If $\overline{\mathrm{a}},\overline{\mathrm{b}},\overline{\mathrm{c}}$ are vectors such that $\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}=0,\left|\overline{\mathrm{a}}\right|=1,\left|\overline{\mathrm{b}}\right|=2,\left|\overline{\mathrm{c}}\right|=3$ then $\overline{\mathrm{a}}\cdot \overline{\mathrm{b}}+\overline{\mathrm{b}}\cdot \overline{\mathrm{c}}+\overline{\mathrm{c}.}\overline{\mathrm{a}}=$

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