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