I have a question regarding the legality and generality of the following method for simplifying a vector expression. The specific expression that arose my question is ##\left[\left(\vec{P}\cdot\hat{r}\right)\left(\hat{r}\times\vec{M}\right) + \left(\vec{M}\cdot\hat{r}\right)\left(\vec{P}\times\hat{r}\right)\right]## where ##\hat{r}## is the spherical unit vector pointing away from the origin. This is part of a longer expression that needs to be integrated over a certain volume. I know that it can be simplified using a lengthy manipulation involving multiple usages of triple product formulas but even then, one still needs to assume that ##\vec{M}\times\vec{P}## lies along some known axis (like the z-axis) to do the integral and then bring it back into coordinate free form (if it is relevant or interesting to someone, this method can be found here). What I propose, however, is to suppose that, for instance, ##\vec{M} = M\hat{z}## and ##\vec{P} = P\hat{y}##. Then, plug these into the above expression, get the final answer (since the expression will be integrated over a volume, there will be no dependence on the spherical coordinates) and only then express the result in terms of ##\vec{M}\times\vec{P}##.(adsbygoogle = window.adsbygoogle || []).push({});

Is there any loss of generality in this approach? (It does give the correct answer, I'm worried about its legality and generality).

By the way, if there is any "test" I can perform to test for generality in such case, I would be glad to know about it.

Any comments/corrections/suggestions will be greatly appreciated!

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# Legality and Generality of a Simplifying Method

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