If \(\overline{\mathrm{a}}, \overline{\mathrm{b}}\) and \(\overline{\mathrm{c}}\) are three non-coplanar vectors, then \((\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}) \cdot[(\overline{\mathrm{a}}+\overline{\mathrm{b}}) \times(\overline{\mathrm{a}}+\overline{\mathrm{c}})]\) equals

\(\begin{aligned} & {[\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}] \cdot[(\overline{\mathrm{a}}+\overline{\mathrm{b}}) \times(\overline{\mathrm{a}}+\overline{\mathrm{c}})]} \\ & =(\overline{\mathrm{a}}+\overline{\mathrm{b}}) \cdot[(\overline{\mathrm{a}}+\overline{\mathrm{b}}) \times(\overline{\mathrm{a}}+\overline{\mathrm{c}})]+\overline{\mathrm{c}} \cdot[(\overline{\mathrm{a}}+\overline{\mathrm{b}}) \times(\overline{\mathrm{a}}+\overline{\mathrm{c}})] \\ & =0+[\overline{\mathrm{c}} \overline{\mathrm{a}}+\overline{\mathrm{b}} \overline{\mathrm{a}}+\overline{\mathrm{c}}] \\ & =[\overline{\mathrm{c}} \overline{\mathrm{a}} \overline{\mathrm{a}}+\overline{\mathrm{c}}]+[\overline{\mathrm{c}} \overline{\mathrm{b}} \overline{\mathrm{a}}+\overline{\mathrm{c}}] \\ & =[\overline{\mathrm{c}} \overline{\mathrm{a}} \overline{\mathrm{a}}]+[\overline{\mathrm{c}} \overline{\mathrm{a}} \overline{\mathrm{c}}]+[\overline{\mathrm{c}} \overline{\mathrm{b}} \overline{\mathrm{a}}]+[\overline{\mathrm{c}} \overline{\mathrm{b}} \overline{\mathrm{c}}] \\ & =0+0+[\overline{\mathrm{c}} \overline{\mathrm{b}} \overline{\mathrm{a}}]+0 \\ & =-[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]\end{aligned}\)