This answer to my previous post confuses me: I have always thought about spherical coordinates as a triad of mutually orthogonal vectors $\hat{r},\hat{\theta},\hat{\phi}$ disposed as in the following figure copied from here
but the answer says that the correct representation of a vector $\vec{r}$ is $r \hat{r}$, not something like this:
$$\left(\begin{array}{c} r\\ \theta\\ \phi \end{array}\right)$$
Can anyone explain in a simple way where my interpretation is wrong?
The vector $\vec{r} = x \hat{x} + y \hat{y} + z \hat{z}$ tells you how to go from the origin to the point $(x,y,z)$. And that's the same thing as going $r$ steps from the origin in the $\hat{r}$ direction, right?
Here $r=r(x,y,z)=\sqrt{x^2+y^2+z^2}$ of course, and “the $\hat{r}$ direction” means $\hat{r}(x,y,z)$, the $\hat{r}$ direction associated with that specific point $(x,y,z)$. You could write the formula as $$ \vec{r}(x,y,z) = r(x,y,z) \, \hat{r}(x,y,z) $$ to make this clearer.