Difference between a vector function and parametric equations What is the difference between a vector function and parametric equations? Both are capable of describing plane/space curves and both can indicate direction. The only notable difference I see is that a vector function includes a measured distance between a given point and the origin (magnitude of position vector that draws out a plane/space curve). 

What do you mean that a vector function includes a measured distance (magnitude)?
Vector functions and parametric equations are essentially the same thing.
Sometimes, if you're working in a different coordinate system, like polar coordinates, you might see that they include the magnitude $r$ in their functions. 
For example, $(x,y)=(\cos(t),\sin(t))$ for $t\in [0,2\pi]$, versus $(r,\theta)=(1,t)$ for $t \in [0,2\pi]$. 
A vector function says the exact same thing except $\vec{v}(t)=\cos(t)\vec{i}+\sin(t)\vec{j}.
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