Dot product of of quaternion-rotated vectors

Question

I'm reading http://people.csail.mit.edu/bkph/articles/Quaternions.pdf and it says "it is easy to show that the operation preserves dot-products." on the page 3. But how is it done? I tried to make a dot product of r' and r' using the formula for r' on that page but I couldn't make it to the conclusion. Could somebody explain how can it be shown?

Answer 1

Using your notation. We know that $q\cdot q=1$ so that $q^*=q^{-1}$ so, using the properties in the previous page, we find: $$ r'\cdot s'=(qrq^{-1})\cdot(qsq^{-1})=(qr)\cdot(qsq^{-1}q)=(qr)\cdot(qs)=(q\cdot q)(r \cdot s)=r \cdot s $$