Question

If $\overline{a}$ and $\overline{b}$ are unit vectors and $\theta$ be the angle between them then $\sin \frac{\theta }{2}=$

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

Answer 1

Answer: (C)

$\frac{1}{2}|\overline{a}-\overline{b}|$

Since a and b are unit vectors $|a|=|b|=1$

then,

$|\overrightarrow{a}-\overrightarrow{b}|=\sqrt{|a{|}^2+|b{|}^2+2|a||b|\cos (180-\theta )}$

Putting the values

$|\overrightarrow{a}-\overrightarrow{b}|=\sqrt{1+1+2\times 1\times 1(-\cos \theta )}$

$=\sqrt{2-2\cos \theta }$

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

$=\sqrt{2\times 2{\left(\frac{\sin \theta }{2}\right)}^2}$

$|\overrightarrow{a}-\overrightarrow{b}|=2\sin \frac{\theta }{2}$

$\sin \frac{\theta }{2}=\frac{1}{2}|\overrightarrow{a}-\overrightarrow{b}|$