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I was astonished by ingenuity of many users who demonstrated reasons for why rotational matrices are not commutative.

However in 3d rotations I'm more puzzled by some other theorem ...

How intuitively to show that

composition of rotations about fixed axes of global frame is equal to composition of the same rotations about their current X,Y,Z axes but made in reverse order.

In the previous question the most interesting to me was example with permutations... Maybe someone also knows such nice examples..

Widawensen
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We'll use three properties of rotations - they are isometries, conformal, and form a group under composition. Letting this group act on the canonical basis vectors we see that it maps them onto other unit vectors being isometries, and that the vectors remain orthogonal, because the map is conformal and so the image is also an orthonormal and linearly independent set of vectors defining another frame of reference. All that is left is to note that in any group $(xyz)^{-1}=z^{-1}y^{-1}x^{-1}$ and we are done.

CyclotomicField
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  • Interesting argumentation (+1) from the abstract algebra point of view, and elements of it can be inspiring but somehow I don't see (up to now) the whole picture. Rotations indeed transform ortonormal set ot vectors into other ortonormal sets and it can be somehow used and I wonder how many such sets we need to illustrate the thesis, 3 is minimum, but probably 4 would be more appropriate. I should add that after 3 years I did not find convincing illustration for this phenomenon, although I was searching. – Widawensen Jul 29 '20 at 06:52