Question

Find all vectors of magnitude \[10\sqrt{3}\] that are perpendicular to the plane of \[\hat{ i }  + 2 \hat{ j }  + \hat{ k } \] and \[- \hat { i }  + 3 \hat{ j }  + 4 \hat{ k } \] .

 

Answer 1

Let \[\vec{a} = \hat{ i }  + 2 \hat{ j } + \hat{ k } \] and \[\vec{b} = - \hat{ i } + 3 \hat{ j }  + 4 \hat{ k } \] .

Unit vectors perpendicular to both \[\vec{a}\] and  \[\vec{b}\] =  \[\pm \frac{\vec{a} \times \vec{b}}{\left| \vec{a} \times \vec{b} \right|}\]

Now,

\[\vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i }  & \hat{ j  } & \hat{ k }  \\ 1 & 2 & 1 \\ - 1 & 3 & 4\end{vmatrix} = 5 \hat{ i }  - 5 \hat{ j } + \hat{ k }  \]

\[ \therefore \left| \vec{a} \times \vec{b} \right| = \left| 5 \hat{ i }  - 5 \hat{ j }  + 5 \hat{ k }  \right| = \sqrt{5^2 + \left( - 5 \right)^2 + 5^2} = \sqrt{75} = 5\sqrt{3}\]

Unit vectors perpendicular to both \[\vec{a}\] and \[\vec{b}\] =  \[\pm \frac{5 \hat{ i } - 5 \hat{ j }  + 5 \hat{ k } }{5\sqrt{3}} = \pm \frac{\hat{ i }  - \hat{ j }  + \hat{ k } }{\sqrt{3}}\]

∴ Required vectors = \[10\sqrt{3}\left( \pm \frac{\hat{ i}  - \hat{ j }  + \hat{ k } }{\sqrt{3}} \right) = \pm 10\left( \hat{ i } - \hat{ j }  + \hat{ k }  \right)\] 

Thus, the vectors of magnitude \[10\sqrt{3}\]  that are perpendicular to the plane of \[\hat{ i }  + 2 \hat{ j }  + \hat{ k } \] and  \[- \hat{ i }  + 3 \hat{ j } + 4 \hat{ k } \] are  \[\pm 10\left( \hat{ i } - \hat{ j }  + \hat{ k }  \right)\] .