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Question
\[\text{ If } \vec{a} = \hat { i } + 3 \hat { j } - 2 \hat { k } \text{ and } \vec{b} = - \hat { i } + 3 \hat { k } , \text{ find } \left| \vec{a} \times \vec{b} \right| .\]
Solution
\[Given: \]
\[ \vec{a} = \hat { i } + 3 \hat { j } - 2 \hat { k } \]
\[ \vec{b} = - \hat { i } + 0 \hat { j } + 3 \hat { k } \]
\[ \therefore \vec{a} \times \vec{b} = \begin{vmatrix}\hat { i } & \hat { j } & \hat { k } \\ 1 & 3 & - 2 \\ - 1 & 0 & 3\end{vmatrix}\]
\[ = \left( 9 + 0 \right) \hat { i } - \left( 3 - 2 \right) \hat { j } + \left( 0 + 3 \right) \hat { k } \]
\[ = 9 \hat { i } - \hat { j } + 3 \hat { k } \]
\[ \Rightarrow \left| \vec{a} \times \vec{b} \right| = \sqrt{9^2 + \left( - 1 \right)^2 + 3^2}\]
\[ = \sqrt{91}\]
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