The cross product of three vectors

Question

Is the cross product of three vectors associative? If not, then how do I determine $A\times B\times C$? Is it a vague statement? What I did was $A\times B$ then $(A\times B)\times C$.

Answer 1

The cross product is not associative; you have to use brackets to disambiguate. Normally one would write $(A\times B)\times C$, but never $A\times B\times C$ unless it is absolutely clear from context (and even then it is frowned upon).

Answer 2

Hint: The double vector product $\space \vec{a} × (\vec{b} × \vec{c}) \space$ results in a vector coplanar to $\space \vec{b} \space$ and $\space \vec{c}, \space$i.e.:

$$\vec{a} × (\vec{b} × \vec{c})=\vec{b}(\vec{a} \vec{c}) - \vec{c}(\vec{a} \vec{b}).$$