I am trying to show that the cross product is not associative, using the vectors $\mathbf{A} = (5, 3, 1), \mathbf{B} = (1, 2, 3), \mathbf{C} = (4, 2, 1)$.
Using the determinants method, I get
$$\mathbf{A} \times \mathbf{B} = [(3)(3) - (1)(2)]\mathbf{\hat{i}} - [(5)(3) - (1)(1)]\mathbf{\hat{j}} + [(5)(2) - (3)(1)]\mathbf{\hat{k}} = (7, 14, 7)$$
$$(\mathbf{A} \times \mathbf{B}) \times \mathbf{C} = [(14)(1) - (7)(2)]\mathbf{\hat{i}} - [(7)(1) - (7)(4)]\mathbf{\hat{j}} + [(7)(2) - (14)(4)]\mathbf{\hat{k}} = (0, 21, -42)$$
According to this document, $(\mathbf{A} \times \mathbf{B}) \times \mathbf{C}$ should be $(−28, 21, 70)$. I would greatly appreciate it if someone would please take the time to explain what I did incorrectly.