Cross product order of operations question

Question

The question I have is:

Is $\overrightarrow{u} \times (\overrightarrow{v} \times \overrightarrow{w})=(\overrightarrow{u} \times \overrightarrow{v}) \times \overrightarrow{w}$? What is the meaning of $\overrightarrow{u} \times \overrightarrow{v} \times \overrightarrow{w}$? Explain. Hint: Try ($\overrightarrow{i} \times \overrightarrow{j}) \times \overrightarrow{k}$.

I don't understand what $\overrightarrow{u}$, $\overrightarrow{v}$, and $\overrightarrow{w}$ are supposed to be? Are they equivalent to the $i,j,k$ unit vectors?

Things I tried:

1. I tried the Hint given above, computing ($\overrightarrow{i} \times \overrightarrow{j}) \times \overrightarrow{k}$. Which I believe gives $0$? Since $\overrightarrow{i} \times \overrightarrow{j} =\overrightarrow{k}$, and $ \overrightarrow{k} \times \overrightarrow{k}=0$?

I'm not sure where else to go on this question, since I cant figure out what what $\overrightarrow{u}$, $\overrightarrow{v}$, and $\overrightarrow{w}$ are supposed to be...

Answer 1

They are supposed to be arbitrary vectors.

To show that cross product is not associative, you need only to find one counter-example, which is what the hint supposed to guide you. But the hint is a wrong one.

Try consider $$\hat{i}\times\hat{i}\times\hat{j}$$ instead.

Answer 2

The objects ${\bf u}, {\bf v}, {\bf w}$ appear to be arbitrary vectors in $\Bbb R^3$.

The hint looks badly misleading to me, since both (as you've computed) the indicated iterated product and the same product carried out in other order (that is, with the parentheses in the other possible location) are $0$. Instead, try comparing $$({\bf i} \times {\bf i}) \times {\bf j} \qquad \textrm{and} \qquad {\bf i} \times ({\bf i} \times {\bf j}) .$$