TITLE: How do quaternions represent rotations?
QUESTION [10 upvotes]: I wonder how $qvq^{-1}$ gives the rotated vector of $v$.
Is there any easy-to-understand proof for it?
I was on Wikipedia, but I could not understand the proof there
because of the conversions.
Why is $uv-vu$ the same as $2(u \times v)$, and why is $uvu$ the same as $v(uu)-2(uv)u$?

REPLY [6 votes]: A proof is outlined here, although I skipped a few computations you should verify.