Question

Given $\overrightarrow{P}.\overrightarrow{Q}=|\overrightarrow{P}\times \overrightarrow{Q}|$ and $\overrightarrow{R}=\overrightarrow{P}+\overrightarrow{Q}$ then $\left|\overrightarrow{R}\right|$ is:

• A. $\sqrt{P^2+Q^2}$
• B. $P+Q$
• C. $\sqrt{P^2+Q^2+\frac{PQ}{2}}$
• D. ${[P^2+Q^2+\sqrt{2}PQ]}^{\frac{1}{2}}$

Answer 1

Answer: (D)

${[P^2+Q^2+\sqrt{2}PQ]}^{\frac{1}{2}}$

$\overrightarrow{R}=\overrightarrow{P}+\overrightarrow{Q}$
$\Rightarrow R=\sqrt{P^2+Q^2+2PQcos\theta }$
Also since, $\overrightarrow{P}.\overrightarrow{Q}=|\overrightarrow{P}\times \overrightarrow{Q}|$
$PQcos\theta =PQsin\theta$
$\Rightarrow \theta ={45}^{\circ }$
$\therefore R=\sqrt{P^2+Q^2+2PQ\times \frac{1}{\sqrt{2}}}$ $=\sqrt{P^2+Q^2+\sqrt{2}PQ}$