Better would be to choose two vectorsinthe plane. 6x- 3y- 2z= 0 is the same as z= 3x- (3/2)y so that <x, y, z>= <x, y, 3x- (3/2)y>= x<1, 0, 3> + y<0, 1, -3/2> or, if you don't like fractions, x<1, 0, 3>- (y/2)<0, 2, -3> so that u= <1, 0, 3> and v= <0, 2, -3> form a basis for the subspace consisting of the plane.

I wouldn't bother with Gram-Schmidt, just project <1, 0, 0> onto each of those two vectors and add. (in fact, since <1, 0, 0>.<0, 2, -3>= 0, just project onto u!)