rotation of spherical coordinate system

Question

If I rotate the coordinate system by spherical angles θ and ϕ, and the vector in the new system is v'=(x′,y′,z′), what is its coordinate (x,y,z) in the original coordinate system? Thank you in advance!

Answer 1

It's not immediately clear what is your coordinate system, and how you define rotations. Let's assume that you have first a rotation around $x$ axis by angle $\theta$, then a rotation around $z$ axis by angle $\phi$. You can then write $$\textrm v'=R_z(\phi)R_x(\theta)\textrm v$$ Note the order of rotations in the above equation. You can find the matrix representation of this rotation on wikipedia for example.

Then just multiplying with the inverses, in reverse order, you get $$R_x^{-1}(\theta)R_z^{-1}(\phi)\textrm v'=R_x^{-1}(\theta)R_z^{-1}(\phi)R_z(\phi)R_x(\theta)\textrm v=\textrm v$$

Note that the inverse of the simple rotation matrices can be written as the transpose or rotation by negative angle: $R^{-1}(\alpha)=R^T(\alpha)=R(-\alpha)$