Question

The sum of two forces $\overrightarrow{P}$ and $\overrightarrow{Q}$ is $\overrightarrow{R}$ such that $|\overrightarrow{R}|=|\overrightarrow{P}|.$ The angle $\theta$ (in degrees) that the resultant of $2\overrightarrow{P}$ and $\overrightarrow{Q}$ will make with $\overrightarrow{Q}$ is _____.

Answer 1

Given,
$|\overrightarrow{R}|=|\overrightarrow{P}|\Rightarrow |\overrightarrow{P}+\overrightarrow{Q}|=|\overrightarrow{P}|$

$P^2+Q^2+2PQ\cos \theta =P^2$

$\Rightarrow Q+2P\cos \theta =0$

$\Rightarrow \cos \theta =-\frac{Q}{2P}---\left(1\right)$

Let $\alpha$ be the angle between the resultant of $(2\overrightarrow{P}+\overrightarrow{Q})$ and $\overrightarrow{Q}$.


$\tan \alpha =\frac{2P\sin \theta }{Q+2P\cos \theta }$

Putting the value from eq.$\left(1\right)$,

$\tan \alpha =\frac{2P\sin \theta }{0}=\infty$

$\Rightarrow \alpha ={90}^{\circ }$

Hence, $90$ is the correct answer.