HI again PG,
Oh you're welcome and im happy to see you are interested in these things.
If you like, you might try to map out a field if you've never done this before. This will give you a lot of insight i think.
What you could do is start with the field described by the function:
F(x,y,)=-y*i+x*j
where i is the unit vector in the direction of x and j is the unit vector in the direction of y. This is actually much easier than we might think at first.
For example, when x=1 and y=1, F(x,y)=-1i+1j, so what we do is draw an arrow that has length sqrt((-1)^2+(1)^2))=sqrt(2) and imagine that we had a new coordinate system x',y' riding on the old system so that on the x y plane where x'=-1 and y'=1 this would be an arrow pointing to the left and upward and centered at the original point (1,1).
The attachment shows this single vector. The length is sqrt(2) and the position is at the original (1,1), and because (1,1) mapped to (-1,1) we see that the new coordinate system on the right has an arrow from (0,0) to (-1,1), and then that arrow is taken over to the left and placed at (1,1). It maintains it's length and direction, but it's position comes from the original coordinate system. So that original point has the magnitude and direction as indicated by the vector.
So the steps involved here were quite simple:
1. Transform the original coordinates into the F function coordinates using the function F(x,y)=-yi+xj (this makes a new set (x',y')).
2. Draw those coordinates as a point in a new coordinate system and an arrow from (0,0) to the point, that's the vector.
3. Transfer that vector to the original coordinate system point that was used to generate it, centering the vector body on the point (this is because the vector represents that point and only that one single point so it is associated with only that one point and not really the whole arrow).
Note that the position came from the original point (x,y) but the arrow came from the new coordinate system (x',y') after (x,y) was mapped using the function F into (x',y') with the tail at (0',0').
If you go ahead and try to map some of the other vectors of that function F you should gain quite a bit of insight into the vector field concept.
For example, try doing the point (-1,-1). Note that this should lead to an arrow that is centered at the original coordinate system's point at (-1,-1).