If \([\bar{a} \bar{b} \bar{c}]=4\), then the volume (in cubic units) of the parallelepiped with \(\overline{\mathrm{a}}+2 \overline{\mathrm{b}}, \overline{\mathrm{b}}+2 \overline{\mathrm{c}}\) and \(\overline{\mathrm{c}}+2 \overline{\mathrm{a}}\) as coterminal edges, is

We have \(\bar{a} \cdot(\bar{b} \times \bar{c})=4\)
Volume of required parallelepiped is
\((\overline{\mathrm{a}}+2 \overline{\mathrm{b}}) \cdot[(\overline{\mathrm{b}}+2 \overline{\mathrm{c}}) \times \overline{\mathrm{c}}+2 \overline{\mathrm{a}}] \)
\( =(\overline{\mathrm{a}}+2 \overline{\mathrm{b}}) \cdot[(\overline{\mathrm{b}} \times \overline{\mathrm{c}})+2(\overline{\mathrm{c}} \times \overline{\mathrm{c}})+2(\overline{\mathrm{b}} \times \overline{\mathrm{a}})+4(\overline{\mathrm{c}} \times \overline{\mathrm{a}})] \)
\( =\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})+\overline{\mathrm{a}}(0)+2 \overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{a}})+4 \overline{\mathrm{a}} \cdot(\overline{\mathrm{c}} \times \overline{\mathrm{a}})+2 \overline{\mathrm{b}} ~\cdot\) \((\overline{\mathrm{b}} \times \overline{\mathrm{c}}) \) \( +4 \overline{\mathrm{b}}(0)+4 \overline{\mathrm{b}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{a}})+8 \overline{\mathrm{b}} \cdot(\overline{\mathrm{c}} \times \overline{\mathrm{a}}) \)
\( =\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})+0+0+0+0+0+0+8 \overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}}) \)
\( =9[\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})]=9(4)=36\)