Question

Two equal vectors $\overline{A}$ and $\overline{B}$ acting at angle $\theta$ with each other then show that there resultant vector bisect this angle $\theta$

Answer 1

Let the angle between two vectors is $\theta$ and the angle between the resultant and any one of the vectors is ${\theta }^{\prime }$.

$\tan {\theta }^{\prime }=\frac{A\sin \theta }{A\cos \theta +A}$

$\tan {\theta }^{\prime }=\frac{\sin \theta }{\left(1+\cos \theta \right)}$

$\tan {\theta }^{\prime }=\frac{2\sin \left(\theta /2\right)\cos \left(\theta /2\right)}{2{\cos }^2\left(\theta /2\right)}$

$\tan {\theta }^{\prime }=\tan \left(\theta /2\right)$

${\theta }^{\prime }=\theta /2$

Thus the resultant vector bisect the angle between the two vectors.