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?

From $$Ax+By+Cz+D=0$$
you get first the normal vector to the plane $n=(A,B,C)$.
then you can take
$$u=(0,C,-B)$$
and
$v$ as the vectorial product of $n$ by $u$.
To get the cartesian equation from two vectors $u,v$ and a point $P$, 
$$det(PM,u,v)=0$$
with $M=(x,y,z)$.
