Question

Which of the following relations is true for two unit vectors $\hat{A}$ and $\hat{B}$ making an angle $\theta$ to each other?

• A. $|\hat{A}+\hat{B}|=|\hat{A}-\hat{B}|\tan \frac{\theta }{2}$
• B. $|\hat{A}-\hat{B}|=|\hat{A}+\hat{B}|\tan \frac{\theta }{2}$
• C. $|\hat{A}+\hat{B}|=|\hat{A}-\hat{B}|\cos \frac{\theta }{2}$
• D. $|\hat{A}-\hat{B}|=|\hat{A}+\hat{B}|\cos \frac{\theta }{2}$

Answer 1

Answer: (B)

$|\hat{A}+\hat{B}|=\sqrt{|\hat{A}{|}^2+|\hat{B}{|}^2+2|\hat{A}\Vert \hat{B}|\cos \theta }$
$=\sqrt{1+1+2\cos \theta }$
$=\sqrt{2(1+\cos \theta )}$
$=\sqrt{2\times 2{\cos }^2\frac{\theta }{2}}$
$=2\cos \frac{\theta }{2}$
$|\hat{A}-\hat{B}|=\sqrt{|\hat{A}{|}^2+|\hat{B}{|}^2-2|\hat{A}||\hat{B}|\cos \theta }$
$=\sqrt{2-2\cos \theta }$
$=2\sin \frac{0}{2}$
$\frac{|\hat{A}+\hat{B}|}{|\hat{A}-\hat{B}|}=\cot \frac{\theta }{2}$