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Equation of a plane passing through three non collinear point - Practice Questions & MCQ

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

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  • 18 Questions around this concept.

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Let P be the plane passing through the points (5, 3, 0), (13, 3, -2), and (1, 6, 2). For \alpha \in N, if the distances of the points \mathrm{A}(3,4, \alpha)\; and \; \mathrm{B}(2, \alpha,a) from the plane \mathrm{P}  are 2 and 3 respectively, then the positive value of a is:

Concepts Covered - 1

Equation of a plane passing through three non collinear point

Let A, B and C be three non-collinear points on the plane with position vectors \vec{\mathbf a},\;\vec{\mathbf b} and \vec{\mathbf c} respectively.

The vectors, \overrightarrow{AB}=\vec{\mathbf b}-\vec{\mathbf a} and \overrightarrow{AC}=\vec{\mathbf c}-\vec{\mathbf a} are in the given plane. Therefore, the vector \overrightarrow{AB}\times \overrightarrow{AC} is perpendicular to the plane containing points A, B and C.

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

Therefore, the equation of the plane passing through OP and perpendicular to the vector \overrightarrow{AB}\times \overrightarrow{AC} is

\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}(\vec{r}-\vec{a}) \cdot(\overrightarrow{\mathrm{AB}} \times \overrightarrow{\mathrm{AC}})=0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left ( \because \overrightarrow{AR}=(\vec{r}-\vec{a}) \right )\\\text{or}\;\;\;\;\;\;\;\;\mathbf{(\vec{r}-\vec{a}) \cdot[(\vec{b}-\vec{a}) \times(\vec{c}-\vec{a})]=0}

This is the equation of the plane in vector form passing through three noncollinear points.

 

Cartesian Form

Let (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) be the coordinates of points A, B and C respectively.

Let P(x, y, z) be any point on the plane.

Then, the vectors \overrightarrow{PA},\;\overrightarrow{BA} and \overrightarrow{CA} are coplanar.
\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\left [ \overrightarrow{\mathbf{PA}}\;\;\overrightarrow{\mathbf{BA}}\;\;\overrightarrow{\mathbf{CA}} \right ]=0\\\\\mathrm{\;\;\;}{\begin{vmatrix} \mathbf{x-x_1 }&\mathbf{y-y_1} &\mathbf{z-z_1} \\ \mathbf{x_2-x_1} &\mathbf{y_2-y_1} &\mathbf{z_2-z_1} \\ \mathbf{x_3-x_1} &\mathbf{y_3-y_1} &\mathbf{z_3-z_1} \end{vmatrix}=0}\\\\\text{which is required equation of the plane .}

 

 

Intercept form of the equation of a plane

Cartesian Form

The equation of a plane having intercepting lengths a, b and c with X-axis, Y-axis and Z-axis, respectively is

\mathbf{\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1}

Let the plane meets X, Y and Z-axes at (a, 0, 0), (0, b, 0), (0, 0, c) respectively and P(x, y, z) be any point on the plane.

Since these three points are non-collinear points.

Then, the vectors \overrightarrow{PA},\;\overrightarrow{BA} and \overrightarrow{CA} are coplanar. where P be any point in the plane ABC.

\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;}\left [ \overrightarrow{\mathbf{PA}}\;\;\overrightarrow{\mathbf{BA}}\;\;\overrightarrow{\mathbf{CA}} \right ]=0\\\\\mathrm{\;\;\;}{\begin{vmatrix} \mathbf{x-a }&\mathbf{y-0} &\mathbf{z-0} \\ \mathbf{0-a} &\mathbf{b-0} &\mathbf{0-0} \\ \mathbf{0-a} &\mathbf{0-0} &\mathbf{c-0} \end{vmatrix}=0}\\\\\Rightarrow\mathbf{(x-a)bc-y(-ac)+z(ba)=0}

\mathbf{\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1}\\\\\text{which is required equation of the plane .}

This is the equation of the plane in cartesian form when the plane makes Intercepts a, b and c on the coordinate axes.

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Equation of a plane passing through three non collinear point

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Equation of a plane passing through three non collinear point

Mathematics for Joint Entrance Examination JEE (Advanced) : Vectors and 3D Geometry

Page No. : 4.29

Line : 22

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