# In the equation for the magnetic field caused by a single moving point charge, is the degree of $|r|$ in the denominator 2 or 3?

You miswrote the first of the two.

$$\vec{B} = \frac{\mu_0}{4\pi} \frac{q\vec{v} \times \hat{r}}{r^2} = \frac{\mu_0}{4\pi} \frac{q\vec{v} \times \vec{r}}{r^3}$$ because $$\hat{r}=\frac{\vec{r}}{r}\implies\frac{\hat{r}}{r^2}=\frac{\vec{r}}{r^3}$$ where $$\hat{r}$$ is the unit vector in the $$r$$ direction.

You've written the first equation wrong, it's actually $$\hat{\mathbf{r}}$$, not $$\vec{\mathbf{r}},$$ i.e. it's the unit vector in the $$\mathbf{r}$$ direction. Clearly, in this case, both equations are the same, if you multiply the first by $$|\mathbf{r}|$$, since $$r\hat{\mathbf{r}} = \vec{\mathbf{r}}$$.

If you have trouble remembering which of the two is "right", dimensional analysis is your friend. Let's say all you remember are Maxwell's Equations (a very good practice, in my opinion). Then you'd know that $$\nabla \times \mathbf{B} = \mu_0 \mathbf{j}.$$ Looking at the above equation dimensionally, you should see that the dimensions of $$\mathbf{B}$$ are:

$$L^{-1}[B] = [\mu_0] [j] = [\mu_0] [I] L^{-2},$$

where I've used the fact that the curl is a derivative with respect to position, and that $$j$$ is the current ($$I$$) density. Now, with the definition of current as being charge per unit time, I'll leave it to you to show that: $$[B] = [\mu_0]\, [q v]\,\ L^{-2},$$

meaning that of the two equations as you've written them in your question, only the second could be dimensionally consistent. (Of course, dimensional consistency is no guarantee that it's the right equation! It just helps filter obviously wrong ones.)