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Question
Find the volume of the parallelopiped with its edges represented by the vectors \[\hat {i} + \hat {j} , \hat {i} + 2 \hat {j} \text { and } \hat {i} + \hat {j} + \pi k .\]
Solution
Let:
\[ \vec{a} = \hat {i} + \hat {j} \]
\[ \vec{b} =\hat { i} + 2 \hat {j} \]
\[ \vec{c} =\hat { i} + \hat {j} + \pi \hat {k} \]
\[\text { We know that the volume of a parallelopiped whose three adjecent edges are }\vec{a} , \vec{b} \text { and } \vec{c} \text { is equal to } \left| \left[ \vec{a} \vec{b} \vec{c} \right] \right| . \]
We have
\[\left[ \vec{a} \vec{b} \vec{c} \right] = \begin{vmatrix}1 & 1 & 0 \\ 1 & 2 & 0 \\ 1 & 1 & \pi\end{vmatrix} = 1\left( 2\pi - 0 \right) - 1\left( \pi - 0 \right) + 0\left( 1 - 2 \right) = 2\pi - \pi = \pi \]
\[ \therefore \text { Volume } = \left| \left[ \vec{a} \vec{b} \vec{c} \right] \right| = \left| \pi \right| = \pi\text { cubic units }\]
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