Linear Algebra and the C Language/a062

 A homogeneous linear system with as many equations 
 as unknowns has a nontrivial  solution if and only 
 if the determinant  of the matrix is zero.

Let us calculate the equation of the plan passing through points P, Q and R:
 
  c1 x  + c2 y  + c3 z  + c4 = 0

This same equation with the points  P(x1,y1,z1) Q(x2,y2,z2) and R(x3,y3,z3):
 
  c1 x1 + c2 y1 + c3 z1 + c4 = 0 
  c1 x2 + c2 y2 + c3 z2 + c4 = 0 
  c1 x3 + c2 y3 + c3 z3 + c4 = 0 
 
The system of four equations:

  c1 x  + c2 y  + c3 z  + c4 = 0
  c1 x1 + c2 y1 + c3 z1 + c4 = 0
  c1 x2 + c2 y2 + c3 z2 + c4 = 0
  c1 x3 + c2 y3 + c3 z3 + c4 = 0

The determinant of the system:

    |x    y    z   1|
    |x1   y1   z1  1| = 0
    |x2   y2   z2  1| 
    |x3   y3   z3  1|

The determinant in C language:

    |1    1    1   1|
    |x1   y1   z1  1| = 0
    |x2   y2   z2  1| 
    |x3   y3   z3  1|

To calculate the coefficients of the equation of the plan, we use the cofactor expansion along the first row.
  
  cof(R1,C1) x + cof(R1,C2) y + cof(R1,C3) Z + cof(R1,C4) = 0

This equation gives us the equation of the plan that passes through the three points P, Q and R.