Why does the Biot-Savart law give the induced velocity of a infinitesimal segment of a vortex filament at point P? My question is about the proof of how the Biot-Savart law can be used for vortex filaments. This is what I have in my textbook:


However, I'm not certain how it was derived - I understand that the Biot-Savart law originally came from electromagnetic theory, but could anyone present the proof for it's use for vortex filaments?

Intuitively though, something seems off - a vortex filament is just made out of lots of individual vortexes put together in a filament, so shouldn't the induced velocity at a point P of an infinitesimal section of the filament be zero unless $d\vec{l}$ and $\vec{r}$ are perpendicular... but clearly it's not the case,
$$
 d\vec{V} = \frac{ d\vec{l} \times  \vec{r} }{ \lvert \vec{r} \lvert ^3}
$$
, which is nonzero even when $d\vec{l}$ and $\vec{r}$ aren't perpendicular
Why would


  the induced velocity at a point P of an infinitesimal section of the filament be zero unless $d\vec{l}$ and $\vec{r}$ are perpendicular


?

The cross product in
$$
 \frac{ d\vec{l} \times  \vec{r} }{ \lvert \vec{r} \lvert ^3}
$$

does not change signs when going over the $\frac{\pi}{2}$ angle in between vectors, so the infinitesimal filament sections $d\vec{l^-}$ (immediately before $d\vec{l}$) and $d\vec{l^+}$ (immediately after $d\vec{l}$) have contributions to $d\vec{V}$ that do not cancel each other out. Therefore, $\vec{V}$ is not just a function of a single filament section, but of the entire filament, hence the integral.

If you check the next section in the source of your image, titled Straight-vortex case you will see this in action.