What is the actual difference between these two: Vector valued functions and Vector Fields?
As far as I can see, both these concepts are alike, they take in $x$ and $y\,\,$(simple case) and give out a vector. But the former gives a vector from origin and the tips of these vectors trace out the function and the latter takes in $x$ and $y$ and associates a vector to each point. Where does the difference(vector from origin, vector from the given point) kick in?
By Vector Function, I mean the following, a function that takes a real number $t$ and gives a vector, whose coordinates are given by three functions $f(t)\, ,g(t) \, $and$ \, h(t)$. All these vectors trace out a curve.(space curve). Please take a look at Position vector valued functions
I saw certain questions regarding the same, but I couldn't understand the author. Please make the answer more physical and geometric instead of set notation definitions if possible.