If $\overline{\mathrm{\upsilon }}$ and $\overline{\mathrm{\nu }}$ are unit vectors inclined at an angle $\mathrm{\alpha }$ and $\overline{\mathrm{a}}$ is a unit vector bisecting the angle between them, then $\overline{\mathrm{a}}$=

$$\begin{gathered} \left|\overline{\mathrm{u}}\right|=1 \left|\overline{\mathrm{v}}\right|=1 \overline{\mathrm{u}}.\overline{\mathrm{v}}=1\times 1\times \cos \mathrm{\alpha } \\ =\cos \mathrm{\alpha } \\ \begin{array}{c}\hat{\mathrm{u}}+\hat{\mathrm{v}}=\overline{\mathrm{u}}+\overline{\mathrm{v}}\\ \left|\hat{\mathrm{u}}+\hat{\mathrm{v}}\right|=\left|\overline{\mathrm{u}}+\overline{\mathrm{v}}\right|\end{array}\overline{)\begin{array}{c}{\left|\overline{\mathrm{u}}+\overline{\mathrm{v}}\right|}^2=1+1+2\mathrm{cos\alpha }\\ =2(1+\cos \mathrm{\alpha })\\ =4{\cos }^2 \mathrm{\alpha }/2\end{array}} \\ \therefore \mathrm{Unit} \mathrm{rector} \mathrm{bisected} \mathrm{angle} \mathrm{v}/\mathrm{w} \mathrm{them} \mathrm{is} \\ \frac{\hat{\mathrm{u}}+\hat{\mathrm{v}}}{\left|\hat{\mathrm{u}}+\hat{\mathrm{v}}\right|} =\frac{\overline{\mathrm{u}}+\overline{\mathrm{v}}}{2 \cos \mathrm{\alpha }/2} \end{gathered}$$