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Scalars and Vectors

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Question

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

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Solution

From triangle law of vector addition , we can say that angle between $(2 \vec{P} + \vec{Q})$ and $\vec{Q}$ is 90º

Alternate solution $|\vec{P} + \vec{Q}| = |\vec{P}|$

$P^{2} + Q^{2} + 2 P Q \cos θ = P^{2} \Rightarrow Q + 2 P \cos θ = 0 \Rightarrow \cos θ = \frac{- Q}{2 P}$

$\tan α = \frac{2 P \sin θ}{2 P \cos θ + Q} = \infty ∵ [2 P \cos θ + Q = 0]$
$α = 90 º$
So angle between $2 \bar{P} + \bar{Q}$ and $\bar{Q}$ is 90º

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