Maths > Three Dimensional Coordinate System > 4.0 Plane
Three Dimensional Coordinate System
1.0 Introduction
2.0 Equation of a line in space
2.1 Vectorial form of a line passing through a given point and parallel to a given vector
2.2 Vectorial form of a line passing through two given points
2.3 Cartesian form of a line passing through a given point and parallel to a given vector
2.4 Cartesian form of a line passing through two given points
3.0 Distance and Angle between lines and points.
3.1 Condition for perpendicularity
3.2 Condition for parallelism
3.3 Co-planarity of two lines
3.4 Shortest distance between two lines
3.5 Distance between two skew lines
3.6 Distance between parallel lines
3.7 Perpendicular distance of a point from a line
3.8 Image of a point in a straight line
4.0 Plane
4.1 Equation of plane in normal form
4.2 Equation of a plane perpendicular to a given vector and passing through a given point
4.3 Equation of plane passing through a given point and parallel to the two given vectors
4.4 Equation of plane passing through three non-collinear points
4.5 Intercept form of plane
4.6 Equation of plane passing through the intersection of two given planes
5.0 Relation between Plane, Line and Point.
5.1 Angle between two planes
5.2 Angle between a line and a plane
5.3 Distance of a point from a plane
5.4 Distance between two parallel planes
6.0 Intersection of a line and a plane
7.0 Image of a point in a plane
4.3 Equation of plane passing through a given point and parallel to the two given vectors
2.2 Vectorial form of a line passing through two given points
2.3 Cartesian form of a line passing through a given point and parallel to a given vector
2.4 Cartesian form of a line passing through two given points
3.2 Condition for parallelism
3.3 Co-planarity of two lines
3.4 Shortest distance between two lines
3.5 Distance between two skew lines
3.6 Distance between parallel lines
3.7 Perpendicular distance of a point from a line
3.8 Image of a point in a straight line
4.2 Equation of a plane perpendicular to a given vector and passing through a given point
4.3 Equation of plane passing through a given point and parallel to the two given vectors
4.4 Equation of plane passing through three non-collinear points
4.5 Intercept form of plane
4.6 Equation of plane passing through the intersection of two given planes
5.2 Angle between a line and a plane
5.3 Distance of a point from a plane
5.4 Distance between two parallel planes
This is just the special case of an above-mentioned method to find the equation of plane when a point through which plane pass and the normal vector to the plane is given.
Here the difference is instead of a normal vector to the plane, two vectors parallel to the plane are given. Our approach is to find the normal vector to the plane from these two vectors.
As we know that the cross product of two vectors in the same plane gives a vector perpendicular to that plane in which two vectors lie.
Therefore, the cross product of two given vectors parallel to the plane will give us the normal vector to the plane.
Let us assume the plane passes through a point $A$ with position vector $\overrightarrow a $ and is parallel to the given vectors $\overrightarrow b $ and $\overrightarrow c $ and $\overrightarrow r $ be the position vector of any point $P$ in the plane.
Therefore, we can write $$\overrightarrow {AP} = \overrightarrow {OP} - \overrightarrow {OA} = \overrightarrow r - \overrightarrow a $$ Since, the vectors $\overrightarrow r - \overrightarrow a ,\;\overrightarrow b ,\;\overrightarrow c $ are coplanar. Thus, $$\begin{equation} \begin{aligned} \left( {\overrightarrow r - \overrightarrow a } \right).\;\left( {\overrightarrow b \times \;\overrightarrow c } \right) = 0 \\ \overrightarrow r .\left( {\overrightarrow b \times \;\overrightarrow c } \right) = \overrightarrow a .\left( {\overrightarrow b \times \;\overrightarrow c } \right) \\\end{aligned} \end{equation} $$
\[\left[ {\begin{array}{c} {\overrightarrow r }&{\overrightarrow b }&{\overrightarrow c } \end{array}} \right] = \left[ {\begin{array}{c} {\overrightarrow a }&{\overrightarrow b }&{\overrightarrow c } \end{array}} \right]\]
which is the required vector equation of plane.
Similarly, the Cartesian equation of plane passing through a given point and parallel to the two given lines whose direction ratios are $\left( {{\alpha _1},{\beta _1},{\gamma _1}} \right)$ and $\left( {{\alpha _2},{\beta _2},{\gamma _2}} \right)$ is
\[\left| {\begin{array}{c} {x - {x_1}}&{y - {y_1}}&{z - {z_1}} \\ {{\alpha _1}}&{{\beta _1}}&{{\gamma _1}} \\ {{\alpha _2}}&{{\beta _2}}&{{\gamma _2}} \end{array}} \right| = 0\]
