This may be a dumb question. If so I apologize... I don't consider myself an expert in Linear Algebra by any means.
If I have the equation of a plane where $Ax+By+Cz=D$ and $D\neq 0$, then how do you find the basis?
Normally, you would solve for $Ax=0$ and determine the basis vectors, thus based on the null space basis. However $D$ is really equivalent to the dot product of a position vector to a known point on the plane and the normal vector. If this known point is the origin, thus a zero vector, then this makes sense. But if it's not, then what? Does $D$ count in somehow? Should I then use $Ax=b$ and derive the basis based on the Column Space instead of the Null Space?
If $D\ne0$ then the plane is not a vector space and so the concept of basis is meaningless.
However, you could find a basis $\def\v#1{{\bf#1}}\{\v v_1,\v v_2\}$ for the plane $Ax+By+Cz=0$; then if $\v u$ is one specific vector on your plane, all vectors on the plane will be given by $$\v u+\lambda_1\v v_1+\lambda_2\v v_2\ ,\quad \lambda_1,\lambda_2\in{\Bbb R}\ .$$