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a derivation of the lorentz transformation for an arbitrary direction of the relative velocity often makes use of writing the spatial position vector of an event as the sum of its component parallel and perpendicular to the velocity vector in one inertial frame and then transforming both components in the following way:

[itex]\vec x'_{⊥}= \vec x_{⊥}[/itex]

[itex]\vec x'_{\parallel}= γ(\vec x_{\parallel}-\vec v t)[/itex]

Now, I understand that the magnitude of the perpendicular component stays the same in both frames and I also know both inertial frames have to agree on the magnitude of the relative velocity. Unfortunately I can't think of arguments why the unit vector of the velocity vector and the perpendicular component should be identical in the two frames, that is why it can't be the case that some kind of rotation of the spatial axis gets involved when performing a general lorentz transformation. But in order to justify the above transformations I need to be confident that the direction and orientation of those vectors are indeed the same.

This is the only part in the derivation that troubles me and I would appreciate any help. Hopefully I stated my problem clear enough so that maybe someone can come up with an argument adressing these issues.

Thanks in advance!

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# General Lorentz transformation, spatial vector components

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