Question

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

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

Answer 1

The Correct Option is A

$\left|\hat{A}-\hat{B}\right|^{2}=\left|\hat{A}\right|^{2}+\left|\hat{B}\right|^{2}-2\,\hat{A}\cdot\hat{B}$ $=2-2cos\theta=2\left(1-cos\theta\right)\,\left(\because \left|\hat{A}\right|=\left|\hat{B}\right|=1\right)$ $=2\left(sin^{2} \frac{\theta}{2}\right)$ $=4\,sin^{2} \frac{\theta}{2}$ or $\left|\hat{A}-\hat{B}\right|=2\,sin \frac{\theta}{2}$