4.4 Equation of plane passing through three non-collinear points

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

3.1 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

4.1 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

5.1 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

To find the vector equation of plane when three points through which it passes are given, first step is to find the position vector of those three given points.

Let us assume the given points be $A$, $B$ and $C$ with position vectors $\overrightarrow a $, $\overrightarrow b $ and $\overrightarrow c $ respectively and $\overrightarrow r $ be the position vector of any arbitrary point $P$ in the plane. Hence, we can say $$\begin{equation} \begin{aligned} \overrightarrow {AP} = \overrightarrow r - \overrightarrow a \\ \overrightarrow {AB} = \overrightarrow b - \overrightarrow a \\ \overrightarrow {AC} = \overrightarrow c - \overrightarrow a \\\end{aligned} \end{equation} $$ are coplanar. Hence, $$\left( {\overrightarrow r - \overrightarrow a } \right).\left\{ {\left( {\overrightarrow b - \overrightarrow a } \right) \times \left( {\overrightarrow c - \overrightarrow a } \right)} \right\} = 0$$ which is the vector equation of plane passing through three non-collinear points. On further simplifying the equation we can write it as

$$\begin{equation} \begin{aligned} \left( {\overrightarrow r - \overrightarrow a } \right).\left\{ {\left( {\overrightarrow b - \overrightarrow a } \right) \times \left( {\overrightarrow c - \overrightarrow a } \right)} \right\} = 0 \\ \left( {\overrightarrow r - \overrightarrow a } \right).\left( {\overrightarrow b \times \overrightarrow c - \overrightarrow b \times \overrightarrow a - \overrightarrow a \times \overrightarrow c + \overrightarrow a \times \overrightarrow a } \right) = 0 \\ \overrightarrow r .\left( {\overrightarrow b \times \overrightarrow c + \overrightarrow a \times \overrightarrow b + \overrightarrow c \times \overrightarrow a } \right) = \overrightarrow a .\left( {\overrightarrow b \times \overrightarrow c } \right) + \overrightarrow a .\left( {\overrightarrow a \times \overrightarrow b } \right) + \overrightarrow a .\left( {\overrightarrow c \times \overrightarrow a } \right) \\\end{aligned} \end{equation} $$

$$\left[ {\overrightarrow r \quad \overrightarrow b \quad \overrightarrow c } \right] + \left[ {\overrightarrow r \quad \overrightarrow a \quad \overrightarrow b } \right] + \left[ {\overrightarrow r \quad \overrightarrow c \quad \overrightarrow a } \right] = \left[ {\overrightarrow a \quad \overrightarrow b \quad \overrightarrow c } \right]$$ which is the required simplified form of vector equation of plane.

To find the cartesian equation of plane when three points through which it passes are given, let us assume the given points be $A({x_1},{y_1},{z_1})$, $B({x_2},{y_2},{z_2})$ and $C({x_3},{y_3},{z_3})$. The co-ordinates of any arbitrary point $P$ on the plane with position vector $\overrightarrow r $ be $(x,y,z)$. Then

$$\begin{equation} \begin{aligned} \overrightarrow {AP} = (x - {x_1})\widehat i + (y - {y_1})\widehat j + (z - {z_1})\widehat k \\ \overrightarrow {AB} = ({x_2} - {x_1})\widehat i + ({y_2} - {y_1})\widehat j + ({z_2} - {z_1})\widehat k \\ \overrightarrow {AC} = ({x_3} - {x_1})\widehat i + ({y_3} - {y_1})\widehat j + ({z_3} - {z_1})\widehat k \\\end{aligned} \end{equation} $$

Substituting the values in the vector equation of plane i.e., $$\overrightarrow {AP} .\left( {\overrightarrow {AB} \times \overrightarrow {AC} } \right) = 0$$$$\left( {\overrightarrow r - \overrightarrow a } \right).\left\{ {\left( {\overrightarrow b - \overrightarrow a } \right) \times \left( {\overrightarrow c - \overrightarrow a } \right)} \right\} = 0$$ We get the cartesian equation of plane passing through three non-collinear points

\[\left| {\begin{array}{c} {x - {x_1}}&{y - {y_1}}&{z - {z_1}} \\ {{x_2} - {x_1}}&{{y_2} - {y_1}}&{{z_2} - {z_1}} \\ {{x_3} - {x_1}}&{{y_3} - {y_1}}&{{z_3} - {z_1}} \end{array}} \right| = 0\]