If $\overline{\mathrm{a}},\overline{\mathrm{b}}\text{ and }\overline{\mathrm{c}}$ are three vectors such that $\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}=\overrightarrow{0,}\left|\overrightarrow{\mathrm{a}}\right|=1,\left|\overrightarrow{\mathrm{b}}\right|=2$ and $\left|\overrightarrow{\mathrm{c}}\right|=3$ then $\overrightarrow{\mathrm{a}}.\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}}.\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}}.\overrightarrow{\mathrm{a}}$is equal to

$$\begin{gathered} Given \left|\overline{a}\right|=1,\left|\overline{b}\right|=2,\left|\overline{c}\right|=3 \\ and \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overline{\mathrm{c}}=\overrightarrow{0} \\ \Rightarrow (\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overline{\mathrm{c}})\cdot (\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overline{\mathrm{c}})=0 \\ \Rightarrow |\overrightarrow{\mathrm{a}}{|}^2+|\overrightarrow{\mathrm{b}}{|}^2+|\overrightarrow{\mathrm{c}}{|}^2+2(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}}\cdot \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}}\cdot \overrightarrow{\mathrm{a}})=0 \\ \Rightarrow 1+4+9++2(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}}\cdot \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}}\cdot \overrightarrow{\mathrm{a}})=0 \\ \Rightarrow 2(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}}\cdot \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}}\cdot \overrightarrow{\mathrm{a}})=-14 \\ \Rightarrow (\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}}\cdot \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}}\cdot \overrightarrow{\mathrm{a}})=-7 \end{gathered}$$