Linear Algebra and the C Language/a062

__NOTOC__ The coefficients of the equation of a plan. Presentation : 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. Application dplan.h .............. Temporary h file Delete this file at the end of the session. c00a.c c00b.c