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). 

I used to think they are the same, but recently I found they are different, at least for the meaning of coplanarity. For example, suppose a vector function $r(t) = <f(t), g(t), h(t)>$ and a paramatric curve $(x, y, z) = (f(t), g(t), h(t))$.
When we say $r(t)$ lies on the same plane, we mean that all vectors determined by $r(t)$ are parallel to the same plane. This means that all points $(f(t), g(t), h(t))$ determined by position vectors of $r(t)$ and the origin lie on the sample plane.
When we say the paramatric curve $(x, y, z) = (f(t), g(t), h(t))$ lie on the same plane, we just mean that all points on the paramatric curve lie on the same plane. The plane deos not need to contain the origin.
Practically, to prove the curve determined by $r(t)$ lies on the same plane with normal vector $n$, one needs to prove $n\cdot r(t)=h$ where $h$ is a constant. To prove $r(t)$ lies on the same plane, one needs to prove  $ n\cdot r(t)=0$.
