I apologize upfront if this question is too vague but it's only because my understanding of it is too vague as well. Anyway, here goes…

I'm currently studying partial derivatives and gradient vectors but am having difficulty visualizing the orientation of the gradient vector relative to its originating function. I've assumed that all vectors originate at (0,0,0) and extend outward into any direction, but if at any given point along an (x,y,z) function there exists a vector indicating the maximum slope and direction at that point, how does that vector originate at (0,0,0) for all possible slopes and directions?

I'm imagining tiny, little arrows pointing here and there scattered throughout the air in a room, all indicating a change in temperature, but, of course, they cannot all be originating from (0,0,0). How can this be?

If anyone can help, thank you very much.

Even if the vectors don't originate at the origin you can express them as if they do. For the sake of my diagram, let go back to two dimensions for a bit:

Consider this situation, we have a vector AB and we want to represent it in terms of vectors from the origin, the following property of vectors can be used:

$\tiny \dpi{300} \fn_cm \underset{AB}{\rightarrow} = \mathbf{b}- \mathbf{a}$

$\tiny \dpi{300} \fn_cm \underset{AB}{\rightarrow} = \underset{OB}{\rightarrow}- \underset{OA}{\rightarrow}$

So back to three dimensions:

$\tiny \dpi{300} \fn_cm \underset{AB}{\rightarrow} = \begin{pmatrix}x_{b} \\ y_{b} \\ z_{b} \end{pmatrix} - \begin{pmatrix}x_{a} \\ y_{a} \\ z_{a} \end{pmatrix}$

Where AB is a direction vector from the point A to the point B and x(a), y(a), z(a) etc. are values for the direction vector from the origin to the points A and B ( therefore also the position vectors of A and B ).

I hope this is what you were asking. =)