# Parallelogram Law of Vector Addition

## Parallelogram Law of Vector Addition

If two vectors acting simultaneously at a point can be represented both in magnitude and direction by the adjacent sides of a parallelogram drawn from a point, then the resultant vector is represented both in magnitude and direction by the diagonal of the parallelogram passing through that point.

If two non-zero vectors are represented by the two adjacent side of a parallelogram then the resultant is given by the diagonal of the parallelogram passing through the point of intersection of the two vectors.

1) Magnitude:

Since, $${{R}^{2}}={{\left( ON \right)}^{2}}+{{\left( CN \right)}^{2}}$$,

$$\Rightarrow {{R}^{2}}={{\left( OA+AN \right)}^{2}}+{{\left( CN \right)}^{2}}$$,

$$\therefore \,\,R=|\overrightarrow{R}|=|\overrightarrow{A}+\overrightarrow{B}|=\sqrt{{{A}^{2}}+{{B}^{2}}+2AB\cos \theta }$$,

Special cases: $$R=A+B$$ when $$\theta ={{0}^{0}}$$,

$$R=A-B$$ When $$\theta ={{180}^{0}}$$,

$$R=\sqrt{{{A}^{2}}+{{B}^{2}}}$$ When $$\theta ={{90}^{0}}$$.

2) Direction:

$$\tan \beta =\frac{CN}{ON}=\frac{B\sin \theta }{A+B\cos \theta }$$,

$$\beta ={{\tan }^{-1}}\left( \frac{B\sin \theta }{A+B\cos \theta } \right)$$.