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Question
Find the area of the parallelogram determined by the vector \[\hat{ i } - 3 \hat{ j } + \hat{ k } \text{ and } \hat{ i } + \hat{ j } + \hat{ k } .\]
Solution
\[\text{ Let: } \]
\[ \vec{a} = \hat{ i } - 3 \hat{ j } + \hat{ k } \]
\[ \vec{b} = \hat{ i } + \hat{ j } + \hat{ k } \]
\[ \vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 1 & - 3 & 1 \\ 1 & 1 & 1\end{vmatrix}\]
\[ = \left( - 3 - 1 \right) \hat{ i } - \left( 1 - 1 \right) \hat{ j } + \left( 1 + 3 \right) \hat{ k } \]
\[ = - 4 \hat{ i } + 0 \hat{ j } + 4 \hat{ k } \]
\[\text{ Area of the parallelogram } =\left| \vec{a} \times \vec{b} \right|\]
\[ = \sqrt{\left( - 4 \right)^2 + 0 + 4^2}\]
\[ = \sqrt{32}\]
\[ = 4\sqrt{2} \text{ sq. units } .\]
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