Right way to define vectors under Galilean transformations? This two questions: Vectors under Galilean transformation and Galilean transformations of velocity seem to tackle the issue but one was closed and the latter did not refer to vectors.
To me a vector is some kind of oriented quantity that has preserves a definite magnitude and direction under some specific linear transformations.
A clear example are 4-vectors in special relativity. These objects have the following properties: a magnitude defined as
$$a^\mu a_\mu\tag{1}\label{eq:1}$$
and an orientation (angles) with respect to other vectors
$$b^\mu a_\mu\;, \tag{2}\label{eq:2} $$
where $a^\mu,b^\mu$ are 4-vectors in Minkowski space. Properties \eqref{eq:1} and \eqref{eq:2} are invariants under Lorentz transformations and rotations, so these objects respect the definition of vector.
Now we could do the same analogy in classical mechanics, two 3D vectors $\mathbf A$ and $\mathbf B$ preserve their magnitude
$$|A|^2=\mathbf A\cdot \mathbf A=A^iA_i\tag{3}\label{eq:3}$$
and orientation
$$\mathbf A\cdot \mathbf B=A^iB_i=|A||B|\cos \theta\tag{4}\label{eq:4}$$
under rotations. But when it comes to Galilean boosts $\mathbf r\to \mathbf r'=\mathbf r-\mathbf V t$, for some speed $\mathbf V$ and time $t$. Quantities \eqref{eq:3} and \eqref{eq:4}  are not invariant as the speed of a particle $\mathbf v$ can become $\mathbf v'=0$ in a different frame. The only vector that seems to preserve its absolute value \eqref{eq:3} is the acceleration. So what are vectors in classical mechanics any way?
The question Galilean transformations of velocity seems to indicate that the right vectors that preserve \eqref{eq:3} and \eqref{eq:4} are not any traditional vectors (velocity, position,...), but differences between traditional vectors (example, two positions $\mathbf r_1$ and $\mathbf r_2$ preserve $|\Delta \mathbf r|^2=|\mathbf r_2-\mathbf r_1|^2=|\mathbf r_2'-\mathbf r_1'|^2$). Only relative vectors conserve \eqref{eq:3} and \eqref{eq:4}.
Is this the right way to define vectors under Galilean transformations?
 A: Yes. Absolute positions and velocities are not invariant under Galilean transformations, but the magnitudes of relative positions and velocities are invariant.
By contrast, absolute time is invariant under Galilean transformations because the $t$ coordinate does not change.
A: Rethinking this, I have come to an answer.
In special relativity, spatial translations that does not conserve either (1) or (2) for all $a^\mu$ and $b^\mu$. But (1) or (2) are conserved as long as we are talking about relative positions, velocities and so on.
Special relativity just has this interesting notion that $a^\mu a_\mu$ is conserved under Lorentz's boost while Galilean boosts do not conserve $|\mathbf{A}|^2$, for classical $\mathbf{A}\neq \ddot{\mathbf{r}}$. Nevertheless conditions (1), (2), (3) and (4 are always conserved for relative vectors (under the appropriate transformation).
A better definition for vectors in physics would specify that a vector are those that conserve their relative quantities (like norm and relative angles) under a given transformation.
