**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: **

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

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