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प्रश्न
Find a vector of magnitude \[\sqrt{171}\] which is perpendicular to both of the vectors \[\vec{a} = \hat{ i } + 2 \hat{ j } - 3 \hat{ k } \] and \[\vec{a} = \hat{ i } + 2 \hat{ j } - 3 \hat{ k } \] .
उत्तर
The given vectors are \[\vec{a} = \hat{ i } + 2 \hat{ j } - 3 \hat{ k } \] \text{ and } \[\vec{b} = 3 \hat{ i } - \hat{ j } + 2 \hat{ k } \] 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,
\[\vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 1 & 2 & - 3 \\ 3 & - 1 & 2\end{vmatrix} = \hat{ i } - 11 \hat{ j } - 7 \hat{ k } \]
\[ \therefore \left| \vec{a} \times \vec{b} \right| = \left| \hat{ i } - 11 \hat{ j } - 7 \hat{ k } \right| = \sqrt{1^2 + \left( - 11 \right)^2 + \left( - 7 \right)^2} = \sqrt{1 + 121 + 49} = \sqrt{171}\]
Unit vectors perpendicular to both \[\vec{a}\] and\[\vec{ b }\] = \[\pm \frac{{i } - 11 \hat{ j } - 7 \hat{ k }} {\sqrt{171}}\]
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