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Consider vectors $\overrightarrow { A } $ and $\overrightarrow { B } $ as operators or vector of operators. If this commutation holds$$[\overrightarrow { A },\overrightarrow { B }]=0$$ Then, is that right to claim that commutation of each components are also zero, or it's necessary to be mentioned that the sum of the components commutations are zero. I mean each commutation holds this $$[ A_{i} , B_{j}]=0$$ or the summation holds it. $$\sum _{ i }\sum _{ j }{[ A_{i} , B_{j}]}=0$$

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    $\begingroup$ That completely depends on what you mean by $[\vec A,\vec B] = 0$. In fact, you might define it to hold iff one of the things you write hold. What's your definition for $[\vec A,\vec B] = 0$? $\endgroup$ – ACuriousMind Jan 30 '16 at 23:27

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