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Question
Find the magnitude of \[\vec{a} = \left( 3 \hat{ k } + 4 \hat{ j } \right) \times \left( \hat{ i } + \hat{ j } - \hat{ k } \right) .\]
Solution
\[\vec{a} = \left( 0 \hat{ i } + 4 \hat{ j } + 3 \hat{ k } \right) \times \left( \hat{ i } +\hat{ j } - \hat{ k } \right)\]
\[ = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 0 & 4 & 3 \\ 1 & 1 & - 1\end{vmatrix}\]
\[ = \hat{ i } \left( - 4 - 3 \right) -\hat{ j } \left( 0 - 3 \right) + \hat{ k } \left( 0 - 4 \right)\]
\[ = - 7 \hat{ i } + 3 \hat{ j } - 4 \hat{ k } \]
\[ \Rightarrow \left| \vec{a} \right| = \sqrt{\left( - 7 \right)^2 + 3^2 + \left( - 4 \right)^2}\]
\[ = \sqrt{74}\]
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