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I'm trying to understand why the 3D rotation matrices are non commutative. I've searched for similar questions such as this one here

Why are rotational matrices not commutative?

Which explains very intuitively, that if you rotate around the X axis by 90 degrees, then around the Y axis by 90 degrees you end up in a different orientation than if you were to first rotate 90 degrees along Y then X.

https://i.sstatic.net/6vvFi.png

This makes total sense, until I get to the Euler/Tait-Bryan Angles with "intrinsic" rotation.

If I rotate my pitch 90 degrees, then roll 90 degrees, I end up in the exact same orientation as if I were to roll 90 degrees then pitch 90 degrees.

What this tells me is that the order of rotations no longer matters, so the associated matrices should then be able to commute? I'm pretty sure this is incorrect, but I'm not sure I'm understanding why.

Jackdaw
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Ryan D.
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    some things might commute. This doesn't mean all things do. – Randall Jan 31 '22 at 18:56
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    @Randall For sure, I'm aware the full SO(3) group doesn't commute, but I'm asking if there is a special case where some matrices do always commute. – Ryan D. Jan 31 '22 at 19:11
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    @RyanD. All rotations that fix the same axis will commute since it reduces the problem to a two dimensional rotation within the same plane. – CyclotomicField Jan 31 '22 at 19:23
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    It is not true that rotating pitch 90 degrees then roll 90 degrees leads to the same orientation as if you did these in the opposite order. Note that after rolling 90 degrees, pitching 90 degrees "upward" is a yaw relative to the initial frame. – Ben Grossmann Jan 31 '22 at 19:54

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