5.3 Distance of a point from a plane

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

Let us consider a vector equation of a plane $$\overrightarrow r .\overrightarrow n = d$$ and the perpendicular is drawn from a point $P$ having position vector $\overrightarrow a $

$PM$ be the length of the perpendicular from $P$ to the plane.

Since the line $PM$ passes through a point $P$ having position vector $\overrightarrow a $ and parallel to the normal vector $\overrightarrow n $ of the plane.

Therefore, the equation of line $PM$ is $$\overrightarrow r = \overrightarrow a + \lambda \overrightarrow n $$

Point $M$ is the intersection of line and plane therefore we can write $$\begin{equation} \begin{aligned} \left( {\overrightarrow a + \lambda \overrightarrow n } \right).\overrightarrow n = d \\ \Rightarrow \overrightarrow a .\overrightarrow n + \lambda \overrightarrow n .\overrightarrow n = d \\ \Rightarrow \lambda = \frac{{d - \overrightarrow a .\overrightarrow n }}{{{{\left| {\overrightarrow n } \right|}^2}}} \\\end{aligned} \end{equation} $$ Put the value of $\lambda $ in the equation of line $PM$ we get, $$\overrightarrow r = \overrightarrow a + \left( {\frac{{d - \overrightarrow a .\overrightarrow n }}{{{{\left| {\overrightarrow n } \right|}^2}}}} \right)\overrightarrow n $$

$$\begin{equation} \begin{aligned} \overrightarrow {PM} = {\text{Position vector of }}M - {\text{Position vector of }}P \\ {\text{ = }}\overrightarrow a + \left( {\frac{{d - \left( {\overrightarrow a .\overrightarrow n } \right)}}{{{{\left| {\overrightarrow n } \right|}^2}}}} \right)\overrightarrow n - \overrightarrow a \\ {\text{ = }}\left( {\frac{{d - \left( {\overrightarrow a .\overrightarrow n } \right)}}{{{{\left| {\overrightarrow n } \right|}^2}}}} \right)\overrightarrow n {\text{ }} \\ \Rightarrow PM = \left| {\overrightarrow {PM} } \right| = \left| {\left( {\frac{{d - \left( {\overrightarrow a .\overrightarrow n } \right)}}{{{{\left| {\overrightarrow n } \right|}^2}}}} \right)\overrightarrow n } \right| \\ \Rightarrow PM = \frac{{\left| {d - \left( {\overrightarrow a .\overrightarrow n } \right)} \right|\left| {\overrightarrow n } \right|}}{{{{\left| {\overrightarrow n } \right|}^2}}} \\ \Rightarrow PM = \frac{{\left| {d - \left( {\overrightarrow a .\overrightarrow n } \right)} \right|}}{{\left| {\overrightarrow n } \right|}} \\\end{aligned} \end{equation} $$

or we can write $$ \Rightarrow PM = \frac{{\left| {\left( {\overrightarrow a .\overrightarrow n } \right) - d} \right|}}{{\left| {\overrightarrow n } \right|}}$$which is the length of perpendicular from a point having position vector ${\overrightarrow a }$ to the plane $\overrightarrow r .\overrightarrow n = d$.

Let us consider the cartesian equation of plane $$ax+by+cz=d$$ and $P({x_1},{y_1},{z_1})$ be the point from which the perpendicular is drawn to the plane. Therefore,

$$\overrightarrow a = {x_1}\widehat i + {y_1}\widehat j + {z_1}\widehat k\,;\;\;\overrightarrow n = a\widehat i + b\widehat j + c\widehat k\;$$ Put these values in the above derived equation of perpendicular distance from a point to the plane, we get

$$\begin{equation} \begin{aligned} PM = \frac{{\left| {\left( {{x_1}\widehat i + {y_1}\widehat j + {z_1}\widehat k} \right).\left( {a\widehat i + b\widehat j + c\widehat k} \right) - d} \right|}}{{\left| {\sqrt {{a^2} + {b^2} + {c^2}} } \right|}} \\ PM = \left| {\frac{{a{x_1} + b{y_1} + c{z_1} - d}}{{\sqrt {{a^2} + {b^2} + {c^2}} }}} \right| \\\end{aligned} \end{equation} $$