Deriving the equation of a plane $Ax+By+Cz=D$. Defining a plane as the span of two linearly independent vectors, I've been trying to derive the equation $$Ax+By+Cz=D$$ without much success. The equation seems to indictate that a vector
$$\vec{v}=\begin{bmatrix} x \\ y \\ z\end{bmatrix}$$
is in the plane if and only if $Ax+By+Cz=D$.

I was wondering if anyone could at least point me in the right direction as to how to prove the two definitions are equivalent.

The span of vectors contains the origin, but in general
$$A\,0+B\,0+C\,0\ne D.$$

A plane can be defined as the affine set
$$\vec p=\lambda\vec a+\mu\vec b+\vec c.$$
We can eliminate $\lambda,\mu$ by forming the dot product with $\vec a\times\vec b$,
$$\vec a\times\vec b\cdot\vec p=\vec a\times\vec b\cdot\vec c,$$
which is of the form
$$Ax+By+Cz=D.$$
