To check whether four points are coplanar (lie in the same plane)
The given points $(x_1,y_1,z_1)$, $(x_2,y_2,z_2)$, $(x_3,y_3,z_3)$, $(x_4,y_4,z_4)$ are coplanar (they lie in the same geometric plane), if the determinant of the matrix M is equal to zero, where $$ M = ( \table 1,x_1,y_1,z_1; 1,x_2,y_2,z_2; 1,x_3,y_3,z_3; 1,x_4,y_4,z_4 ) $$