Question

If $\overline{a}$ and $\overline{b}$ 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

$\begin{array}{c}Given,\\ \overline{a}and\overline{b}areunitvectorsand\theta istheanglebetweenthen,\\ \left|\overline{a}-\overline{b}\right|=\sqrt{{\left|a\right|}^2+{\left|b\right|}^2-2\left|a\right|\left|b\right|\cos \theta }\\ \left|\overline{a}-\overline{b}\right|=\sqrt{{\left|1\right|}^2+{\left|1\right|}^2-2\left|1\right|\left|1\right|\cos \theta }\\ \left|\overline{a}-\overline{b}\right|=\sqrt{1+1-2\left|1\right|\left|1\right|\cos \theta }\\ \left|\overline{a}-\overline{b}\right|=\sqrt{2-2\cos \theta }\\ \left|\overline{a}-\overline{b}\right|=\sqrt{2(1-\cos \theta )}(cos\theta =1-2\frac{{\sin }^2\theta }{2})\\ \left|\overline{a}-\overline{b}\right|=2\sin \frac{\theta }{2}\\ \therefore \sin \frac{\theta }{2}=\frac{1}{2}\left|\overline{a}-\overline{b}\right|\end{array}$