Proving invariance of scalar product

Gabor
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Hi everyone,

How would I go about proving that the scalar product of two four-vectors (A,B) is invariant under a Lorentz transformation?
 
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As a warmup, you might try to prove that the scalar product of two vectors in the plane is invariant under a rotation.
 
Okay... I could do that for 2 vectors (x1, x2) and (y1, y2) in a plane.

As for the four-vector proof, I'm not even sure I'm doing it right... My understanding is that I have to take the scalar product of the two vectors A and B. Then I have to apply Lorentz transform to both vectors and calculate the scalar product of A' and B'. For invariance, these two scalar products should be equal?
 
yes.

How did you do the problem for the dot product of vectors in the plane?
 
I figured out the proof for the four-vectors. Now I see the similarity between that and the rotation proof. Turns out I was using the wrong transformation formulas for my vectors and that's why things didn't add up. Thanks for your help!
 

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