The many ways in which to express a plane There are many ways to express a plane of $R^3$. I am focusing on two of them.
The first is the cartesian equation $Ax + By + Cz + D = 0$.
The second is to give two direction vectors $u$ and $v$ and a point $P$ of the plane.
My question is: how can I obtain two ortogonal direction vectors $u$ and $v$ and a point $P$ from the cartesian equation $Ax + By + Cz + D = 0$? How can I obtain the cartesian equation from the direction vectors and a point of the plane?

As other answers point out, you know the normal vector $n=[A,B,C]$. Suppose you have one vector $u\perp n$, $|u|\neq 0$, then clearly you can find $v\perp u$, $v\perp n$ using $v=u\times n$.
So the problem reduces to finding a single nonzero vector perpendicular to $n$.
In 3D, there is no single (non-branching) formula that will do this: see this question.
