Question: The title is worded a bit confusingly. I apologize, I just couldn't think of how to phrase it.

Either way, say you have the plane $x+2y-4z=8$

The normal line will have direction $x=1, y=2, z=-4$

So they are very closely linked. However, I'm having trouble wrapping my head around *why* that is. How come you can list the normal vector and (along with a scalar) define a plane? I understand that it works, I just wonder how come?

I hope someone can explain this. Thanks!

Answer: 
For a really down-to-earth explanation, how about this:

Imagine a satellite dish with a central aerial that you aim at a distant point (the aerial is the normal), and imagine that the "dish" is actually rather flat (this is your plane). Any plane passing through the place where you are can be achieved by rotating your dish appropriately, and you can get to any parallel plane just by moving a fixed distance along the direction your aerial points along.

So, you just need 2 pieces of information to specify your plane: 

* the direction of the aerial (= the normal)
* the distance you displace
View it *geometrically*. For example, let the normal be the $z$-axis. Geometrically, this makes no difference, since any line could be picked as the $z$-axis. Now you can *see* what planes have the $z$-axis as a normal. They are just the planes parallel to the $x$-$y$ plane.

Any such plane can be fully identified if you know in addition a single number, namely the height at which the plane meets the $z$-axis.