Question

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

• A. 0
• B. $\pi /2$
• C. $\left|\sin \frac{1}{2}\varphi \right|$
• D. $\left|\cos \frac{1}{2}\varphi \right|$

Answer 1

The correct option is C $\left|\sin \frac{1}{2}\varphi \right|$

$\frac{1}{2}|x-y|=\frac{1}{2}\sqrt{x^2+y^2-2|x||y|\cos \varphi }$

$=\frac{1}{2}\sqrt{2(1-\cos \varphi )}=\frac{1}{2}\sqrt{4{\sin }^2\frac{\varphi }{2}}=|\sin \frac{\varphi }{2}|$