I'm solving the following problem from the book "Analytical Mechanics for Relativity and Quantum Mechanics" by Oliver Davis Johns:
Let $R(t)=R_a(t)R_b(t)$ be a rotation matrix. Show that the angular velocity associated to the product is \begin{gather*} \vec{w}=\vec{w_a}+R_a(t)\vec{w_b}(t) \end{gather*}
So, my attempt, which seems correct, is:
\begin{gather*}
\vec{x}(t)=R(t)\vec{x}(0)\tag{1}
\\
R(t)=R_a(t)R_b(t)\tag{2}
\\
\dot{R}(t)=\dot{R_a}(t)R_b(t)+R_a(t)\dot{R_b}(t)\tag{3}
\\
\dot{\vec{x}}(t)=\dot{R}(t)R^{t}(t)\vec{x}(t)\tag{4}
\end{gather*}
Using some matrices properties, we have
\begin{gather*}
\dot{R}(t)R^{t}(t)=(\dot{R_a}(t)R_b(t)+R_a(t)\dot{R}_b(t))(R_b^{t}(t)R_a^{t}(t)) \tag{5}
\\
=\dot{R_a}(t)IR_a(t)+R_a(t)\dot{R}_b(t)R_b(t)R_a^{t}(t)\tag{6}
\\
=\dot{R_a}(t)R_a(t)+T(t)
\\
where
\\
T(t)=R_a(t)\dot{R}_b(t)R_b(t)R_a^{t}(t)
\end{gather*}
On equation 6, how do I manipulate the term $T(t)$ so when mulplitying it by $x(t)$ I get $\vec{w_b}(t)$?
