Question

If ${\overrightarrow{e}}_1$ & ${\overrightarrow{e}}_2$ are two unit vectors and $\theta$ is the angle between them , then $\cos \left(\frac{\theta }{2}\right)$ 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 A $\frac{1}{2}|{\overrightarrow{e}}_1+{\overrightarrow{e}}_2|$

$\cos \theta =\overrightarrow{e_1}.\overrightarrow{e_2}$
Also, $\cos \theta =2{\cos }^2\frac{\theta }{2}-1,$
$\Rightarrow \cos \frac{\theta }{2}=\sqrt{\frac{1+\cos \theta }{2}}$
$\cos \frac{\theta }{2}=\sqrt{1+{\overrightarrow{e}}_1.{\overrightarrow{e}}_2}\times \sqrt{0.5}=\sqrt{2+2{\overrightarrow{e}}_1.{\overrightarrow{e}}_2}\times \sqrt{0.25}$
$\cos \frac{\theta }{2}=\sqrt{1+1+2{\overrightarrow{e}}_1.{\overrightarrow{e}}_2}\times \sqrt{0.25}$
$\cos \frac{\theta }{2}=\frac{|\overrightarrow{e_1}+\overrightarrow{e_2}|}{2}$