A high-level reason that $u \cdot (v \times w) = (u \times v) \cdot w$? I can do the algebra to show that for $u, v, w \in \mathbb{R}^3$, this identity is true:
$$u \cdot (v \times w) = (u \times v) \cdot w$$
But is there a more high-level reason? I didn't expect the cross and dot product to be connected in this surprising way.

$u \times v$ can be defined as the vector that makes 
$$\langle w, u \times v \rangle=\det(w,u,v) \quad  \forall w.$$
That there is such a vector can be seen by evaluating the above equality with the canonical basis. That it is unique can be seen in the same way. Now your result follows easily.
