Question

Let $R$ be the resultant of $P$ and $Q$ and if $\frac{P}{3}=\frac{Q}{7}=\frac{R}{5}$, then the angle between $P$ and $R$ is

• A. ${\cos }^{-1}\left(\frac{11}{14}\right)$
• B. ${\cos }^{-1}\left(\frac{-11}{14}\right)$
• C. $\frac{2\pi }{3}$
• D. $\frac{5\pi }{6}$

Answer 1

The correct option is B

${\cos }^{-1}\left(\frac{-11}{14}\right)$

Explanation of correct option

Given, $\frac{P}{3}=\frac{Q}{7}=\frac{R}{5}$

Let, $\theta$ be the angle between $P$ and $R$

We need to find the angle$\left(\theta \right)$ between $P$ and $R$

Let, $\frac{P}{3}=\frac{Q}{7}=\frac{R}{5}=\beta \left(say\right)$

This Gives, $P=3\beta ,Q=7\beta ,andR=5\beta$

Since $R$ is the resultant of $P$ and $Q$:

By Using the Formula $R^2=P^2+Q^2+2PQ\cos \theta$ and substituting the values $PandR$ in the above formula we get,

$$\begin{gathered} 25{\beta }^2=9{\beta }^2+49{\beta }^2+42{\beta }^2\cos \theta \\ \Rightarrow 25{\beta }^2=58{\beta }^2+42{\beta }^2\cos \theta \\ \Rightarrow \cos \theta =\frac{-11}{14} \end{gathered}$$

Therefore $\theta ={\cos }^{-1}\left(\frac{-11}{14}\right)$

Hence, the correct answer is option (B)