Question

If $\vec{a}$ and $\vec{b}$ are unit vectors and $\theta$ is the angle between $\vec{a}$ and $\vec{b}$ then sin $\frac {\theta}{2}$

• A. $\left|\vec{a}+\vec{b}\right|$
• B. $\frac {\left|\vec{a}+\vec{b}\right|}{2}$
• C. $\frac {\left|\vec{a}-\vec{b}\right|}{2}$
• D. $\left|\vec{a}-\vec{b}\right|$

Answer 1

Answer: (C)

$|\vec{a}-\vec{b}|^{2}=|\vec{a}|^{2}+|\vec{b}|^{2}-2|\vec{a}||\vec{b}| \cos \theta$
$=2(1-\cos \theta) \,\,\,(\because|\vec{a}|=|\vec{b}|=1)$
$\Rightarrow |\vec{a}-\vec{b}|^{2}=2\left(2 \sin ^{2} \frac{\theta}{2}\right)$
$\therefore \sin \frac{\theta}{2}=\frac{|\vec{a}-\vec{b}|}{2}$