Question

The angle between two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ is $\theta$. Vector $\overrightarrow{R}$ is the resultant of these two vectors. If $\overrightarrow{R}$ makes an angle $\frac{\theta }{2}$ with $\overrightarrow{A}$, then

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

Answer 1

The correct option is C

A = B

The angle $\alpha$ which the resultant $\overrightarrow{R}$ makes with $\overrightarrow{A}$ is given by
$tan\alpha =\frac{Bsin\theta }{A+Bcos\theta }$
Here, $\alpha =\frac{\theta }{2}$.
Hence, $tan\left(\frac{\theta }{2}\right)=\frac{Bsin\theta }{A+Bcos\theta }$
$\Rightarrow \frac{sin\left(\frac{\theta }{2}\right)}{cos\left(\frac{\theta }{2}\right)}=\frac{2Bsin\left(\frac{\theta }{2}\right)cos\left(\frac{\theta }{2}\right)}{A+Bcos\theta }$
$\Rightarrow 2Bco{s}^2\left(\frac{\theta }{2}\right)=A+Bcos\theta =A+B[2co{s}^2\left(\frac{\theta }{2}\right)-1]$
$\Rightarrow A=B$
Hence, the correct choice is (c).