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In physics, if properties have the same units, they can be added, although this isn't always physically descriptive, it can be done. I understand this is because the same units have the same span. More often than not, in math, units are withheld for generalization of values that have units, (coordinate space can be length or angular).

So what happens if I evaluate $$ \vec{i}\times\vec{j} - \vec{k}$$ Without a second thought we get $\vec{0}$, but if we ask a different but similar question where it's obvious one of the vectors represents rotation in a way the other does not $$\vec{\nabla}\times\vec{F}(\vec{r}) + \vec{F}(\vec{r})$$ I don't expect to be able to say anything about the answer, but what does it mean that these values can be added? We know the curl of F represents the rotation of F, but without units, it seems that nothing stops one from adding these quantities, so what should they mean?

Orion Yeung
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    Note that there's a very subtle implicit conversion going on here, with a very important consequence - the expressions you've written aren't coordinate-invariant. In particular, cross products transform differently under transformations of the coordinate system than vectors do, so while you can write expressions like the ones you've written, you can't write them in a fashion that isn't implicitly tied to your coordinates. – Steven Stadnicki Dec 12 '16 at 20:14
  • (I wrote a little bit about this at http://math.stackexchange.com/questions/62318/origin-of-the-dot-and-cross-product/62370#62370 , though that doesn't really talk as much about the behavior under transformations as it could.) – Steven Stadnicki Dec 12 '16 at 20:16
  • Thanks a bunch! It might not have been extensive on transformations, it did clear up that I can just add vectors and cross products willy nilly just because they share notation and it's convenient to call them vectors. – Orion Yeung Dec 12 '16 at 20:30

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