# $\mathbf{U}\times(\mathbf{V}\times\mathbf{W})=(\mathbf{U}\cdot\mathbf{W})\mathbf{V}-(\mathbf{U}\cdot\mathbf{V})\mathbf{W}$ Quaternion Proof

Prove the identity $$\mathbf{U}\times(\mathbf{V}\times\mathbf{W})=(\mathbf{U}\cdot\mathbf{W})\mathbf{V}-(\mathbf{U}\cdot\mathbf{V})\mathbf{W}$$ given three vectors $\mathbf{U},\mathbf{V}$ and $\mathbf{W}$ by a quaternion calculus.

I am quite unsure of what specifically the above question asks me to do. What is the meaning of quaternion calculus?

Any help or hint would be greatly appreciated. Thank you.

$\textbf{EDIT}$: Basically, my understanding is that we use quaternions to avoid the tedious derivations of composition of rotations which can be computed by applying the Rodrigues formula given an angle $\theta$ and a unit vector $\mathbf{u}$. Following the Hamilton rules we define the Quaternion algebra and then proceed to prove the correspondence between quaternions and rotations.

Basically, I don't understand the wording of the above question; I am not asking for a solution.

• What do you know about quaternion? – user99914 Sep 9 '18 at 23:15
• @JohnMa Not much. I was introduced to the definition of Quaternions and their basic properties in the context of rotations as part of a Dynamics course. – johnny09 Sep 9 '18 at 23:17
• It might be better if you describe more your knowledge about that, in particular those related to rotations, so that others can write an answer that you understand. – user99914 Sep 9 '18 at 23:44
• @JohnMa I have edited my question. Not sure if my EDIT does it better but I appreciate your suggestion. – johnny09 Sep 9 '18 at 23:58
• Hmm, I thought it might fall out of the definition of the Hamilton product, but it's not so obvious. Incidentally, there is also this (unanswered) question. Honestly, I think "calculus" here should probably just be algebra ... unless derivatives of quaternions could somehow be used... – user3658307 Sep 26 '18 at 4:54