I'm interested in coordinate free (non-matrix based) approaches to geometry. What I'd like to do is to show that every Galilean transformation can be written uniquely as the composition of a rotation, a translation, and a uniform motion.
Here's where I'm stuck. It's easy enough to write down formulas for translation and uniform motion: $$T(t,\mathbf p) = (t + s,\mathbf p + \vec{\mathbf a}) \\ UM(t,\mathbf p) = (t,\mathbf p + t\vec{\mathbf v})$$ but I can't think of a way to describe a given rotation except with matrices.
Question: What's a coordinate-free way (ideally a formula like those above) of specifying a given rotation in $\Bbb R \times \Bbb E^3$?
$\newcommand{\Pt}{\mathbf{p}}$Additon and (real) scalar multiplication don't effect non-trivial rotation, so you can't express a rotation in a form completely analogous to your formulas for translation and uniform motion.
One approach is to introduce a quarter-turn rotation $J$ (which is "secretely" complex multiplication by $i$). Counterclockwise rotation by $\theta$ about the point $\Pt_{0}$ can then be expressed as $$ R_{\Pt_{0}}(\theta, \Pt) = \Pt_{0} + (\cos\theta)(\Pt - \Pt_{0}) + (\sin\theta) J(\Pt - \Pt_{0}). $$