Question

If ${\overrightarrow{e}}_1$ & ${\overrightarrow{e}}_2$ are two unit vectors and $\theta$ is the angle between them, then $\sin \left(\frac{\theta }{2}\right)$ is

• A. $\frac{1}{2}|{\overrightarrow{e}}_1+{\overrightarrow{e}}_2|$
• B. $\frac{1}{2}|{\overrightarrow{e}}_1-{\overrightarrow{e}}_2|$
• C. $\frac{{\overrightarrow{e}}_1.{\overrightarrow{e}}_2}{2}$
• D. $\frac{|{\overrightarrow{e}}_1\times {\overrightarrow{e}}_2|}{2\left|{\overrightarrow{e}}_1\right|\left|{\overrightarrow{e}}_2\right|}$

Answer 1

The correct option is B $\frac{1}{2}|{\overrightarrow{e}}_1-{\overrightarrow{e}}_2|$

$\sin \left(\frac{\theta }{2}\right)$

$=\frac{1}{2}\sqrt{4{\sin }^2\frac{\theta }{2}}$

$=\frac{1}{2}\sqrt{2(1-\cos \theta )}$

$=\frac{1}{2}\sqrt{1+1-2.\cos \theta }$
$=\frac{1}{2}\sqrt{{\overrightarrow{e_1}}^2+{\overrightarrow{e_2}}^2-2|\overrightarrow{e_1}||\overrightarrow{e_2}|\cos \theta }=\frac{1}{2}\sqrt{(\overrightarrow{e_1}-\overrightarrow{e_2}{)}^2}$

$=\frac{1}{2}|\overrightarrow{e_1}-\overrightarrow{e_2}|$