Question

If $\theta$ is the angle between unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, then $\sin \left(\frac{\theta }{2}\right)$ equals

• A. $\frac{1}{2}|\overrightarrow{a}+\overrightarrow{b}|$
• B. $\frac{1}{2}|\overrightarrow{a}\times \overrightarrow{b}|$
• C. $\frac{1}{2}\sqrt{1-(\overrightarrow{a}.\overrightarrow{b})}$
• D. $\frac{1}{2}|\overrightarrow{a}-\overrightarrow{b}|$

Answer 1

The correct option is D $\frac{1}{2}|\overrightarrow{a}-\overrightarrow{b}|$

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

$=1+1-2\left(1\right)\left(1\right)cos\theta$

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

$\Rightarrow {|\overrightarrow{a}-\overrightarrow{b}|}^2=2\left(2si{n}^2\theta /2\right)$

$\Rightarrow si{n}^2\theta /2=\frac{{|\overrightarrow{a}-\overrightarrow{b}|}^2}{4}\Rightarrow sin\theta /2=\frac{|\overrightarrow{a}-\overrightarrow{b}|}{2}\left(D\right)$