Question

If $a$ and $b$ are two unit vectors and $\varphi$ is the angle between them, then $\frac{1}{2}|a-b|$ is equal to

• A. $0$
• B. $\frac{\pi }{2}$
• C. $\left|\sin \frac{\varphi }{2}\right|$
• D. $\left|\cos \frac{\varphi }{2}\right|$

Answer 1

The correct option is C $\left|\sin \frac{\varphi }{2}\right|$

${|a-b|}^2=(a-b).(a-b)={\left|a\right|}^2{\left|b\right|}^2-2\left|a\right|\left|b\right|\cos \varphi$
$=2(1-\cos \varphi )=4{\sin }^2\left(\frac{\varphi }{2}\right)$
Hence $\left(\frac{1}{2}\right)|a-b|=\left|\sin \frac{\varphi }{2}\right|$