If $\overline{\mathrm{a}},\overline{\mathrm{b}},\overline{\mathrm{c}}$ are three mutually perpendicular vectors such that $\left|\overline{\mathrm{a}}\right|=\left|\overline{\mathrm{b}}\right|=\left|\overline{\mathrm{c}}\right|,$ then 

$\left(\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}, \overline{\mathrm{a}}\right) =$

$$\begin{gathered} \left|\overline{\mathrm{a}}\right|=\left|\overline{\mathrm{b}}\right|=\left|\overline{\mathrm{c}}\right|=\mathrm{\lambda } \overline{\mathrm{a}}.\overline{\mathrm{b}}=0=\overline{\mathrm{b}}.\overline{\mathrm{c}}=\overline{\mathrm{c}}.\overline{\mathrm{a}} \\ \cos \left(\underline{\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}},\overline{\mathrm{a}}}\right)=\frac{(\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}).\overline{\mathrm{a}}}{(\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}})\left|\overline{\mathrm{a}}\right|} \\ {\left|\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}\right|}^2={\mathrm{\lambda }}^2+{\mathrm{\lambda }}^2+{\mathrm{\lambda }}^2+0+0+0=3{\mathrm{\lambda }}^2 \\ \cos \left(\underline{\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}},\overline{\mathrm{a}}}\right)=\frac{{\mathrm{\lambda }}^2+0+0}{\sqrt{3}\mathrm{\lambda }.\mathrm{\lambda }}=\frac{1}{\sqrt{3}} \\ \left(\underline{\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}},\overline{\mathrm{a}}}\right)={\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right) \end{gathered}$$