cross products under rotations
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A brief note by Phil dated 10.18.11 with an update from 2.1.17. It sets up Q = B x C in index form, Qa = εabc Bb Cc, and reduces the claim that Q transforms as a vector to the identity εabc Rbb' Rcc' = εa'b'c' Raa', which holds when R is a rotation. It refers to an N-dimensional proof in his Matrix Theorems Addendum and a simpler proof in Appendix A of his Frames document.
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Cross Products under Rotations PhL 10.18.11
(a) Set up the Problem.
Consider
Q = B x C Qa = εabcBbCc
Presumably if B and C transform as vectors, so does Q. How do we show this fact.
Q'a = εabcB'bC'c = εabc(Rbb'Bb') (Rcc'Bc') = (εabc Rbb' Rcc') Bb'Bc'
On the other hand, we expect to have
Q'a = Raa'Qa' = Raa' εa'bcBbCc
Thus, we need to show that
(εabc Rbb' Rcc') Bb'Bc' = Raa' εa'bcBbCc = Raa' εa'b'c'Bb'Cc'
Thus we need to show that
(εabc Rbb' Rcc') = εa'b'c' Raa'
This last equation is true as long as R is a rotation. I have solved this problem in N dimensions in Section 5 of document Matrix Theorems Addendum.doc in the matrix folder! It took a long time.
Update 2.1.17.
I just installed a simpler proof into Appendix A of Frames doc.