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