Question

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors and $\theta$ is the angle between them, then show that $|\overrightarrow{a}-\overrightarrow{b}|=2\sin \frac{\theta }{2}$.

Answer 1

$\overrightarrow{a}.\overrightarrow{b}=|\overrightarrow{a}||\overrightarrow{b}|\cos \theta$
$=1.1.\cos \theta$
$=\cos \theta$
Now,

$|\overrightarrow{a}-\overrightarrow{b}|=(\overrightarrow{a}-\overrightarrow{b}{)}^2$
$=|\overrightarrow{a}{|}^2+|\overrightarrow{b}{|}^2-2\overrightarrow{a}.\overrightarrow{b}$
$=1+1-2\cos \theta$
$=2(1-\cos \theta )$
$\Rightarrow |\overrightarrow{a}-\overrightarrow{b}{|}^2=2.2{\sin }^2\frac{\theta }{2}$
$\Rightarrow |\overrightarrow{a}-\overrightarrow{b}|=2\sin \frac{\theta }{2}$