<s>[INST] Finding the conditions for $\mathbf{A}×(\mathbf{B}×\mathbf{C}) = (\mathbf{A}×\mathbf{B})×\mathbf{C}$. I wanted to find the conditions in which $\mathbf{A}×(\mathbf{B}×\mathbf{C})$ is equal to $(\mathbf{A}×\mathbf{B})×\mathbf{C}$.
On solving $\mathbf{A}×(\mathbf{B}×\mathbf{C})-(\mathbf{A}×\mathbf{B})×\mathbf{C}=\mathbf{0}$, I got
$$
\mathbf{A}(\mathbf{B}\cdot\mathbf{C})- \mathbf{C}(\mathbf{A}\cdot\mathbf{B})=\mathbf{0}\tag{1}
$$
In the solution manual, it's given that this is possible if and only if either $\mathbf{A}$ is parallel to $\mathbf{C}$ or $\mathbf{B}$ is perpendicular to $\mathbf{A}$ and $\mathbf{C}$.
But as per what I have learnt, this means that $(1)$ will be $\mathbf{0}$ only when both terms are in it are $\mathbf{0}$. But here why don't we consider the equality of those two terms?
I am beginning vector analysis, So it will be really helpful if someone could help me out.
 [/INST] The Jacobi identity is
$$
\mathbf{A}\times(\mathbf{B}\times\mathbf{C})+
\mathbf{B}\times(\mathbf{C}\times\mathbf{A})+
\mathbf{C}\times(\mathbf{A}\times\mathbf{B})=\mathbf{0}
$$
Using anticommutativity, we get
$$
\mathbf{A}\times(\mathbf{B}\times\mathbf{C})-
(\mathbf{A}\times\mathbf{B})\times\mathbf{C}=
-\mathbf{B}\times(\mathbf{C}\times\mathbf{A})
$$
If $\mathbf{A}$ is parallel to $\mathbf{C}$, then $\mathbf{C}\times\mathbf{A}=\mathbf{0}$. If $\mathbf{B}$ is perpendicular to both $\mathbf{A}$ and $\mathbf{C}$, then it is parallel to $\mathbf{C}\times\mathbf{A}$ and therefore $\mathbf{B}\times(\mathbf{C}\times\mathbf{A})=\mathbf{0}$.
What about the converse? Suppose the two triple products are equal and that $\mathbf{A}$ is not parallel to $\mathbf{C}$. Then, in order for $\mathbf{B}\times(\mathbf{C}\times\mathbf{A})=\mathbf{0}$ we need that $\mathbf{B}$ is parallel to $\mathbf{C}\times\mathbf{A}$.
</s>