Question

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

• A. $\sin \frac{\theta }{2}$
• B. $\sin \theta$
• C. $2\sin \theta$
• D. $\sin 2\theta$

Answer 1

The correct option is A $\sin \frac{\theta }{2}$

Since $a$ and $b$ are unit vector $|a|=|b|=1$

$|a-b{|}^2=|a{|}^2+|b{|}^2-2|a||b|\cos \theta$

$1+1-2.|.|.\cos \theta$

$=2-2\cos \theta$

$|a-b{|}^2=2(1-\cos \theta )$

$|a-b|=2(1-\cos \theta {)}^{1/2}$

$|a-b|={\left(2v2{\sin }^2\frac{\theta }{2}\right)}^{1/2}$

$|a+b|=2\sin \frac{\theta }{2}$

now $\frac{|a-b|}{2}=\frac{2\sin \frac{\theta }{2}}{2}=\sin \frac{\theta }{2}$