Question

The angle between two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ is $\theta$. The resultant of these vectors $\overrightarrow{R}$ makes an angle of $\frac{\theta }{2}$ with $A$. Which of the following is true?

• A. $A=2B$
• B. $A=\frac{B}{2}$
• C. $A=B$
• D. $AB=1$

Answer 1

Answer: (C)

$A=B$

$\tan \frac{\theta }{2}=\frac{B\sin \theta }{A+B\cos \theta }=\frac{B(2\times \sin \frac{\theta }{2}\cos \frac{\theta }{2})}{A+B(2{\cos }^2\frac{\theta }{2}-1)}$

$A\tan \frac{\theta }{2}+B(2{\cos }^2\frac{\theta }{2}-1)\times \tan \frac{\theta }{2}=2B\sin \frac{\theta }{2}\cos \frac{\theta }{2}$

$A\tan \frac{\theta }{2}+2B\cos \frac{\theta }{2}\sin \frac{\theta }{2}-B\tan \frac{\theta }{2}=2B\sin \frac{\theta }{2}\cos \frac{\theta }{2}$

$\Rightarrow A\tan \frac{\theta }{2}=B\tan \frac{\theta }{2}$

$A=B$