Condition for Coplanarity of 4 Points

1. Four Points \(A\) (\(x_1,y_1,z_1\)), \(B\) (\(x_2,y_2,z_2\)), \(C\) (\(x_3,y_3,z_3\)), and \(D\) (\(x_4,y_4,z_4\)) having Position Vectors \(\vec{A}\), \(\vec{B}\), \(\vec{C}\) and \(\vec{D}\) respectively are Coplanar if:
   
   \(\vec{AB}\cdot(\vec{AC} \times \vec{AD})= \begin{vmatrix} x_1-x_2 & y_1-y_2 & z_1-z_2 \\ x_1-x_3 & y_1-y_3 & z_1-z_3\\ x_1-x_4 & y_1-y_4 & z_1-z_4\end{vmatrix}=\begin{vmatrix} x_1 & y_1 & z_1 & 1 \\ x_2 & y_2 & z_2 & 1\\ x_3 & y_3 & z_3 & 1\\ x_4 & y_4 & z_4 & 1\end{vmatrix}=0\)
   
   where
   
   \(\vec{AB} =\vec{A}-\vec{B} = (x_1-x_2)\hat{i} + (y_1-y_2)\hat{j} + (z_1-z_2)\hat{k}\)
   
   \(\vec{AC} =\vec{A}-\vec{C} = (x_1-x_3)\hat{i} + (y_1-y_3)\hat{j} + (z_1-z_3)\hat{k}\)
   
   \(\vec{AD} =\vec{A}-\vec{D} = (x_1-x_4)\hat{i} + (y_1-y_4)\hat{j} + (z_1-z_4)\hat{k}\)