5.4 Distance between two parallel planes

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 assume the vector equation of two parallel planes be $$\overrightarrow r .\overrightarrow n = {d_1}$$ and $$\overrightarrow r .\overrightarrow n = {d_2}$$

Since the normal vector $\overrightarrow n $ is same for both the vectors means they are parallel. Therefore, the distance between two parallel planes is $$d = \frac{{\left| {{d_1} - {d_2}} \right|}}{{\left| {\overrightarrow n } \right|}}$$

If the equations of two parallel planes is given in cartesian form i.e., $$ax + by + cz + {d_1} = 0$$ and $$ax + by + cz + {d_2} = 0$$ then the distance between two parallel planes is given by $$d = \left| {\frac{{{d_1} - {d_2}}}{{\sqrt {{a^2} + {b^2} + {c^2}} }}} \right|$$