I understand that for any plane equation $(n1, n2, n3)·(x, y, z) = c$, $c/|\mathbf n|$ exactly equals the distance you go along $\hat n$ to reach the plane (so $c\hat n/|\mathbf n|$ is on the plane). I don't understand why we need to deal with $c/|\mathbf n|$, if we require that the standard plane equation is written using the unit normal vector to n instead of any n then c is exactly the distance instead of $c/|\mathbf n|$ which is more intuitive.
I believe that math generally makes these definition to ease some calculation (maybe it's just to not deal with unit vectors). Is it so the standard definition isn't as rigid, or to not deal with awkward numbers, or is there something else useful about using any n that I'm missing?