The implicit equation for a plane perpendicular to a given vector at the origin is $ax + by + cz = 0$.
I can write this in parametric form as $x = t, y = u, z = -\frac{at + bu}{c}$.
The only problem is that I can't use this equation when $c = 0$, and $a$, $b$ and $c$ are each going to be zero at different times so I can't avoid division by zero by shuffling around the coordinates. Is there a way to formulate this plane parametrically without restricting values of $a$, $b$ and $c$?