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

Answer 1

Answer: (C)

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