I am reading a book on the math behind game engine. At the end of the chapter that talks about quaternions they are some exercises. I need to prove that $||q_1q_2||^2 = ||q_1||^2||q_2||^2$
It should be farely simple but I'm stuck on probably the dumbest thing. How I proceed :
Let $q_1 = v_1 + s_1$ and $q_2 = v_2 + s_2$. V is a 3d vector and s a scalar.
Then $||q_1q_2||^2 = (v_1\times v_2 +s_1v_2+s_2v_1)^2+(s_1s_2-v_1\cdot v_2)^2$
I can already see the LaGrange identity simplifying the annoying part but I apparently clearly don't understand how the distribution works in this case.
For example if I take the last part I would like to expand it as $(s_1s_2)^2+(v_1\cdot v_2)^2-2 (s_1s_2(v_1\cdot v_2))$ but the book expands at as $(s_1s_2)^2+(v_1\cdot v_2)^2$.
Of course how I distribute the terms makes me unable to give the proper answer but I don't understand why it doesn't work how I think. I am clearly missing some knowledge or I am very tired but I would need help to understand why I can't do what I want here.
Screenshot of the official answer :
