Question

If $\overline{a}$ and $\overline{b}$ are two unit vectors and $\theta$ be the angle between them, then $\sin \left(\frac{\sin \theta }{2}\right)=$.

• 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

The correct option is C $\frac{1}{2}|\overline{a}-\overline{b}|$

The rectangle representative of the vector of their reserve $\tan A$

$|\overrightarrow{a}|=|\overrightarrow{s}|=1$

$|\overrightarrow{a}-\overrightarrow{3}{|}^2=|\overrightarrow{a}{|}^2+|\overrightarrow{-b}{|}^2+2|\overrightarrow{a}|.|\overrightarrow{-b}|.\cos ({180}^o-\theta )$

$=1^2+1^2+2,1,1,-\cos \theta$

$=2-2\cos \theta =2\left(1.\cos \theta \right)$

$=2\left(2{\sin }^2\theta /2\right)=4\left({\sin }^2\theta /2\right)$

$\Rightarrow |\overrightarrow{a}-\overrightarrow{b}|=\sqrt{4{\sin }^2\theta /2}=2\sin \theta /2$

$\Rightarrow \sin \theta /2=\frac{1}{2}|\overrightarrow{a}-\overrightarrow{3}|$