Question

If $\theta$ is the angle between the unit vectors $\hat{a}$ and $\hat{b},$ then $\cos \frac{\theta }{2}=$

• A. $\frac{1}{2}|\hat{a}-\hat{b}|$
• B. $\frac{1}{2}|\hat{a}+\hat{b}|$
• C. $\left|\frac{\hat{a}-\hat{b}}{\hat{a}+\hat{b}}\right|$
• D. $\left|\frac{\hat{a}+\hat{b}}{\hat{a}-\hat{b}}\right|$

Answer 1

The correct option is B $\frac{1}{2}|\hat{a}+\hat{b}|$

We have

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

Putting values

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

$\Rightarrow {|\overrightarrow{a}+\overrightarrow{b}|}^2=2+2{\cos }^2\frac{\theta }{2}$

$\Rightarrow \cos \frac{\theta }{2}=\frac{1}{2}|\overrightarrow{a}+\overrightarrow{b}|$