Question

If unit vectors $\hat{A}$ and $\hat{B}$ are inclined at an angle $\theta$, then $|\hat{A}-\hat{B}|$ is equal to

• A. $2\cos \frac{\theta }{2}$
• B. $\sin \theta$
• C. $2\sin \frac{\theta }{2}$
• D. $\cos \theta$

Answer 1

The correct option is C $2\sin \frac{\theta }{2}$

We can write,
$|\hat{A}-\hat{B}{|}^2=|\hat{A}{|}^2+|\hat{B}{|}^2-2|\hat{A}||\hat{B}|\cos \theta$
$\Rightarrow |\hat{A}-\hat{B}{|}^2=1+1-2\cos \theta$
$\Rightarrow |\hat{A}-\hat{B}{|}^2=2(1-\cos \theta )$
$\Rightarrow |\hat{A}-\hat{B}{|}^2=4{\sin }^2\frac{\theta }{2}(\because 1-\cos \theta =2{\sin }^2\frac{\theta }{2})$
$\Rightarrow |\hat{A}-\hat{B}|=2\sin \frac{\theta }{2}$