Question

Write the number of vectors of unit length perpendicular to both the vectors \[\vec{a} = 2 \hat{ i } + \hat{ j }  + 2 \hat{ k }  \text{ and }  \vec{b} = \hat{ j }  + \hat{ k } \] .

 

Answer 1

Unit vectors perpendicular to \[\vec{a}\] and \[\vec{b}\]  are \[\pm \left( \frac{\vec{a} \times \vec{b}}{\left| \vec{a} \times \vec{b} \right|} \right)\] .

\[\vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i }& \hat{ j }  & \hat{ k } \\ 2 & 1 & 2 \\ 0 & 1 & 1\end{vmatrix} = - \hat{ i }  - 2 \hat{ j } + 2 \hat{ k } \]

∴ Unit vectors perpendicular to \[\vec{a}\] and \[\vec{b}\]  are  \[\pm \frac{- \hat{ i } - 2 \hat{ j } + 2 \hat{ k } }{\sqrt{\left( - 1 \right)^2 + \left( - 2 \right)^2 + \left( 2 \right)^2}} = \pm \left( - \frac{1}{3} \hat{ i }  - \frac{2}{3} \hat{ j } + \frac{2}{3} \hat{ k } \right)\] 

Thus, there are two unit vectors perpendicular to the given vectors.