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Equation of a Plane Passing Through a Given Point and Parallel to Two Given Vectors - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Equation of a Plane Passing Through a Given Point and Parallel to Two Given Vectors is considered one of the most asked concept.

  • 13 Questions around this concept.

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QuestionEASY

A plane containing the point (3, 2, 0) and the line  \frac{x-1}{1}=\frac{y-2}{5}=\frac{z-3}{4}  also contains the point :

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Concepts Covered - 1

Equation of a Plane Passing Through a Given Point and Parallel to Two Given Vectors

Vector Form

Let a plane passes through a point A with position vector \vec{\mathbf a} and parallel to two vectors \vec{\mathbf b} and \vec{\mathbf c}.

Let \vec{\mathbf r} be the position vector of any point P on the plane.

\overrightarrow{AP}=\overrightarrow{OP}-\overrightarrow{OA}=\vec{r}-\vec a

Since, \overrightarrow{AP} lies in the plane, hence, \vec{\mathbf r}-\vec {\mathbf a} ,\;\vec{\mathbf b}\text{ and }\vec{\mathbf c} are coplanar.

We have,  

\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\left ( \vec{\mathbf r}-\vec {\mathbf a} \right )\cdot\left ( \vec{\mathbf b} \times \vec{\mathbf c} \right )=0\\\mathrm{or\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\left ( \vec{\mathbf r} \right )\cdot\left ( \vec{\mathbf b} \times \vec{\mathbf c} \right )=\left ( \vec {\mathbf a} \right )\cdot\left ( \vec{\mathbf b} \times \vec{\mathbf c} \right )\\\mathrm{or\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\left [ \vec{\mathbf r} \;\;\vec{\mathbf b}\;\;\vec{\mathbf c} \right ]=\left [ \vec{\mathbf a} \;\;\vec{\mathbf b}\;\;\vec{\mathbf c} \right ]

Which is required equation of plane.

 

Cartesian Form

From \left ( \vec{\mathbf r}-\vec {\mathbf a} \right )\cdot\left ( \vec{\mathbf b} \times \vec{\mathbf c} \right )=0 \text{ we have, } \left [ \vec{\mathbf r}- \vec{\mathbf a} \;\;\vec{\mathbf b}\;\;\vec{\mathbf c} \right ]

\Rightarrow \;\;\;\;\;\;\;\;\;\;\;\;\begin{vmatrix} x-x_1 &y-y_1 &z-z_1 \\ x_2 &y_2 &z_2 \\ x_3 & y_3 &z_3 \end{vmatrix}=0

Which is required equation of plane in cartesian form where \vec{\mathbf{b}}=x_{2} \hat{\mathbf{i}}+y_{2} \hat{\mathbf{j}}+z_{2} \hat{\mathbf{k}} and \vec{\mathbf{c}}=x_{3} \hat{\mathbf{i}}+y_{3} \hat{\mathbf{j}}+z_{3} \hat{\mathbf{k}}.

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Equation of a Plane Passing Through a Given Point and Parallel to Two Given Vectors

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