If the volume of a tetrahedron whose conterminous edges are $\bar{a}+\bar{b}, \bar{b}+\bar{c},…

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
  1. 48 cubic units
  2. 144 cubic units
  3. 72 cubic units
  4. 10 cubic units

Solution

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

Asked in: MHT CET 2021 (20 Sep Shift 1)

Practice more Vectors questions on Aicharya