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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
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The correct answer is:
36
We have $\bar{a} \cdot(\bar{b} \times \bar{c})=4$
Volume of required parallelepiped is
$$
\begin{aligned}
& (\overline{\mathrm{a}}+2 \overline{\mathrm{b}}) \cdot[(\overline{\mathrm{b}}+2 \overline{\mathrm{c}}) \times \overline{\mathrm{c}}+2 \overline{\mathrm{a}}] \\
& =(\overline{\mathrm{a}}+2 \overline{\mathrm{b}}) \cdot[(\overline{\mathrm{b}} \times \overline{\mathrm{c}})+2(\overline{\mathrm{c}} \times \overline{\mathrm{c}})+2(\overline{\mathrm{b}} \times \overline{\mathrm{a}})+4(\overline{\mathrm{c}} \times \overline{\mathrm{a}})] \\
& =\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})+\overline{\mathrm{a}}(0)+2 \overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{a}})+4 \overline{\mathrm{a}} \cdot(\overline{\mathrm{c}} \times \overline{\mathrm{a}})+2 \overline{\mathrm{b}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}}) \\
& +4 \overline{\mathrm{b}}(0)+4 \overline{\mathrm{b}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{a}})+8 \overline{\mathrm{b}} \cdot(\overline{\mathrm{c}} \times \overline{\mathrm{a}}) \\
& =\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})+0+0+0+0+0+0+8 \overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}}) \\
& =9[\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})]=9(4)=36
\end{aligned}
$$
Volume of required parallelepiped is
$$
\begin{aligned}
& (\overline{\mathrm{a}}+2 \overline{\mathrm{b}}) \cdot[(\overline{\mathrm{b}}+2 \overline{\mathrm{c}}) \times \overline{\mathrm{c}}+2 \overline{\mathrm{a}}] \\
& =(\overline{\mathrm{a}}+2 \overline{\mathrm{b}}) \cdot[(\overline{\mathrm{b}} \times \overline{\mathrm{c}})+2(\overline{\mathrm{c}} \times \overline{\mathrm{c}})+2(\overline{\mathrm{b}} \times \overline{\mathrm{a}})+4(\overline{\mathrm{c}} \times \overline{\mathrm{a}})] \\
& =\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})+\overline{\mathrm{a}}(0)+2 \overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{a}})+4 \overline{\mathrm{a}} \cdot(\overline{\mathrm{c}} \times \overline{\mathrm{a}})+2 \overline{\mathrm{b}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}}) \\
& +4 \overline{\mathrm{b}}(0)+4 \overline{\mathrm{b}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{a}})+8 \overline{\mathrm{b}} \cdot(\overline{\mathrm{c}} \times \overline{\mathrm{a}}) \\
& =\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})+0+0+0+0+0+0+8 \overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}}) \\
& =9[\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})]=9(4)=36
\end{aligned}
$$
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