Any 3 Planes in 3D Cartisean Space can be related in following ways
All 3 Planes are Parallel
All 3 Planes are Coincident
2 Planes are Coincident and the 3rd Plane is Parallel to the Coincident Planes
1 Plane Intersects 2 Coincident Planes
1 Plane Intersects 2 Parallel Planes
All 3 Planes Intersect on a Single Line
All 3 Planes Intersect on 3 Different Lines
All 3 Planes Intersect at a Single Point
The following gives Explicit Equation of 3 Planes \(P_1\), \(P_2\) and \(P_3\)
\(P_1:\hspace{2mm}A_1x + B_1y + C_1z = D_1 \)
\(P_2:\hspace{2mm}A_2x + B_2y + C_2z = D_2 \)
\(P_3:\hspace{2mm}A_3x + B_3y + C_3z = D_3 \)
The Normal Vectors to the Planes \(P_1\), \(P_2\) and \(P_2\) represented by Vectors \(\vec{P_1}\), \(\vec{P_2}\) and \(\vec{P_3}\) respectively are given by Co-efficients of the Coordinate Variables of the Equation of Planes as follows
The following gives the Steps to Find the Relation between Planes \(P_1\), \(P_2\) and \(P_3\)
If \(\vec{P_1}\cdot(\vec{P_2}\times\vec{P_3})=0\) and \(\vec{P_1}\times\vec{P_2}=0\) and \(\vec{P_2}\times\vec{P_3}=0\) 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.
To find out further, Calculate Distances \(D_{12}\), \(D_{23}\) and \(D_{13}\) between Planes \(P_1\) and \(P_2\), \(P_2\) and \(P_3\), \(P_1\) and \(P_3\) respectively. Now,
If \(D_{12}\neq0\), \(D_{23}\neq0\) and \(D_{13}\neq0\) then the 3 Planes are Parallel. System of Linear Equations formed by the Equations of these 3 Planes has No Solutions.
If any 2 of the Distances \(D_{12}\), \(D_{23}\) and \(D_{13}\) are 0 then the 3 Planes are Coincident. System of Linear Equations formed by the Equations of these 3 Planes has Infinitely Many Solutions.
If any 2 of the Distances \(D_{12}\), \(D_{23}\) and \(D_{13}\) are Non Zero and 1 of them is 0 then the 2 Planes Corresponding to 0 Distance are Coincident and the 3rd is Parallel. System of Linear Equations formed by the Equations of these 3 Planes has No Solutions.
If \(\vec{P_1}\cdot(\vec{P_2}\times\vec{P_3})=0\) and 2 of of the Vectors \(\vec{P_1}\times\vec{P_2}\), \(\vec{P_2}\times\vec{P_3}\) and \(\vec{P_1}\times\vec{P_3}\) are Not Zero and 1 is Zero, then 1 Plane Intersects 2 Parallel or 2 Coincident Planes.
If the Distance \(D\) between the Planes having Cross Product of Normals as Zero is 0, then the Corresponding Planes are Coincident and the System of Linear Equations formed by the Equations of these 3 Planes has Infinitely Many Solutions.
Otherwise they are Parallel and the System of Linear Equations formed by the Equations of these 3 Planes has No Solutions.
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
the 3 Planes Intersect on a Single Line if the System of Linear Equations formed by the Equations of 3 Planes has Infinitely Many Solutions or
the 3 Planes Intersect on 3 Different Lines if the System of Linear Equations formed by the Equations of 3 Planes has No Solution.
If \(\vec{P_1}\cdot(\vec{P_2}\times\vec{P_3})\neq0\) then All the 3 Planes Intersect at a Single Point. This Point of Intersection
of 3 Planes can be found out by Solving the System of Linear Equations formed by the Equations of 3 Planes, which gives a Unique Solution.