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Question
Find the volume of the parallelopiped whose coterminous edges are represented by the vector:
\[\vec{a} = 11 \hat{i} , \vec{b} = 2 \hat{j} , \vec{c} = 13 \hat{k}\]
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Solution
Given:
\[ \vec{a} = 11 \hat{i} \]
\[ \vec{b} = 2 \hat{j} \]
\[ \vec{c} = 13 \hat{k} \]
\[\text { We know that the volume of a parallelopiped whose three adjacent edges are } \vec{a} , \vec{b} , \vec{c} is equal to \left| \left[ \vec{a} \vec{b} \vec{c} \right] \right| . \]
Here,
\[\left[ \vec{a} \vec{b} \vec{c} \right] = \begin{vmatrix}11 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 13\end{vmatrix} = 11 \left( 26 - 0 \right) - 0\left( 0 - 0 \right) + 0\left( 0 - 0 \right) = 286\]
\[\text{Volume of the parallelopiped }= \left| \left[ \vec{a} \vec{b} \vec{c} \right] \right| = \left| 286 \right| = 286 \text{ cubic units }\]
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