Question

The angle between $\overrightarrow{P}$ and $\overrightarrow{Q}$ is $\theta$. $\overrightarrow{R}=\overrightarrow{P}+\overrightarrow{Q}$ makes angle $\theta /2$ with $\overrightarrow{P}$. Which of the following is true?

• A. $P=2Q$
• B. $2P=Q$
• C. $PQ=1$
• D. none of the above

Answer 1

Answer: (D)

none of the above

$\tan \frac{\theta }{2}=\frac{Q\sin \theta }{P+Q\cos \theta }$

$\frac{\sin \left(\theta /2\right)}{\cos \left(\theta /2\right)}=\frac{Q\sin \theta }{P+Q\cos \theta }$

$\frac{2\sin \left(\theta /2\right)\cos \left(\theta /2\right)}{2\cos \left(\theta /2\right)\cos \left(\theta /2\right)}=\frac{\sin \left(\theta \right)}{2{\cos }^2\left(\theta /2\right)}=\frac{\sin \theta }{1+\cos \theta }=\frac{\sin \theta }{\frac{P}{Q}+\cos \theta }$

On comparing: $\frac{P}{Q}=1orP=Q$