Let \(\mathcal R=(\mathbb R,V_{\mathbb R},v)\) be the \(3\)-dimensional affine space with the associated normed vector space \((V_{\mathbb R},||~||)\) over the field of real numbers \(\mathbb R\). Any non-empty affine bounded subset \(\mathcal I\subset\mathcal R\) is also called a frame of reference.
Please note that any frame of reference \(\mathcal I\) has a norm \(||~||\) induced by the associated vector space, i.e. we can measure the distance between any two points \(P,Q\in\mathcal I\) by measuring the norm of the corresponding vector \[||\overrightarrow{PQ}||\ge 0\in\mathbb R.\]
Please also note that the boundedness of \(\mathcal I\) is chosen in the definition for technical reasons to avoid the necessity to deal with infinite (unbounded) frames of reference to model the physics of real world.
Frames of reference can be:
Definitions: 1