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.

  • $\begingroup$ While vector fields should be clear to most people, can you give an example of the former? Your description doesn't make much sense. Please make it into the post instead of commenting. $\endgroup$ – Lee David Chung Lin Apr 27 at 7:05
  • $\begingroup$ @LeeDavidChungLin I've added some text please take a look. $\endgroup$ – Aravindh Vasu Apr 27 at 7:21
  • $\begingroup$ The explanation you added is good. $\endgroup$ – Lee David Chung Lin Apr 27 at 7:32
  • $\begingroup$ The domain of the Vector Function as defined is just the $1$-dim real line, no matter how many components it has. A vector field has a domain of $2$-dim plane, $3$-dim space, or higher. They are just totally different stuff. The Youtube clip you linked to made an extra step of assigning (pairing) the components of the Vector Function to the $2$-dim (or $3$-dim) unit vectors. Only after this extra step does one treat the Vector Function the same as a vector field. $\endgroup$ – Lee David Chung Lin Apr 27 at 7:35
  • $\begingroup$ So after the assignment, both are the same stuff? $\endgroup$ – Aravindh Vasu Apr 27 at 7:40

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