Since everything turns out to generalize directly from $\mathbb R^3$, it helps to think geometrically. As Joel says, what you're looking for is the equations for two planes such that their intersection is the line. There are many solutions to this; any two different planes that contain your two points will work.
You know how to find an equation for a plane given a normal vector and a point in it, right? So the task is just to find to non-parallel vectors perpendicular to the line.
The direction of the line is $D=P_1-P_2=(3,1,0)$. It is easy enough to find some perpendiculars by guessing, such as $n_1=(1,-3,0)$ and $n_2=(0,0,1)$.
If we don't want to guess (in a general field it might not be immediately clear that our guesses are not parallel) we can be a bit more systematic by finding some vector $e$ that is simply not parallel to $D$ and then using the cross product to get normals $n_1=d\times e$, $n_2=d\times n_1$.