If \(\bar{a}, \bar{b}, \bar{c}\) are mutually perpendicular vectors having magnitudes 1,2,3 respectively, then \([\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}} \overline{\mathrm{b}}-\overline{\mathrm{a}} \overline{\mathrm{c}}]=\)

\(\begin{aligned} & {[\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}} \quad \overline{\mathrm{b}}-\overline{\mathrm{a}} \quad \overline{\mathrm{c}}]} \\ & =(\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}) \cdot[(\overline{\mathrm{b}}-\overline{\mathrm{a}}) \times \overline{\mathrm{c}}] \\ & =(\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}) \cdot[(\overline{\mathrm{b}} \times \overline{\mathrm{c}})-(\overline{\mathrm{a}} \times \overline{\mathrm{c}})] \\ & =\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})-\overline{\mathrm{b}} \cdot(\overline{\mathrm{a}} \times \overline{\mathrm{c}})=\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})+\overline{\mathrm{b}} \cdot(\overline{\mathrm{c}} \times \overline{\mathrm{a}}) \\ & =2 \overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})=2(1)(2)(3)=12\end{aligned}\)