Question 31 of 121

The sum of two forces $\vec{P}$ and $\vec{Q}$ is $\vec{R}$ such that $\left|\vec{R}\right|=\left|\vec{P}\right|$. Find the angle between resultant of $2\vec{P}$ and $\vec{Q}$ and $\vec{Q}$ , ________

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Answer: 90

So angle between $\left(2\vec{P}+\vec{Q}\right)$ and $\vec{Q}$ is ${90}^{o}$

Alternate solution

$\left|\vec{\text{P}}+\vec{\text{Q}}\right|=\left|\vec{\text{P}}\right|$
${P}^{2}+{Q}^{2}+2PQ\cos ⁡\theta ={P}^{2}$
$\Rightarrow Q+2PQ\cos ⁡\theta =O$
$\Rightarrow \cos ⁡\theta =-\dfrac{Q}{2P}$

$\tan ⁡\alpha =\dfrac{2P\sin ⁡\theta }{2P\cos ⁡\theta +Q}=\infty$
$∵\left[2P\cos ⁡\theta +Q=0\right]$
$\alpha ={90}^{o}$