Orthogonality of vectors to a plane

Question

Let´s consider $g$ as a symmetry to the plane $ax + by + cz = 0$ How would you prove that for all vectors $v$ in R3, $g(v) - v$ is orthogonal to the to the symmetry plane?

Answer 1

Up to an invertible linear change of coordinates we can assume that the plane is defined by $z=0$, so that the simmetry is $g(x, \, y, \, z)=(x, \, y, \, -z)$.

Then if $v=(v_1, \, v_2, \, v_3)$ we have $g(v)-v=(0, \, 0, \, -2v_3)$ and we are done.