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Vectors are said to be independent of coordinate system. They remain the same object but its the description of them that changes with the different coordinate system. But velocity, which is a vector, does change with the frame of reference and since each frame of reference has a coordinate system to give a measure of the vectors the velocity vector does change between these different coordinate systems. How is this possible?

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You're confusing different kinds of coordinate transformations. If I rotate the Cartesian axes or translate the origin, I'll preserve velocity but change its components. But a reference frame shift can change the velocity.

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  • $\begingroup$ I have come to understand that the change of basis or the change of coordinate won't change the vector itself just its components are changed in such a way so as to keep the vector itself invariant. Here rotation and translation of coordinate axis u mentioned are, from what I understand, a change of basis. What does a reference frame shift mean?? $\endgroup$ – GRANZER Jun 15 '18 at 8:49
  • $\begingroup$ @GRANZER A reference frame shift is a time-dependent change in spatial coordinates, obtaining a new coordinate system that moves relative to the original. Laws of physics are invariant under these, so two observers in relative motion see the same laws. $\endgroup$ – J.G. Jun 15 '18 at 9:08
  • $\begingroup$ The physical laws remain the same in all inertial reference frames but velocity vectors itself is not a physical law. So do observes in different frame agree on the velocity? $\endgroup$ – GRANZER Jun 15 '18 at 10:36
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    $\begingroup$ @GRANZER That's basically it. In special relativity, these different kinds of coordinate transformation are subsumed into the Lorentz transformations, giving "4-vectors" that are invariant under all of them. $\endgroup$ – J.G. Jun 15 '18 at 11:02
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    $\begingroup$ @GRANZER From the context, that's a discussion of translating the origin, which doesn't change the inertial frame. $\endgroup$ – J.G. Jun 15 '18 at 11:24

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