The way that I was taught to find the equation of a plane was to take the cross product of basis vectors (v1 X v2)

to get a vector that is normal to the plane, and then take the dot product between that vector and P-P0, with P = <x,y,z> and P0 being the position vector of

another point on the plane. The result would be the equation.

However, given the basis of the plane (v1, v2).. P-P0 should be expressible as the sum of the basis vectors:

P - P0 = a*v1 + b*v2

Is it possible to use this system to find the equation of the plane (z = f(x,y)) with some kind of traditional method (Gaussian elimination)? I think I'm missing a basic idea. Shouldn't this system be a sufficient restriction? Why is the cross product/dot product method necessary?

Thanks in advance for any help.