Context
I am studying spherical multipole radiation. I am seeking identities pertaining to the same.
Given the properties and orthogonality relation of the vector spherical harmonics [1], what do the following integrals evaluate to?
\begin{align} \int\mathbf{Y}_{\ell m}\times \mathbf{Y}^*_{\ell'm'}\,d\Omega . \\ \int\mathbf{\Psi}_{\ell m}\times \mathbf{\Psi}^*_{\ell'm'}\,d\Omega. \\ \int\mathbf{\Phi}_{\ell m}\times \mathbf{\Phi}^*_{\ell'm'}\,d\Omega . \\ \int\mathbf{Y}_{\ell m}\times \mathbf{\Psi}^*_{\ell'm'}\,d\Omega . \\ \int\mathbf{Y}_{\ell m}\times \mathbf{\Phi}^*_{\ell'm'}\,d\Omega . \\ \int\mathbf{\Psi}_{\ell m}\times \mathbf{\Phi}^*_{\ell'm'}\,d\Omega . \end{align}
Bibliography
[1] https://en.wikipedia.org/wiki/Vector_spherical_harmonics