Consider a system of linear equation in x,y and z
a1x + b1y + c1z = 0,
a2x + b2y + c2z = 0,
a3x + b3y + c3z = 0;
and the vectors a' = (a1, b1, c1), b' = (a2, b2, c2), and c' = (a3, b3, c3). (we position all vectors so that their starting points are at the origin)
Prove that this system has a nontrivial solution ((x,y,z) ≠ (0,0,0)) if and only if the vectors a', b', and c' lie in the same plane (Hint: for the vector n = (x,y,z), we have 0 = a1x + b1y + c1z = a'.n, and similarly for b' and c'.)
Any answers or explanations would be much appreciated.