I have a question that follows like this:
Let $U = (0,1,2)$ and $V = (1,-1,-2)$. Assume that $V \times W = (3,1,1)$
Is there enough information provided in order to determine $(U \times V) \times W$ and $W \times (V \times W)$?
According to the answer sheet, there is no enough information to determine the mentioned cross products according to the question. But I do not know how to explain this algebraically nor geometrically.
Here is an algebraic way to see this:
Setting $\vec{w}=\langle w_1,w_2,w_3\rangle$, $\vec{v}\times\vec{w}=\langle2w_2-w_3,-2w_1-w_3,w_1+w_2\rangle=\langle3,1,1\rangle$, so
solving the system $2w_2-w_3=3, \;\;2w_1+w_3=-1,\;\;w_1+w_2=1$ gives
$\;\;\;\vec{w}=\langle\frac{1}{2}-t,\; \frac{3}{2}+t,\; 2t\rangle$.
In particular, taking $t=\frac{1}{2}\;$ gives $\vec{w}=\langle -1,2,1\rangle$, and taking $t=-\frac{1}{2}\;$ gives $\vec{w}=\langle0,1,-1\rangle$.
Now it is easy to show that these values of $\vec{w}$ give different values for
$(\vec{u}\times\vec{v})\times\vec{w}=\langle0,2,-1\rangle\times\vec{w}$, and for $\vec{w}\times(\vec{v}\times\vec{w})=\vec{w}\times\langle3,1,1\rangle$.