec{q}$

This method for the addition of two vectors is called head to tail rule of addition. Since $\vec{p}, \vec{q}$

, $q$

and $\vec{p} + \vec{q}$

+ $q$

are along the sides of a triangle XYZ, so this method of addition of vectors is also known as triangle law of addition.

Parallelogram Law of Addition

Consider any parallelogram PQRS. Let $\overrightarrow{PQ} = \vec{p}$

= $p$

and $\overrightarrow{QR} = \vec{q}$

= $q$

Since the vector $\overrightarrow{QR}$

has the same magnitude and direction as that of $\overrightarrow{PS}$

. Similarly, $\overrightarrow{SR}$

has the same magnitude and direction as that $\overrightarrow{PQ}$

. Therefore:

\overrightarrow{PQ} = \overrightarrow{SR} \text{ and } \overrightarrow{QR} = \overrightarrow{PS}

Using triangle law of addition, we have:

\overrightarrow{PQ} + \overrightarrow{QR} = \overrightarrow{PR}

\Rightarrow \vec{p} + \vec{q} = \vec{P R}

Showing that the diagonal vector $\overrightarrow{PR}$

of the parallelogram is the sum of the vectors of $\vec{p}$

and $\vec{q}$

. This is known as parallelogram law of addition.

Furthermore: From $Δ P Q R$ , $\overrightarrow{PR} = \vec{p} + \vec{q}$

= $p$

+ $q$

. From $Δ P S R$ , $\overrightarrow{PR} = \vec{q} + \vec{p}$

= $q$

+ $p$

. $\therefore \vec{p} + \vec{q} = \vec{q} + \vec{p}$

+ $q$

= $q$

+ $p$

. This shows that vector addition is commutative.

For any vector $\vec{a}$

: $\vec{a} + \vec{0} = \vec{0} + \vec{a} = \vec{a}$

+ $0$

= $0$

+ $a$

= $a$

and $\vec{a} + (-\vec{a}) = (-\vec{a}) + \vec{a} = \vec{0}$

+ $(- a$

)= $(- a$

)+ $a$

= $0$