Question

If $x$ and $y$ are two unit 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 $\left|\sin \frac{1}{2}\varphi \right|$

$\frac{1}{2}|x-y|=\frac{\sqrt{(x-y).(x-y)}}{2}=\frac{\sqrt{{\left|x\right|}^2-2x.y+{\left|y\right|}^2}}{2}=\frac{\sqrt{2-2\left|x\right|\left|y\right|\cos \varphi }}{2}=\left|\sin \frac{\varphi }{2}\right|$