Relation Between 2 Planes

1. Any 2 Planes in 3D Cartisean Space can be related in following ways
   1. The 2 Planes can be Parallel
   2. The 2 Planes can be Coincident
   3. The 2 Planes can Intersect with each other on a Line
2. The following gives Explicit Equation of 2 Planes \(P_1\) and \(P_2\)
   
   \(P_1:\hspace{2mm}A_1x + B_1y + C_1z = D_1 \)
   
   \(P_2:\hspace{2mm}A_2x + B_2y + C_2z = D_2 \)
   
   The Normal Vectors to the Planes \(P_1\) and \(P_2\) represented by Vectors \(\vec{P_1}\) and \(\vec{P_2}\) respectively are given by Co-efficients of the Coordinate Variables of the Equation of Planes as follows
   
   \(\vec{P_1}=\begin{bmatrix}A_1 \\ B_1 \\ C_1\end{bmatrix}\hspace{6mm}\vec{P_2}=\begin{bmatrix}A_2 \\ B_2 \\ C_2\end{bmatrix}\)
   
   The following gives the Steps to Find the Relation between Planes \(P_1\) and \(P_2\)
   1. If \(\vec{P_1}\times\vec{P_2}=0\), then Find a Point on One Plane. If this Point has a Distance 0 from the Other Plane or the Point Satisfies the Equation of the Other Plane then the Planes are Coincident. Otherwise the Planes are Parallel.
   2. If \(\vec{P_1}\times\vec{P_2}\neq0\) then the Planes Intersect with each other on a Line whose Direction Vector is given by \(\vec{P_1}\times\vec{P_2}\). Following steps can be used to Find a Point on the Line
      1. Calulate the following Determinants
         
         \(D_1=\begin{vmatrix} A_1 & B_1 \\A_2 & B_2 \end{vmatrix} \hspace{.5cm} D_2=\begin{vmatrix} B_1 & C_1 \\B_2 & C_2 \end{vmatrix} \hspace{.5cm} D_3=\begin{vmatrix} A_1 & C_1 \\A_2 & C_2 \end{vmatrix} \)
      2. If \(D_1\neq0\) then the Line Passes through a Point having its Z coordinate as 0. Setting Z coordinate as 0 in the two Equations of Planes and solving for X coordinate and Y coordinate the Point can be determined.
      3. If \(D_1=0\) and \(D_2\neq0\) then the Line Passes through a Point having its X coordinate as 0. Setting X coordinate as 0 in the two Equations of Planes and solving for Y coordinate and Z coordinate the Point can be determined.
      4. If \(D_1=0\) and \(D_2=0\) and \(D_3\neq0\) then the Line Passes through a Point having its Y coordinate as 0. Setting Y coordinate as 0 in the two Equations of Planes and solving for X coordinate and Z coordinate the Point can be determined.