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

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