Question

If $\overrightarrow{a}and\overrightarrow{b}$are unit vectors and $\theta$ is the angle between them, show that $\sin \frac{\theta }{2}=\frac{1}{2}|\overrightarrow{a}-\overrightarrow{b}|$ ?

Answer 1

$\begin{array}{c}\left|\overline{a}\right|=1=\left|\overline{b}\right|and\theta =\left(\overline{a,}\overline{b}\right)\\ weknowthat,\\ {\left|\overline{a}-\overline{b}\right|}^2={\left|a\right|}^2+{\left|b\right|}^2-2\left|a\right|\left|b\right|\\ =1+1-2\left|a\right|\left|b\right|\cos \theta \\ =2-2\left|a\right|\left|b\right|\cos \theta \\ =2(1-\cos \theta )\\ =2\times 2si{n}^2\frac{\theta }{2}\\ {\left|\overline{a}-\overline{b}\right|}^2=4si{n}^2\frac{\theta }{2}\\ \left|\overline{a}-\overline{b}\right|=2\sin \frac{\theta }{2}\\ \therefore \sin \frac{\theta }{2}=\frac{1}{2}\left(\left|\overline{a}-\overline{b}\right|\right)Ans.\end{array}$