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I have two 3D rotation matrices, A and B. The columns of these rotation matrices can of course be interpreted as representing the x-, y- an z-bases of two new coordinate frames. What I want to know is, how can I define the z-basis of coordinate frame B in spherical coordinates defined in coordinate frame A.

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The notation $R^P_Q$ denotes the rotation matrix whose columns represent the coordinate axes of the frame $Q$ expressed in the coordinate frame $P$. If I understand your statements correctly, you have $R^E_A$ and $R^E_B$, where $E$ is some reference coordinate frame, and you seek $R^A_B$. The z-axis of the frame $B$ expressed in frame $A$ is simply the third column of this matrix. $$ R^A_B = R^A_E R^E_B = (R^E_A)^{-1}R^E_B $$

Since you already have the right hand side of this equation, $R^A_B$ is readily obtained and you can extract its third column which I denote by $(x,y,z)$. Now, you can simply convert it into spherical coordinates using the relevant formulas for coordinate transformations. See this for instance.

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