$\mathbf{U}\times(\mathbf{V}\times\mathbf{W})=(\mathbf{U}\cdot\mathbf{W})\mathbf{V}-(\mathbf{U}\cdot\mathbf{V})\mathbf{W}$ 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.
 A: I wrote a derivation on the page (mentioned by user3658307) Is there a way to prove vector triple product from quaternion multiplication? .
As to your question: "What is the meaning of the quaternion calculus?"
First, the word "calculus" here does not refer to any infinitesimal calculus, as one learns in a calculus course.
It means a way of calculating. (That is also the origin of the more common use of the word "calculus").
The quaternions are a 4-dimensional algebra over the reals, basically real 4-vectors together with multiplication operation, so that the product of two quaternions is again a quaternion.  It is a 4-dimensional extension of the complex numbers.  (There is no 3-dimensional extension.)
Multiplication by quaternions describe orthogonality-preserving transformations in 4-space. This makes them the best way of describing certain important physical processes.
Multiplication by a quaternion can be regarded as a change of orientation in 3-space: imagine a pilot of an aircraft, which can perform a maneuver involving rotation in both pitch and yaw axes.
Such a maneuver changes the pilot's orientation.  A change of orientation is more than a simple rotation -- and it isn't very well represented even by two simple rotations.
Any orthogonality-preserving transformation of geometrical 4-space (that is, any element of O(4) ) can be written as one quaternion multiplication on the left, and another multiplication on the right.
A specific use of quaternions is to describe solid rotations in 3-space.  (Any rotation of 3-space is just a multiplication on the left by a quaternion, and on the right by the inverse of the same quaternion.)
In these applications, it is true that the quaternion calculations take much simpler form than the equivalents involving multiple three-dimensional calculations, but more importantly, the three-dimensional analogs introduce nasty singularities, which the more natural quaternion calculations do not have.  Those singlularites have bad practical consequences.  Search for "gimbal lock".
The quaternions are complicated, because orthogonal transformations of four dimensions are complicated.  But they provide a faithful reflection of those 4-space transformations --- that is, they are no more and no less complicated than orthogonal transformations of 4-space.
