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Question
If \[\vec{a} = 2 \hat{ i } + \hat{ k } , \vec{b} = \hat { i } + \hat{ j } + \hat{ k } ,\] find the magnitude of \[\vec{a} \times \vec{b} .\]
Solution
\[\text{ Given } : \]
\[ \vec{a} = 2 \hat{ i } + 0 \hat{ j } + \hat{ k } \]
\[ \vec{b} = \hat{ i } + \hat{ j } +\hat{ k } \]
\[ \vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 2 & 0 & 1 \\ 1 & 1 & 1\end{vmatrix}\]
\[ = \left( 0 - 1 \right) \hat{ i } - \left( 2 - 1 \right) \hat{ j } + \left( 2 - 0 \right) \hat{ k } \]
\[ = - \hat{ i } - \hat{ j } + 2 \hat{ k } \]
\[ \Rightarrow \left| \vec{a} \times \vec{b} \right| = \sqrt{\left( - 1 \right)^2 + \left( - 1 \right)^2 + 2^2}\]
\[ = \sqrt{6}\]
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