Condition for Collinearity and Concurrency of Planes

1. A Set of \(N\) Planes (where \(N \geq 3\)) are said to be Collinear if they All Pass through a Single Common Line of Intersection.
2. A Set of \(N\) Planes (where \(N \geq 3\)) are said to be Concurrent if they All Pass through a Single Common Point of Intersection.
3. The following gives Explicit Equation of a Set of \(N\) Planes \(P_1\), \(P_2\), \(P_3\), ..., \(P_N\)
   
   \(P_1:\hspace{2mm}A_1x + B_1y + C_1z = D_1 \)
   
   \(P_2:\hspace{2mm}A_2x + B_2y + C_2z = D_2 \)
   
   \(P_3:\hspace{2mm}A_3x + B_3y + C_3z = D_3 \)
   
   \(\vdots\)
   
   \(P_N:\hspace{2mm}A_Nx + B_Ny + C_Nz = D_N \)
   
   The Normal Vectors to the Planes \(P_1\), \(P_2\), \(P_3\), ..., \(P_N\) represented by Vectors \(\vec{P_1}\), \(\vec{P_2}\), \(\vec{P_3}\), ..., \(\vec{P_N}\) respectively are given by Co-efficients of the Coordinate Variables of the Equation of Planes as follows
   
   \(\vec{P_1}=\begin{bmatrix}A_1 \\ B_1 \\ C_1\end{bmatrix}\hspace{6mm}\vec{P_2}=\begin{bmatrix}A_2 \\ B_2 \\ C_2\end{bmatrix}\hspace{6mm}\vec{P_3}=\begin{bmatrix}A_3 \\ B_3 \\ C_3\end{bmatrix} \hspace{6mm}\cdots\hspace{6mm}\vec{P_N}=\begin{bmatrix}A_N \\ B_N \\ C_N\end{bmatrix}\)
   
   The Set of \(N\) Planes as given above are Collinear if
   1. The Cross Products of All Possible Combinations of Vectors \(\vec{P_1}\), \(\vec{P_2}\), \(\vec{P_3}\), ..., \(\vec{P_N}\) are Non Zero and are Oriented in either Same or Opposite Direction.
   2. System of Linear Equations formed by the Equations of the Planes as given above has Infinitely Many Solutions.
   The Set of \(N\) Planes as given above are Concurrent if
   1. The Scaler Tripple Products of All Possible Combinations of Vectors \(\vec{P_1}\), \(\vec{P_2}\), \(\vec{P_3}\), ..., \(\vec{P_N}\) are Non Zero.
   2. System of Linear Equations formed by the Equations of the Planes as given above has a Unique Solution.