The sum of two forces

Question:

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

Solution:

(90) Given,

$|\vec{R}|=|\vec{P}| \Rightarrow|\vec{P}+\vec{Q}|=|\vec{P}|$

$P^{2}+Q^{2}+2 P Q \cdot \cos \theta=P^{2}$

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

$\Rightarrow \cos \theta=-\frac{Q}{2 P}$                 ...(i)

$\tan \alpha=\frac{2 P \sin \theta}{Q+2 P \cos \theta}=\infty(\because 2 \mathrm{P} \cos \theta+Q=0)$

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

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