We all know that if we consider a mono-atomic molecule, it has $3$ translational degrees of freedom only, along the $3$ principal coordinates of the Cartesian coordinate system.
In case of a diatomic molecule, it has $6$ total degrees of freedom, $3$ translational, $2$ rotational and finally $1$ vibrational degree of freedom.
I'm thinking that aren't there infinite directions in which a molecule may move (other than the $3$ principal coordinates) like in the direction of $$(x~\hat i+y~\hat j +z~\hat k)$$ in general?
These directions can also become the directions of translation or the axes of rotation for the molecules. They can also vibrate. So why aren't these taken into account? Why aren't rotations about an arbitrary axis or translation along arbitrary directions taken into account while finding energy?
Or is it that in such cases of translation, the direction, and hence the kinetic energy is already taken component-wise?
Is the axis of rotation also found in component form?
If so, how is the energy of a molecule while rotation about an arbitrary axis determined? I'm not sure if it can be added component-wise.
Any help is appreciated.