So as it is, I'm now starting to cover vector-valued functions in my Calculus III class. While studying the topic, I noticed that it seemed to be the exact same thing as parametric equations. I know that I am probably missing an important difference between the two topics, but I can't seem to figure it out.
So the question is:
What is the difference between a set of parametric equations and a vector-valued function?
The difference is that a parametrization has some extra properties. A vector valued function is a map $$f:U\subset\mathbb R^m\to V\subset\mathbb R^n$$
And parametric equations for a [portion of a] submanifold $M$ in Euclidean space (it's rare to parametrize things other than manifolds) is a map $$\varphi:U\subset\mathbb R^m\to M\subset\mathbb R^n$$ Where:
What we could say then, is that a parametrization is always in the form of a vector valued function, but conversely, we use vector valued functions with nice properties to parametrize varieties.
They are basically the same (different ways of expressing the same idea).
In the equation below, the LHS is a vector-valued function, and the RHS is a parametric form of the same function.
$$\vec{v} = \pmatrix{\sin{(t)} \\ \cos{(t)} \\ t} \equiv \matrix{x(t) = \sin{(t)} \\ y(t) = \cos{(t)} \\ z(t) = t}$$
