I may deviate from the main subject of this thread but wanted to ask, can spherical waves be build from fourier integrals of plane waves? My main intuitive difficulty in understanding that is that in spherical waves the wave vector is not constant but changes from point to point (I know it always points in the radial direction though). In contrast all plane waves have constant wave vectors, so I believe any fourier integral of plane waves will have a constant wave vector as well.

Yes, you can! You can expand any em. wave in terms of vector spherical harmonics (aka multipole expansion; equivalent to the energy-angular momentum single-photon states) as well as plane waves (equivalent to the energy-momentum single-photon states in QED).

As the name suggests the scalar product maps two vectors to a scalar, which is independent of any basis. Also vectors are independent of the choice of bases you might use to decompose them into components. In a Euclidean vector space the fact that two vectors are prependicular to each other, i.e., their scalar product vanishes, is independent of the choice of basis.

Well, Euclidean geometry can be defined by the axioms of a Euclidean affine manifold. That's way more convenient for the purposes of physics than the original way Euclid defined it in ancient times. Of course, it's still the same mathematical thing.

Well, I was wrong about dot product and the perpendicularity staff. Thanks everybody for your help.
Is the quote: "However, in inhomogenous, non-linear, or isotropic media, the E and B fields may not be perpendicular, e.g. in a crystal (which is isotropic)." right? Apart from the "isotropic" (that should be "anisotropic").