# Relation Between 3 Planes

Given equation of 2 Planes having normal vectors as $$\vec{P_1}$$, $$\vec{P_2}$$ and $$\vec{P_3}$$ and the following equations

$$A_1x + B_1y + C_1z = D_1$$
$$A_2x + B_2y + C_2z = D_2$$
$$A_3x + B_3y + C_3z = D_3$$

Following can be the relation between these 3 Planes:

1. If $$\vec{P_1}\cdot(\vec{P_2}\times\vec{P_3})\neq0$$ then the planes intersect at a single point
2. If $$\vec{P_1}\cdot(\vec{P_2}\times\vec{P_3})=0$$ and $$\vec{P_1}\times\vec{P_2}=0$$ and $$\vec{P_2}\times\vec{P_3}=0$$ and $$\vec{P_1}\times\vec{P_3}=0$$ then either all 3 planes are parallel, or all 3 planes are coincident, or 2 planes are coincident and 1 plane is parallel to the coincident planes
3. If $$\vec{P_1}\cdot(\vec{P_2}\times\vec{P_3})=0$$ and atleast 2 of $$\vec{P_1}\times\vec{P_2}$$, $$\vec{P_2}\times\vec{P_3}$$ and $$\vec{P_1}\times\vec{P_3}$$ are not zero,then 1 plane intersects 2 parallel or 2 coincident planes
4. If $$\vec{P_1}\cdot(\vec{P_2}\times\vec{P_3})=0$$ and $$\vec{P_1}\times\vec{P_2}\neq0$$ and $$\vec{P_2}\times\vec{P_3}\neq0$$ and $$\vec{P_1}\times\vec{P_3}\neq0$$ then either all 3 planes intersect on a single line or 3 planes intersect on 3 different lines. In such cases a point on the line of intersection of 2 planes can be found. If the point of intersection satisfies the equation of third plane then 3 planes intersect on a single line. Otherwise the planes intersect on 3 separate lines.
Following steps can be used to find a point of intersection of plane when $$\vec{P_1}\cdot(\vec{P_2}\times\vec{P_3})\neq0$$
1. Calulate the following determinants

$$D=\begin{vmatrix} A_1 & B_1 & C_1 \\A_2 & B_2 & C_2 \\A_3 & B_3 & C_3 \end{vmatrix} \hspace{.5cm} D_1=\begin{vmatrix} D_1 & B_1 & C_1 \\D_2 & B_2 & C_2 \\D_3 & B_3 & C_3 \end{vmatrix} \hspace{.5cm} D_2=\begin{vmatrix} A_1 & D_1 & C_1 \\A_2 & D_2 & C_2 \\A_3 & D_3 & C_3 \end{vmatrix} \hspace{.5cm} D_3=\begin{vmatrix} A_1 & B_1 & D_1 \\A_2 & B_2 & D_2 \\A_3 & B_3 & D_3 \end{vmatrix}$$
2. The coordinates of point of intersection is given by following

$$x=\frac{D_1}{D} \hspace{.5cm} y=\frac{D_2}{D} \hspace{.5cm} z=\frac{D_3}{D}$$