Q.29 Find the volume of the parallelopiped whose vertices are \(A(3, 2, -1)\), \(B(-2, 2, -3)\), \(C(3, 5, -2)\) and \(D(-2, 5, 4)\).
Best answer
Answer: 120 cubic units
The coterminous edges from A are
\(\overline{AB} = (-2 - 3)\hat{i} + (2 - 2)\hat{j} + (-3 + 1)\hat{k} = -5\hat{i} + 0\hat{j} - 2\hat{k}\)
\(\overline{AC} = 0\hat{i} + 3\hat{j} - \hat{k}\)
\(\overline{AD} = -5\hat{i} + 3\hat{j} + 5\hat{k}\)
Volume \(= \left|\left[\overline{AB}\ \overline{AC}\ \overline{AD}\right]\right|\), where
\(\left[\overline{AB}\ \overline{AC}\ \overline{AD}\right] = \begin{vmatrix} -5 & 0 & -2 \\ 0 & 3 & -1 \\ -5 & 3 & 5 \end{vmatrix} = -5(15 + 3) - 0(0 - 5) + (-2)(0 + 15) = -90 - 30 = -120\).
\(\therefore\) Volume of the parallelopiped \(= |-120| = 120\) cubic units.
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