Question

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

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

Answer 1

Answer: (C)

The angle $\alpha$ which the resultant $R$ makes with A is given by
$\tan \,\alpha=\frac{B \sin \theta}{A+B \cos \theta}$
or $\frac{\sin (\theta / 2)}{\cos (\theta / 2)}=\frac{2 B \sin (\theta / 2) \cos (\theta / 2)}{A+B \cos \theta}$
which gives $A+B \cos \theta=2 B \cos ^{2}\left(\frac{\theta}{2}\right)$
or $A+B\left[2 \cos ^{2}\left(\frac{\theta}{2}\right)-1\right]=2 B \cos ^{2}\left(\frac{\theta}{2}\right)$
$A=B$