If $\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{b}}\text{ and }\overrightarrow{\mathrm{c}}$  are three vectors of   equal magnitude and the angle between each pair of vectors is $\frac{\mathrm{\pi }}{3}$ such that  $|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}|=\sqrt{6}$ . then $\left|\overrightarrow{\mathrm{a}}\right|=$

$\begin{array}{l}|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}{|}^2=6\\ \text{ or }|\overrightarrow{\mathrm{a}}{|}^2+|\overrightarrow{\mathrm{b}}{|}^2+|\overrightarrow{\mathrm{c}}{|}^2\\ +2(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}}\cdot \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}}\cdot \overrightarrow{\mathrm{a}})=6\\ \Rightarrow \left|\overrightarrow{\mathrm{a}}\right|=\left|\overrightarrow{\mathrm{b}}\right|=\left|\overrightarrow{\mathrm{c}}\right|\text{ and }\\ \overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{b}}=\left|\overrightarrow{\mathrm{a}}\right|\left|\overrightarrow{\mathrm{b}}\right|\cdot \cos \frac{\mathrm{\pi }}{3}\\ \text{ i.e., }\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{b}}=\frac{1}{2}|\overrightarrow{\mathrm{a}}{|}^2\\ \therefore 3|\overrightarrow{\mathrm{a}}{|}^2+3|\overrightarrow{\mathrm{a}}{|}^2=6\\ \Rightarrow \left|\overrightarrow{\mathrm{a}}{|}^2\text{ or }\right|\overrightarrow{\mathrm{a}}|=1\end{array}$