I found these thing in an exercise 1.5.6 in the book Differential Geometry of curves and surfaces - Do Carmo.
"Show that the vector product of 2 vectors is invariant under orthogonal transformation with positive determinant. Is the assertion still true if we drop the condition on positive determinant ?"
Denote A is the orthogonal matrix in our orthogonal transformation, I can deduce that $Au\times Av) = A(u\times v)$. Can we call this invariant ? Is there any thing wrong in that exercise ?
I think the orthogonal transformation does not preserve the vector product, because $Au\times Av = \det(A)A(u\times v)$, so when $\det(A) =-1$: $Au\times Av= -A(u\times v)$. So it is never invariant ? Is that true ?
Thanks for your reply :)