Question

If ${\hat{n}}_1$, ${\hat{n}}_2$ are two unit vectors and $\theta$ is the angle between them, then $\cos \frac{\theta }{2}=$

• A. $\frac{1}{2}|{\hat{n}}_1-{\hat{n}}_2|$
• B. $\frac{1}{2}({\hat{n}}_1.{\hat{n}}_2)$
• C. $\frac{1}{2}\left(|{\hat{n}}_1+{\hat{n}}_2|\right)$
• D. $\frac{1}{2}\left(|{\hat{n}}_1\times {\hat{n}}_2|\right)$

Answer 1

The correct option is C $\frac{1}{2}\left(|{\hat{n}}_1+{\hat{n}}_2|\right)$

${\hat{n}}_1+{\hat{n}}_2$ are unit vectors
we have
${\left(|{\hat{n}}_1+{\hat{n}}_2|\right)}^2={\left|{\hat{n}}_1\right|}^2+{\left|{\hat{n}}_2\right|}^2+2\left|{\hat{n}}_1\right|\left|{\hat{n}}_2\right|\cos \theta$
$=2(1+\cos \theta )$

By rearranging $\cos \theta$ in terms of $\cos \theta$ we get
$=4{\cos }^2\frac{\theta }{2}$

$\therefore$ $\cos \frac{\theta }{2}=\frac{1}{2}|{\hat{n}}_1+{\hat{n}}_2|$