Coordinate systems/Derivation of formulas/Cartesian from spherical unit vectors

From Wolfram Mathworld[1], we have the following relations for the unit vectors in a Spherical coordinate system:

We first write this as a matrix equation:

We can then find the inverse of the matrix to make the subjects.

Which was what we set out to show.

  1. http://mathworld.wolfram.com/SphericalCoordinates.html