If $\widehat{{\mathrm{e}}_1}, \widehat{{\mathrm{e}}_2}\mathrm{and}\widehat{{\mathrm{e}}_3}$ are three unit vectors such that  $\widehat{{\mathrm{e}}_3}=\widehat{{\mathrm{e}}_1}+\widehat{{\mathrm{e}}_2}$ , Then the value of $\left(3\widehat{{\mathrm{e}}_1}.\widehat{{\mathrm{e}}_3}+2\widehat{{\mathrm{e}}_1}.\widehat{{\mathrm{e}}_2}\right)$  is

From the vector triangle, 

$\begin{array}{l}3\widehat{{\mathrm{e}}_1}.\widehat{{\mathrm{e}}_3}+2\widehat{{\mathrm{e}}_1}.\widehat{{\mathrm{e}}_2}=3.1.1.\cos {60}^{\mathrm{o}}+2.1.1.\cos {120}^{\mathrm{o}}\\ =0.5\end{array}$