Question

Find a unit vector perpendicular to the plane containing the vectors  \[\vec{a} = 2 \hat{ i } + \hat{ j }  + \hat{ k } \text{ and }  \vec{b} = \hat{ i } + 2 \hat{ j }  + \hat{ k } .\]

 

Answer 1

\[ \text{ Given } : \]
\[ \vec{a} = 2\hat{ i } + \hat{ j }  + \hat{ k }  \]
\[ \vec{b} =  \hat{ i }  +2 \hat {  j }  +\text{ k } \]
\[ \therefore \vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i }  & \hat{ j } & \hat{ k }  \\ 2 & 1 & 1 \\  1 & 2& 1\end{vmatrix}\]
\[ = \left( 1 - 2 \right) \hat{ i }  - \left(  2 - 1 \right) \hat{ j }  + \left( 4 - 1 \right) \hat{ k }  \]
\[ = - \hat{ i }  - \hat{ j }  + 3 \hat{ k }  \]
\[ \Rightarrow \left| \vec{a} \times \vec{b} \right| = \sqrt{1 + 1 + 9 }\]
\[ = \sqrt{11}\]

\[\text{ Unit vector perpendicular to the plane containing vectors } \vec{a} \text{ and }  \vec{b} = \pm \frac{\vec{a} \times \vec{b}}{\left| \vec{a} \times \vec{b} \right|}\]
\[\text{ Unit vector perpendicular to the plane containing vectors }  \vec{a} \text{ and } \vec{b} = \pm \frac{1}{\sqrt{11}}\left( - \hat{ i }  - \hat{ j }  + 3 \hat{ k }  \right)\]