The graph of a function f: X -> Y is the set {(x, f(x)) | x in X}. It is much more clear and precise to associate elements of the graph with vectors such that the 0 vector is identified with the R^2 origin, and then points in R^2 are identified with vectors. Then there is a mapping between vectors in this vector space to the graph, i.e., to points (x, f(x)).
> In physics, a vector is often more specifically something with magnitude and direction.
Physics is sloppy. :) This is not a general description of a vector, where vector is an element of a general vector space. Not all vector spaces have a norm, which is required for magnitude to make any sense.
> but mathematically, if you translate them away from the origin they're still the same vector.
Right, and you cannot always translate vectors without more machinery, such as parallel transport.
> Physics is sloppy. :) This is not a general description of a vector, where vector is an element of a general vector space. Not all vector spaces have a norm, which is required for magnitude to make any sense.
Fine, then let's be precise. An element of the vector space R^n is nothing more than a function from {1,...,n} to R. And every n-dimensional R-vector space is isomorphic to it.
How you wanna draw this in a coordinate system is up to you. It is customary to identify vectors with points on that coordinate system and then equivalently with an arrow pointing from the origin to that point. In that case the zero vector is the origin, or an arrow from the origin to itself.
It is equally customary to use vectors as displacement, i.e. a directed difference between two points. In that case, the vector that you can't actually draw because it has zero length is your zero element. Now your arrows don't have to be anchored at the origin anymore.
Both of these spaces are of course isomorphic.
If your vector space is infinite-dimensional it's of course not gonna be really possible to draw anymore.
The graph of a function f: X -> Y is the set {(x, f(x)) | x in X}. It is much more clear and precise to associate elements of the graph with vectors such that the 0 vector is identified with the R^2 origin, and then points in R^2 are identified with vectors. Then there is a mapping between vectors in this vector space to the graph, i.e., to points (x, f(x)).
> In physics, a vector is often more specifically something with magnitude and direction.
Physics is sloppy. :) This is not a general description of a vector, where vector is an element of a general vector space. Not all vector spaces have a norm, which is required for magnitude to make any sense.
> but mathematically, if you translate them away from the origin they're still the same vector.
Right, and you cannot always translate vectors without more machinery, such as parallel transport.