My professor claims, that
$\vec{r} \times \vec{\nabla} = \vec{e}_{\varphi} \frac{\partial}{\partial \vartheta} - \vec{e}_{\vartheta} \frac{1}{r\sin \vartheta} \frac{\partial}{\partial \varphi}$
in which $\vec{r}$ is an arbitrary position vector and $\vec{\nabla}$ is the nabla operator in spherical coordinates, which is
$ \vec{\nabla} = \vec{e}_r \frac{\partial}{\partial r} + \vec{e}_{\vartheta} \frac{1}{r} \frac{\partial}{\partial \vartheta} + \vec{e}_{\varphi} \frac{1}{r \sin \vartheta} \frac{\partial}{\partial \varphi} $ .
He used the first claim to derivate the operator $ \left( \vec{r} \times \vec{\nabla} \right)^2$ of the eigenvalue equation of the spherical harmonic function $Y_{lm}(\vartheta,\varphi)$ ( $\left( \vec{r} \times \vec{\nabla} \right)^2 Y_{lm}(\vartheta,\varphi) = -l(l+1)Y_{lm}(\vartheta,\varphi)$ ).
I couldn't find the proof of his first claim and I also tried to find the answer of my own. It seems like I forgot something, but I have no clue what. Perhaps someone in here can help. Thx :-)