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Question
Find a unit vector perpendicular to the plane containing the vectors \[\vec{a} = 2 \hat{ i } + \hat{ j } + \hat{ k } \text{ and } \vec{b} = \hat{ i } + 2 \hat{ j } + \hat{ k } .\]
Solution
\[ \text{ Given } : \]
\[ \vec{a} = 2\hat{ i } + \hat{ j } + \hat{ k } \]
\[ \vec{b} = \hat{ i } +2 \hat { j } +\text{ k } \]
\[ \therefore \vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 2 & 1 & 1 \\ 1 & 2& 1\end{vmatrix}\]
\[ = \left( 1 - 2 \right) \hat{ i } - \left( 2 - 1 \right) \hat{ j } + \left( 4 - 1 \right) \hat{ k } \]
\[ = - \hat{ i } - \hat{ j } + 3 \hat{ k } \]
\[ \Rightarrow \left| \vec{a} \times \vec{b} \right| = \sqrt{1 + 1 + 9 }\]
\[ = \sqrt{11}\]
\[\text{ Unit vector perpendicular to the plane containing vectors } \vec{a} \text{ and } \vec{b} = \pm \frac{\vec{a} \times \vec{b}}{\left| \vec{a} \times \vec{b} \right|}\]
\[\text{ Unit vector perpendicular to the plane containing vectors } \vec{a} \text{ and } \vec{b} = \pm \frac{1}{\sqrt{11}}\left( - \hat{ i } - \hat{ j } + 3 \hat{ k } \right)\]
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