I What is the Role of Direct Calculation in Planar Rotation Commutativity?

[IN SU CHUNG]

Commutativity for planar rotations follows from a direct calculation.
What does 'direct calculation' mean?

 

[member 587159]

Take two rotations $a,b$ and show that $ab=ba$.

In the usual setting, this will be showing that two linear maps commute or that two rotation matrices commute.

 

[RPinPA]

I think it means actually calculating $ab$ and $ba$, working out the products of the rotation matrices for general rotations.

 

[StoneTemplePython]

A nice way to do this, is to recognize that you are in effect verifying that complex numbers commute (and that this holds even when they are represented as 2x2 matrices). So consider a complex number

$a_1 + b_1 i$, given as

$\begin{bmatrix} a_1 & -b_1\\ b_1 & a_1 \end{bmatrix}= a_1 \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} + b_1 \begin{bmatrix} 0 & -1\\ 1 & 0 \end{bmatrix} = a_1 \mathbf I + b_1 \mathbf i$

now multiply by some other complex number $a_2 + b_2 i$ and see that

$\big(a_1 \mathbf I + b_1 \mathbf i\big)\big(a_2 \mathbf I + b_2 \mathbf i\big) = \big(a_2 \mathbf I + b_2 \mathbf i\big) \big(a_1 \mathbf I + b_1 \mathbf i\big)$

because $\mathbf i$ commutes with scaled forms of itself and the identity matrix $\mathbf I$ commutes with everything. Since you are talking about rotation matrices, you are constraining yourself to a determinant of 1 here (aka complex numbers on the unit circle).
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edit: cleaned up some table formatting issues based on below hint

 

[jim mcnamara]

@StoneTemplePython
This was created by PF5, it "thought" you were making a table, so it wrapped TABLE HTML tags around the area.
New feature. If you get them and do not want them, toogle into bbcode (gear-like icon on the toolbar, far right).
Remove the two tags - most HTML tags have start and end like this [STARTME] ...blah blah [/STARTME].

You can do this on your next post, I think the one post above is old enough to have locked you out of edit. If you want I can clean them up, PM me.

 

[WWGD]

Notice commutativity applies only in one dimension, meaning on the circle. Once you go into higher dimensions things become more complicated , where you have to define your axis of rotation.