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Question
The unit vector perpendicular to the plane passing through points \[P\left( \hat{ i } - \hat{ j } + 2 \hat{ k } \right), Q\left( 2 \hat{ i } - \hat{ k } \right) \text{ and } R\left( 2 \hat{ j } + \hat{ k } \right)\] is
Options
\[2 \hat{ i } + \hat{ j } + \hat{ k } \]
\[\sqrt{6}\left( 2 \hat{ i } + \hat{ j } + \hat{ k } \right)\]
\[\frac{1}{\sqrt{6}}\left( 2 \hat{ i } + \hat{ j } + \hat{ k } \right)\]
\[\frac{1}{6}\left( 2 \hat{ i } + \hat{ j } + \hat{ k } \right)\]
Solution
\[\frac{1}{\sqrt{6}}\left( 2 \hat{ i } + \hat{ j } + \hat{ k } \right)\]
\[\text{ The vector } \vec{PQ} \times \vec{PR} \text{ is perpendicular to the vectors } \vec{PQ} \text{ and } \vec{PR} . \]
\[ \therefore \text{ Required unit vector} = \frac{\vec{PQ} \times \vec{PR}}{\left| \vec{PQ} \times \vec{PR} \right|}\]
\[\text{ Now } , \]
\[ \vec{PQ} = P . V . \text{ of } Q - P . V . of P\]
\[ = \hat{ i } + \hat{ j } - 3 \hat{ k } \]
\[ \vec{PR} = P . V . \text{ of } R - P . V . of P\]
\[ = - \hat{ i } + 3 \hat{ j } - \hat{ k } \]
\[ \therefore \vec{PQ} \times \vec{PR} = \begin{vmatrix}\text{ i } & \text{ j } & \text{ k } \\ 1 & 1 & - 3 \\ - 1 & 3 & - 1\end{vmatrix}\]
\[ = 8 \hat{ i } + 4 \hat{ j } + 4 \text{ k } \]
\[ = 4 \left( 2 \hat{ i } + \hat{ j } + \hat{ k } \right)\]
\[ \Rightarrow \left| \vec{PQ} \times \vec{PR} \right| = \sqrt{64 + 16 + 16}\]
\[ = \sqrt{96}\]
\[ = 4\sqrt{6}\]
\[\text{ Required unit vector } = \frac{\vec{PQ} \times \vec{PR}}{\left| \vec{PQ} \times \vec{PR} \right|}\]
\[ = \frac{4 \left( 2 \hat{ i } + \hat{ j } + \hat{ k } \right)}{4\sqrt{6}}\]
\[ = \frac{1}{\sqrt{6}}\left( 2 \hat{ i } + \hat{ j } + \hat{ k } \right)\]
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