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If $\bar{a}+\bar{b}, \bar{b}+\bar{c}, \bar{c}+\bar{a}$ are coterminous edges of a parallelepiped, then its volume is
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The correct answer is:
$2[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]$
The volume of required parallelepiped
$\begin{aligned}
& =(\bar{a}+\bar{b}) \cdot[(\bar{b}+\bar{c}) \times(\bar{c}+\bar{a})] \\
& =(\bar{a}+\bar{b}) \cdot[(\bar{b} \times \bar{c})+(\bar{b} \times \bar{a})+(\bar{c} \times \bar{c})+(\bar{c} \times \bar{a})] \\
& =[\bar{a} \cdot(\bar{b} \times \bar{c})]+[\bar{a} \cdot(\bar{b} \times \bar{a})]+0+[\bar{a} \times(\bar{c} \times \bar{a})] \\
& +[\bar{b} \cdot(\bar{b} \times \bar{a})]+0+[\bar{b} \cdot(\bar{c} \times \bar{a})] \\
& =[\bar{a} \cdot(\bar{b} \times \bar{c})]+0+0+0+0+0+0+[\bar{b} \cdot(\bar{c} \times \bar{a})] \\
& =2 \bar{a} \cdot(\bar{b} \times \bar{c})
\end{aligned}$
$\begin{aligned}
& =(\bar{a}+\bar{b}) \cdot[(\bar{b}+\bar{c}) \times(\bar{c}+\bar{a})] \\
& =(\bar{a}+\bar{b}) \cdot[(\bar{b} \times \bar{c})+(\bar{b} \times \bar{a})+(\bar{c} \times \bar{c})+(\bar{c} \times \bar{a})] \\
& =[\bar{a} \cdot(\bar{b} \times \bar{c})]+[\bar{a} \cdot(\bar{b} \times \bar{a})]+0+[\bar{a} \times(\bar{c} \times \bar{a})] \\
& +[\bar{b} \cdot(\bar{b} \times \bar{a})]+0+[\bar{b} \cdot(\bar{c} \times \bar{a})] \\
& =[\bar{a} \cdot(\bar{b} \times \bar{c})]+0+0+0+0+0+0+[\bar{b} \cdot(\bar{c} \times \bar{a})] \\
& =2 \bar{a} \cdot(\bar{b} \times \bar{c})
\end{aligned}$
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