The scalar \(\overline{\mathrm{a}} \cdot[(\overline{\mathrm{b}}+\overline{\mathrm{c}}) \times(\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}})]\) equals

\(\begin{aligned} & \overline{\mathrm{a}}[(\overline{\mathrm{b}} \times \overline{\mathrm{c}}) \times(\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}})] \\ & =\overline{\mathrm{a}}[(\overline{\mathrm{b}} \times \overline{\mathrm{a}}+\overline{\mathrm{b}} \times \overline{\mathrm{b}}+\overline{\mathrm{b}} \times \overline{\mathrm{c}}+\overline{\mathrm{c}} \times \overline{\mathrm{a}}+\overline{\mathrm{c}} \times \overline{\mathrm{b}}+\overline{\mathrm{c}} \times \overline{\mathrm{c}})] \\ & =\overline{\mathrm{a}}(\overline{\mathrm{b}} \times \overline{\mathrm{a}})+\overline{\mathrm{a}}(\overline{\mathrm{b}} \times \overline{\mathrm{c}})+\overline{\mathrm{a}}(\overline{\mathrm{c}} \times \overline{\mathrm{a}})+\overline{\mathrm{a}}(\overline{\mathrm{c}} \times \overline{\mathrm{b}}) \\ & =\left[\begin{array}{lll}\overline{\mathrm{a}} & \overline{\mathrm{b}} & \overline{\mathrm{c}}\end{array}\right]+\left[\begin{array}{lll}\overline{\mathrm{a}} & \bar{c} & \overline{\mathrm{~b}}\end{array}\right] \\ & =\left[\begin{array}{lll}\overline{\mathrm{a}} & \overline{\mathrm{b}} & \overline{\mathrm{c}}\end{array}\right]-\left[\begin{array}{lll}\overline{\mathrm{a}} & \overline{\mathrm{b}} & \overline{\mathrm{c}}\end{array}\right] \\ & =0\end{aligned}\)