If the volume of a tetrahedron whose conterminous edges are \(\bar{a}+\bar{b}, \bar{b}+\bar{c}, \bar{c}+\bar{a}\) is 24 cubic units, then the volume of parallelepiped whose coterminous edges are \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) is

As per data given, we write
\(24=\frac{1}{6}\{(\overline{\mathrm{a}}+\overline{\mathrm{b}}) \cdot[(\overline{\mathrm{b}}+\overline{\mathrm{c}}) \times(\overline{\mathrm{c}}+\overline{\mathrm{a}})]\} \)
\( =\frac{1}{6}\{(\overline{\mathrm{a}}+\overline{\mathrm{b}}) \cdot[(\overline{\mathrm{b}} \times \overline{\mathrm{c}})+(\overline{\mathrm{b}}+\overline{\mathrm{a}})+(\overline{\mathrm{c}} \times \overline{\mathrm{a}})]\} \quad\) \(\ldots[\because \overline{\mathrm{c}} \times \overline{\mathrm{c}}=0] \)
\( \therefore 144=[\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})]+[\overline{\mathrm{b}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})]+[\overline{\mathrm{b}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{a}})]~+\) \([\overline{\mathrm{a}} \cdot(\overline{\mathrm{c}} \times \overline{\mathrm{a}})]+[\overline{\mathrm{b}} \cdot(\overline{\mathrm{c}} \times \overline{\mathrm{a}})] \)
\( =[\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})]+0+0+0+0[\overline{\mathrm{b}}(\overline{\mathrm{c}} \times \overline{\mathrm{a}})] \)
\( =2[\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})] \quad \ldots[\because \overline{\mathrm{b}}(\overline{\mathrm{c}} \times \overline{\mathrm{a}})=\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})] \)
\( \therefore \overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})=72, \text { i.e. volume of parallelepiped with } \)
\( \text { conterminous edges } \overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}.\)