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I'm trying to find a general rule that expresses the product of two rotation matrices as a new matrix.

I'm adopting the topological model of the rotation group, so any rotation which is specified by an angle [itex]\phi[/itex] and an axis [itex]\hat{n}[/itex] is written [itex]R(\hat{n}\phi)= R(\vec{\phi})[/itex].

So, given two rotation matrices [itex]R(\vec{\phi}_1) , R(\vec{\phi}_2) \in SO(3)[/itex], what can be said about the product [itex]R(\vec{\phi}_1)R(\vec{\phi}_2)[/itex]?

What is are the axis and the amount of degrees that specifies this rotation matrix?