If $\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}$ are unit vectors such that $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{0}$ and $\left(\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{b}}\right)=\frac{\mathrm{\pi }}{3},$ then $\left|\overline{\mathrm{a}}\times \overline{\mathrm{b}}\right|+\left|\overline{\mathrm{b}}\times \overline{\mathrm{c}}\right|+\left|\overline{\mathrm{c}}\times \overline{\mathrm{a}}\right|=$
 

$$\begin{gathered} \left|\overline{\mathrm{a}}\right|=\left|\overline{\mathrm{b}}\right|=\left|\overline{\mathrm{c}}\right|=1 |\overline{\mathrm{a}}\times \overline{\mathrm{b}}|1\times 1\times \raisebox{1ex}{$\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ \overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}=\overline{0} \Rightarrow 0+\overline{\mathrm{a}}\times \overline{\mathrm{b}}+\overline{\mathrm{a}}\times \overline{\mathrm{c}}=0 \\ \Rightarrow \overline{\mathrm{a}}\times \overline{\mathrm{b}}=\overline{\mathrm{c}}\times \overline{\mathrm{a}} \\ \overline{\mathrm{b}}\times \overline{\mathrm{a}}+0+\overline{\mathrm{b}}\times \overline{\mathrm{c}}=0 \Rightarrow \overline{\mathrm{b}}\times \overline{\mathrm{c}}=\overline{\mathrm{a}}\times \overline{\mathrm{b}} \\ |\overline{\mathrm{a}}\times \overline{\mathrm{b}}|+|\overline{\mathrm{b}}\times \overline{\mathrm{c}}|+|\overline{\mathrm{c}}\times \overline{\mathrm{a}}|=3|\overline{\mathrm{a}}\times \overline{\mathrm{b}}| \\ =\frac{3\sqrt{3}}{2} \end{gathered}$$