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Question
Find the area of a parallelogram whose adjacent sides are represented by the vectors\[2 \hat{i} - 3 \hat{k} \text { and } 4 \hat{j} + 2 \hat{k} .\]
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Solution
\[\text { Let } \vec{a} = 2 \hat{i} - 3 \hat{k} \text { and } \vec{b} = 4 \hat {j}+ 2 \hat{k}. \]
\[ \vec{a} \times \vec{b} = \begin{vmatrix}i & j & k \\ 2 & 0 & - 3 \\ 0 & 4 & 2\end{vmatrix}\]
\[ = 12 \hat{i} - 4 \hat{j} + 8 \hat{k} \]
\[\text { Thus }, \left| \vec{a} \times \vec{b} \right| = \sqrt{\left( 12 \right)^2 + \left( - 4 \right)^2 + \left( 8 \right)^2} = \sqrt{224} = 4\sqrt{14}\]
∴ Area of the parallelogram = \[\left| \vec{a} \times \vec{b} \right| = 4\sqrt{14}\] square units
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