Thank you for the
replyreplies.
Quote by brydustin
[S,T] = ST  TS (by definition)
Start with [AB,C] = ABC  CAB (+ ACB  ACB )
= ABC  ACB + ACB  CAB
= A(BC  CB) + (AC  CA)B
= A[B,C] + [A,C]B
Therefore we conclude [AB,C] = A[B,C] + [A,C]B

It seems to me that you're being pretty cavalier about vector multiplication, what with the way you're just putting vectors in a row next to each other without any dots or parentheses. For instance, what do you mean when you write “
ABC”, when
A,
B and
C are vector operators?
I would think that you should define [itex][{\bf{\hat S}},{\bf{\hat T}}] = {\bf{\hat S}} \cdot {\bf{\hat T}}  {\bf{\hat T}} \cdot {\bf{\hat S}}[/itex], and therefore start your derivation with
[tex][{\bf{\hat A}} \cdot {\bf{\hat B}},{\bf{\hat C}}] = ({\bf{\hat A}} \cdot {\bf{\hat B}}){\bf{\hat C}}  {\bf{\hat C}}({\bf{\hat A}} \cdot {\bf{\hat B}}).[/tex]
But from there, I'm not sure how you can safely proceed, if you're being rigorous with your dots and parens. For instance — and correct me if I'm wrong on this — but I don't think [itex]({\bf{\hat A}} \cdot {\bf{\hat C}}){\bf{\hat B}}[/itex] is equal to [itex]{\bf{\hat A}}({\bf{\hat C}} \cdot {\bf{\hat B}})[/itex], so your next step seems iffy.