# Given generic position vector $r=\langle x,y,z\rangle$, does the notation “$g=r$” define $g$ as a vector field, or as an identical vector?

Consider a generalised position vector in $$\mathbb{R}^3$$, e.g. $$\mathbf{r} = \langle x,y,z \rangle$$. When I write "$$\mathbf{g} = \mathbf{r}$$", do I define a vector field $$\mathbf{g}$$, or do I simply define another position vector $$\mathbf{g}$$?

To my taste, "$$\mathbf{g}(\mathbf{r}) = \mathbf{r}$$" would rather indicate definition of the vector field, whereas "$$\mathbf{g} = \mathbf{r}$$" would rather define another identical position vector.

If it is really ambiguous, maybe the definition sign, i.e. "$$:=$$", could help to keep those two possibilities apart?

If we write $$a=b$$, then we are saying that $$a$$ and $$b$$ are exactly the same object. If $$\renewcommand{vec}[1]{\mathbf{#1}}\vec{r} = \langle x, y, z \rangle$$ is a general position vector, and we write $$\vec{g} = \vec{r}$$, then we are not saying that $$\vec{g}$$ is a vector field, and we are not saying that $$\vec{g}$$ is an identical copy of $$\vec{r}$$. We are saying that $$\vec{g}$$ is precisely, exactly, identically the same mathematical object as $$\vec{r}$$. There should be no ambiguity here.
If you want $$\vec{g}$$ to denote a vector field, then it would be appropriate to either say "$$\vec{g}$$ is a vector field" or to write $$\vec{g} : \mathbb{R}^n \to \mathbb{R}^n,$$ then specify the action of $$\vec{g}$$ on a general vector $$\vec{r}$$, for example $$\vec{g}(\vec{r}) = \vec{r}$$.