Derivation of ideal magnetic dipole field strength

[RedDeer44]

For reference, this is from Griffiths, introduction to quantum mechanics electrodynamics, p253-255

When deriving the ideal magnetic dipole field strength, if we put the moment m at origin and make it parallel to the z-axis,
the book went from the vector potential A

$$
A= \frac{\mu_0}{4\pi}\frac{\vec{m}\times \hat{r}}{r^2}
$$
to
$$
A = \frac{\mu_0}{4\pi}\frac{m\sin{\theta}}{r^2}\hat{\phi}
$$

Can someone explain how the single $\phi$ component come about? This to me seems to indicate $r$ has non-zero $\theta$ component and zero $\phi$ component. But I thought $r$ is any point?

Also, for a point in spherical coordinates, is $\phi$ value defined when $\theta = 0$? Or when $r=0$?

 

[vanhees71]

You have $\vec{m}=(0,0,m)$ and thus
$$\vec{m} \times \hat{r} = \begin{pmatrix} 0\\0\\m \end{pmatrix} \times \begin{pmatrix} \cos \varphi \sin \vartheta \\ \sin \varphi \sin \vartheta \\ \cos \vartheta \end{pmatrix} =m \sin \vartheta \begin{pmatrix} -\sin \varphi \\ \cos \varphi \\ 0 \end{pmatrix} = m \sin \vartheta \hat{\varphi}.$$