Commutation of abstract $O(3)$ generators and vectors [closed]

I've been given the following problem, and I'm quite lost with it.

Let $$L_1$$, $$L_2$$, and $$L_3$$ denote the abstract $$o(3)$$ algebras. You are given that $$\vec{A} = (A_1, A_2, A_3)$$ and $$\vec{B} = (B_1, B_2, B_3)$$ transform as vector operators of $$o(3)$$.

Show that $$[L_j, \vec{A} \cdot \vec{B}] = 0$$

I know that $$L_j = \varepsilon_{jlm} q_l p_m$$, and I can obviously determine the dot product, but I'm not sure where to go from there.

I do, however, know that $$\vec{A} = \frac{1}{Ze^{2}\mu}(\vec{L} \times \vec{p}) + (\frac{1}{r})\vec{r}$$, but I'm not sure how to integrate that into this problem.

• You don't need to know the specific form of $\vec{A}$ or representation of $\vec{L}$, only that $\vec{A},\vec{B}$ transform like vectors. That tells you the commutators $[L_j, A_i]$ and $[L_j, B_i]$, and from that you can compute $[L_j, \vec{A}\cdot\vec{B}]$. Also note the homework tag applies even if this is not assigned coursework - it's close enough. :) Sep 18 '13 at 8:31
• @MichaelBrown I just saw this; I probably shouldn't have written my response given this comment. My apologies. Sep 18 '13 at 8:42
• @joshphysics No worries. The only reason I didn't post the commutators is that I can never remember the sign. :) Sep 18 '13 at 9:03
• This question appears to be off-topic because it is purely a math question with no reference to physics whatsoever. Dec 30 '14 at 16:49

A collection $\{V_1, V_2, V_3\}$ of operators on a vector space $V$ is called an $\mathfrak{o}(3)$ vector operator with respect to a representation $\rho$ of $\mathfrak{o}(3)$ acting on $V$ provided \begin{align} [V_i, L_j] = i\epsilon_{ijk}V_k \end{align} where $L_1, L_2, L_3$ are the representatives of the standard basis on $\mathfrak{o}(3)$. This means, in your case, that merely knowing that $A_i$ and $B_i$ are the components of vector operators tells us that \begin{align} [A_i, L_j]=i\epsilon_{ijk}A_k, \qquad [B_i, L_j]=i\epsilon_{ijk}B_k \end{align} These commutation relations alone are sufficient to demonstrate the desired result using commutator identities.
• I know this is probably really stretching my luck, but how do I go about applying those commutation relations?? Should I be using $[A, BC] = [A, B]C + B[A, C]$??