Question

The resultant of two vectors $\overrightarrow{P}$ and $\overrightarrow{Q}$ is $\overrightarrow{R}$. If $Q$ is doubled, the new resultant is perpendicular to $\overrightarrow{P}$. Then $R$ equals to

• A. $P$
• B. $(P+Q)$
• C. $Q$
• D. $(P-Q)$

Answer 1

The correct option is C $Q$

Vectors $\overrightarrow{P},\overrightarrow{Q}$ and $\overrightarrow{R}$ are shown in diagram.

From diagram,

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

from question,

$\alpha ={90}^{\circ }$ and $Q$ is doubled. So, we get

$\Rightarrow \tan {90}^{\circ }=\frac{2Q\sin \theta }{P+2Q\cos \theta }$

$\Rightarrow \frac{1}{0}=\frac{2Q\sin \theta }{P+2Q\cos \theta }$

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

$\Rightarrow \cos \theta =-\frac{P}{2Q}$

Using formula of resultant vector,

$\therefore |\overrightarrow{R}|=\sqrt{P^2+Q^2+2PQ\cos \theta }$

$\Rightarrow |\overrightarrow{R}|=\sqrt{P^2+Q^2-2.PQ.\frac{P}{2Q}}$

$\Rightarrow |\overrightarrow{R}|=Q$

Hence, option $\left(c\right)$ is correct.