If I have a state space representation of a physical system $$ x=\begin{pmatrix}x_1 \\ \vdots\\ x_n \end{pmatrix} $$ I may wish to select state variables that correspond to physical quantities that may have different units from each other. For example, I may select the position of a mass and its velocity. In which case, we would use units like meters and meters per second.
In such a case, we could rescale one of the dimensions of the state variable while maintaining the same mapping to a physical system (e.g. we could use m/min instead of m/s by scaling up the corresponding state variable by 60).
Since we can perform this scaling arbitrarily, it appears it would be impossible to define an inner product in this space, since angles between vectors can be arbitrarily changed via a unit scaling. It's almost like everything one knows about vector spaces breaks down when trying to apply typical state space techniques to a physical system with proper units.
Is there any way to deal with this? I can define arbitrary scaling factors to eliminate the units of a system. But those scaling factors are arbitrary, and any two arbitrary scaling factors are not guaranteed to maintain the direction of vectors. Is there a proper way to deal with this? I ideally wish to be able to take dot products!