Question

 Find a unit vector perpendicular to both the vectors  \[4 \hat{ i } - \hat{ j }  + 3 \hat{ k } \text{ and }  - 2 \hat{ i  } + \hat{ j }  - 2 \hat{ k }  .\]

 

Answer 1

\[ \text{ Given } : \]
\[ \vec{a} = 4 \hat { i } - \hat{ j }  + 3 \hat{ k }  \]
\[ \vec{b} = - 2 \hat{ i } + \hat{ j }  - 2 \hat{ k } \]
\[ \therefore \vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i } & \hat{ j }  & \hat{ k}  \\ 4 & - 1 & 3 \\ - 2 & 1 & - 2\end{vmatrix}\]
\[ = \left( 2 - 3 \right) \hat{ i } - \left( - 8 + 6 \right) \hat{ j } + \left( 4 - 2 \right) \hat{ k } \]
\[ = - \hat{ i } + 2 \hat{ j }  + 2 \hat{ k }  \]
\[ \Rightarrow \left| \vec{a} \times \vec{b} \right| = \sqrt{1 + 2^2 + 2^2}\]
\[ = \sqrt{9}\]
\[ = 3\]
\[ \text{ Unit vector perpendicular to }  \vec{a} \text{ and  } \vec{b} =\frac{\vec{a} \times \vec{b}}{\left| \vec{a} \times \vec{b} \right|} = \frac{- \hat{ i } + 2 \hat{ j }  + 2 \hat{ k } }{3}\]