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 

 

विकल्प

• \[2 \hat{ i } + \hat{ j } + \hat{ k } \]

• \[\sqrt{6}\left( 2 \hat{ i }  + \hat{ j }  + \hat{ k }  \right)\]

• \[\frac{1}{\sqrt{6}}\left( 2 \hat{ i } + \hat{ j }  + \hat{ k }  \right)\]

• \[\frac{1}{6}\left( 2 \hat{ i }  + \hat{ j }  + \hat{ k }  \right)\]

Answer 1

\[\frac{1}{\sqrt{6}}\left( 2 \hat{ i } + \hat{ j }  + \hat{ k }  \right)\] 

\[\text{ The vector } \vec{PQ} \times \vec{PR} \text{ is perpendicular to the vectors } \vec{PQ} \text{ and }  \vec{PR} . \]

\[ \therefore \text{ Required unit vector}  = \frac{\vec{PQ} \times \vec{PR}}{\left| \vec{PQ} \times \vec{PR} \right|}\]

\[\text{ Now } , \]

\[ \vec{PQ} = P . V . \text{ of } Q - P . V . of P\]

\[ = \hat{ i  } + \hat{ j }  - 3 \hat{ k }  \]

\[ \vec{PR} = P . V . \text{ of }  R - P . V . of P\]

\[ = - \hat{ i }  + 3 \hat{ j }  - \hat{ k }  \]

\[ \therefore \vec{PQ} \times \vec{PR} = \begin{vmatrix}\text{ i } & \text{ j }  & \text{ k }  \\ 1 & 1 & - 3 \\ - 1 & 3 & - 1\end{vmatrix}\]

\[ = 8 \hat{ i }  + 4 \hat{ j }  + 4 \text{ k }  \]

\[ = 4 \left( 2 \hat{ i  } + \hat{ j }  + \hat{ k }  \right)\]

\[ \Rightarrow \left| \vec{PQ} \times \vec{PR} \right| = \sqrt{64 + 16 + 16}\]

\[ = \sqrt{96}\]

\[ = 4\sqrt{6}\]

\[\text{ Required unit vector }  = \frac{\vec{PQ} \times \vec{PR}}{\left| \vec{PQ} \times \vec{PR} \right|}\]

\[ = \frac{4 \left( 2 \hat{ i  } + \hat{ j }  + \hat{ k }  \right)}{4\sqrt{6}}\]

\[ = \frac{1}{\sqrt{6}}\left( 2 \hat{ i }  + \hat{ j }  + \hat{ k } \right)\]