Lorentz Transformation with an Arbitrary Direction of Motion

we assumed that the S’frame
moves along the xdirection
relative to the Sframe.
Now suppose that’s not the case, and that the S’frame
moves with an arbitrary (but constant) velocity v relative to S. Find the Lorentz Transformation formula between the two frames, i.e., for an event (r, t) in the Sframe,
what is the corresponding expression for the same event (r’, t’) in the S’frame?


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