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Question
Find the volume of the parallelopiped whose coterminous edges are represented by the vector:
\[\vec{a} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k} , \vec{b} =\hat{ i} + 2 \hat{j} - \hat{k} , \vec{c} = 3 \hat{i} - \hat{j} + 2 \hat{k}\]
Solution
Given:
\[ \vec{a} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \]
\[ \vec{b} =\hat{ i} + 2 \hat{j} - \hat{k} \]
\[ \vec{c} = 3 \hat{i} - \hat{j} + 2 \hat{k} \]
\[\text{We know that the volume of a parallelopiped whose three adjacent edges are }\vec{a} , \vec{b} , \vec{c}\text{ is equal to } \left| \left[ \vec{a} \vec{b} \vec{c} \right] \right| . \]
Here,
\[\left[ \vec{a} \vec{b} \vec{c} \right] = \begin{vmatrix}2 & 3 & 4 \\ 1 & 2 & - 1 \\ 3 & - 1 & 2\end{vmatrix} = 2 \left( 4 - 1 \right) - 3\left( 2 + 3 \right) + 4\left( - 1 - 6 \right) = - 37\]
\[\text{Volume of the parallelopiped }= \left| \left[ \vec{a} \vec{b} \vec{c} \right] \right| = \left| - 37 \right| = 37 \text{cubic units }\]
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