Question

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

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

Answer 1

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

We have ${|\overrightarrow{a}-\overrightarrow{b}|}^2={\left|\overrightarrow{a}\right|}^2+{\left|b\right|}^2-2(\overrightarrow{a}.\overrightarrow{b})$

$=1+1-2\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta =2-2\cos \theta =4{\sin }^2\frac{\theta }{2}$

$\Rightarrow {\left|\frac{\overrightarrow{a}-\overrightarrow{b}}{2}\right|}^2={\sin }^2\frac{\theta }{2}\Rightarrow \left|\frac{\overrightarrow{a}-\overrightarrow{b}}{2}\right|=\sin \frac{\theta }{2}$