The unit vector perpendicular to the plane passing through points \[P\left( \hat{ i } - \hat{ j } + 2 \hat{ k } \right), Q\left( 2 \hat{ i } - \hat{ k } \right) \text{ and } R\left( 2 \hat{ j } + \hat{ k } \right)\] is

\[\frac{1}{\sqrt{6}}\left( 2 \hat{ i } + \hat{ j } + \hat{ k } \right)\] \[\text{ The vector } \vec{PQ} \times \vec{PR} \text{ is perpendicular to the vectors } \vec{PQ} \text{ and } \vec{PR} . \] \[ \therefore \text{ Required unit vector} = \frac{\vec{PQ} \times \vec{PR}}{\left| \vec{PQ} \times \vec{PR} \right|}\] \[\text{ Now } , \] \[ \vec{PQ} = P . V . \text{ of } Q - P . V . of P\] \[ = \hat{ i } + \hat{ j } - 3 \hat{ k } \] \[ \vec{PR} = P . V . \text{ of } R - P . V . of P\] \[ = - \hat{ i } + 3 \hat{ j } - \hat{ k } \] \[ \therefore \vec{PQ} \times \vec{PR} = \begin{vmatrix}\text{ i } & \text{ j } & \text{ k } \\ 1 & 1 & - 3 \\ - 1 & 3 & - 1\end{vmatrix}\] \[ = 8 \hat{ i } + 4 \hat{ j } + 4 \text{ k } \] \[ = 4 \left( 2 \hat{ i } + \hat{ j } + \hat{ k } \right)\] \[ \Rightarrow \left| \vec{PQ} \times \vec{PR} \right| = \sqrt{64 + 16 + 16}\] \[ = \sqrt{96}\] \[ = 4\sqrt{6}\] \[\text{ Required unit vector } = \frac{\vec{PQ} \times \vec{PR}}{\left| \vec{PQ} \times \vec{PR} \right|}\] \[ = \frac{4 \left( 2 \hat{ i } + \hat{ j } + \hat{ k } \right)}{4\sqrt{6}}\] \[ = \frac{1}{\sqrt{6}}\left( 2 \hat{ i } + \hat{ j } + \hat{ k } \right)\]

The Unit Vector Perpendicular to the Plane Passing Through Points P ( ^ I − ^ J + 2 ^ K ) , Q ( 2 ^ I − ^ K ) and R ( 2 ^ J + ^ K ) is - Mathematics

प्रश्न

The unit vector perpendicular to the plane passing through points $P (\hat{i} - \hat{j} + 2 \hat{k}), Q (2 \hat{i} - \hat{k}) and R (2 \hat{j} + \hat{k})$ is

विकल्प

$2 \hat{i} + \hat{j} + \hat{k}$
$\sqrt{6} (2 \hat{i} + \hat{j} + \hat{k})$
$\frac{1}{\sqrt{6}} (2 \hat{i} + \hat{j} + \hat{k})$
$\frac{1}{6} (2 \hat{i} + \hat{j} + \hat{k})$

MCQ

उत्तर

$\frac{1}{\sqrt{6}} (2 \hat{i} + \hat{j} + \hat{k})$

$The vector \vec{P Q} \times \vec{P R} is perpendicular to the vectors \vec{P Q} and \vec{P R} .$

$∴ Required unit vector = \frac{\vec{P Q} \times \vec{P R}}{| \vec{P Q} \times \vec{P R} |}$

$Now,$

$\vec{P Q} = P . V . of Q - P . V . o f P$

$= \hat{i} + \hat{j} - 3 \hat{k}$

$\vec{P R} = P . V . of R - P . V . o f P$

$= - \hat{i} + 3 \hat{j} - \hat{k}$

$∴ \vec{P Q} \times \vec{P R} = | \begin{matrix} i & j & k \\ 1 & 1 & - 3 \\ - 1 & 3 & - 1 \end{matrix} |$

$= 8 \hat{i} + 4 \hat{j} + 4 k$

$= 4 (2 \hat{i} + \hat{j} + \hat{k})$

$\Rightarrow | \vec{P Q} \times \vec{P R} | = \sqrt{64 + 16 + 16}$

$= \sqrt{96}$

$= 4 \sqrt{6}$

$Required unit vector = \frac{\vec{P Q} \times \vec{P R}}{| \vec{P Q} \times \vec{P R} |}$

$= \frac{4 (2 \hat{i} + \hat{j} + \hat{k})}{4 \sqrt{6}}$

$= \frac{1}{\sqrt{6}} (2 \hat{i} + \hat{j} + \hat{k})$

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Product of Two Vectors - Vector (Or Cross) Product of Two Vectors

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अध्याय 25: Vector or Cross Product - MCQ [पृष्ठ ३५]

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अध्याय 25 Vector or Cross Product
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The Unit Vector Perpendicular to the Plane Passing Through Points P ( ^ I − ^ J + 2 ^ K ) , Q ( 2 ^ I − ^ K ) and R ( 2 ^ J + ^ K ) is - Mathematics

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