Answer
$({\bf i}\times{\bf j})\times{\bf j}={\bf -i}$
${\bf i}\times({\bf j}\times{\bf j})={\bf 0}$
The cross product is not associative.
Work Step by Step
${\bf i}\times{\bf j}=\left|\begin{array}{lll}
{\bf i} & {\bf j} & {\bf k}\\
1 & 0 & 0\\
0 & 1 & 0
\end{array}\right|=(0-0){\bf i}-(0-0){\bf j}+(1-k){\bf k}$
$={\bf k}$
$({\bf i}\times{\bf j})\times{\bf j}={\bf k}\times{\bf j}=\left|\begin{array}{lll}
{\bf i} & {\bf j} & {\bf k}\\
0 & 0 & 1\\
0 & 1 & 0
\end{array}\right|$
$=(0-1){\bf i}-(0-0){\bf j}+(0-1){\bf k}$
$=-{\bf i}$
${\bf(j\times j)}={\bf 0}$, and
${\bf i}\times({\bf j}\times{\bf j})={\bf i}\times{\bf 0}={\bf 0}$
The cross product is not associative.
If it were associative, then it would be that
$({\bf u}\times{\bf v}) \times{\bf w}={\bf u}\times({\bf v}\times{\bf w})$
and, in this case, this is not so.
