Why is the scalar equation of a given plane unique? There are an infinite number of vector and parametric equations for a given plane. Why is the scalar equation of a given plane unique?
 A: If by scalar equation you mean something like $ax + by + cz = d$, then it is not unique on the nose. But it is unique up to non-zero scalar.
If you think of the equation $ax + by + cz = d$ as being described by the tuple $(a,b,c,d)$, then $(\lambda a, \lambda b, \lambda c, \lambda d)$ will describe the same plane, for any $\lambda \not = 0$. So one says that the equation is unique up to scalar.
Now, if you think about - the collection of these tuples $(a,b,c,d)$ that give you an equation for a plane are exactly those were some one of $a$,$b$ or $c$ is not zero. (other choices will have either "too many" (a 3-space of solutions to the equation $0 = 0$) or too few solutions (no solutions to something like $0 = 1$).)
So this gives you some sense of how many planes there are in 3-space. We get to pick 4 numbers, as long as one of the first three is not zero. That is basically 4 dimensions of choices (we are observing some collection of points in $\mathbb{R}^4$). But then we counted an extra dimension because we counted the same plane multiple times for each scalar $\lambda$. So the "space of planes in $R^3$" is some 3 dimensional space. (What shape does $(\lambda a, \lambda b, \lambda c, \lambda d)$ have as we run through all $\lambda$? It is a line - and you can think that we are counting certain lines in $\mathbb{R}^4$. Indeed this is what is going on. The line $(0,0,0,\lambda)$ that we miss, because the corresponding equation appears to have no solution in $\mathbb{R}^3$, corresponds to a plane "at infinity." This all makes sense in the correct setting...)
Note that I'm allowing planes that don't pass through the origin in this count. Otherwise, what is the dimension? 
You can play the same game with subspaces of fixed dimension (3 dimensional planes in 5 space, for example), but it is harder. The terms to google are Grassmanian and Projective Space.
Hope that helps clarify what is going on...
