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.

A plane containing the origin is the span of two linearly independent vectors $\vec{a}$ and $\vec{b}$.  The plane normal is $\vec{n} = \vec{a}\times \vec{b}$.  If $\vec{x}$ is a point of the plane, $\vec{x}=u\vec{a} + v\vec{b}$ and $\vec{n}\cdot \vec{x} = 0$ since $\vec{n}\cdot \vec{a}=0$ and $\vec{n}\cdot \vec{b}=0$.  To transform the equation $\vec{n}\cdot\vec{x}=0$ to the more general equation, simply apply a translation.  More specifically, $\vec{n}\cdot (\vec{x}-\vec{P}) = 0$ is the equation of a plane containing the point $P$.
