5.1 Angle between two 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

As we know that to find the equation of plane we need a point through which it passes and a normal vector to the plane.

Therefore, the angle between two planes is nothing but the angle between their normals.

Note: If $\theta $ is the acute angle between the two planes then ${180^ \circ } - \theta $ is the obtuse angle between them. Here we are considering the acute angle as the angles between the two planes.

Let us assume the vector equation of two planes be $$\overrightarrow r .\overrightarrow {{n_1}} = {d_1}$$ and $$\overrightarrow r .\overrightarrow {{n_2}} = {d_2}$$ where $\overrightarrow {{n_1}} $ and $\overrightarrow {{n_2}} $ are the normal vectors of the planes respectively and $\theta $ be the angle between the normals. We can use the dot product of two vectors to find the angle between them i.e., $$\begin{equation} \begin{aligned} \overrightarrow {{n_1}} .\overrightarrow {{n_2}} = \left| {\overrightarrow {{n_1}} } \right|\left| {\overrightarrow {{n_2}} } \right|\cos \theta \\ \cos \theta = \frac{{\overrightarrow {{n_1}} .\overrightarrow {{n_2}} }}{{\left| {\overrightarrow {{n_1}} } \right|\left| {\overrightarrow {{n_2}} } \right|}} \\\end{aligned} \end{equation} $$

If two planes are perpendicular then their normal vectors are also perpendicular to each other. Therefore, the condition for two planes to be perpendicular is $${\overrightarrow {{n_1}} .\overrightarrow {{n_2}} = 0}$$

If two planes are parallel then their normal vectors are also parallel to each other. Therefore, the condition for two planes to be parallel is $${\overrightarrow {{n_1}} = \lambda \overrightarrow {{n_2}} }$$ where $\lambda $ is any scalar quantity.

Let us assume the cartesian equation of two planes be

$${a_1}x + {b_1}y + {c_1}z + {d_1} = 0$$ and $${a_2}x + {b_2}y + {c_2}z + {d_2} = 0$$ Using the direction ratios of normal vectors to the plane ${a_1}$, ${b_1}$, ${c_1}$, ${d_1}$, ${a_2}$, ${b_2}$, ${c_2}$, ${d_2}$, we can write $$\cos \theta = \frac{{{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2}}}{{\sqrt {{a_1}^2 + {b_1}^2 + {c_1}^2} \sqrt {{a_2}^2 + {b_2}^2 + {c_2}^2} }}$$

If two planes are perpendicular then their normal vectors are also perpendicular to each other. Therefore, $$\theta = {90^ \circ }$$$$ \Rightarrow \cos \theta = \cos {90^ \circ } = 0$$ The condition for two planes to be perpendicular is $${a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2} = 0$$

If two planes are parallel then their normal vectors are also parallel to each other. Therefore, the condition for two planes to be parallel is $$\frac{{{a_1}}}{{{a_2}}} = \frac{{{b_1}}}{{{b_2}}} = \frac{{{c_1}}}{{{c_2}}}$$