What is the alternative representation of SO(2) and its significance?

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SUMMARY

The discussion centers on the alternative representation of the special orthogonal group SO(2), specifically two matrices that represent rotations in a two-dimensional space. The first matrix, \(\begin{pmatrix} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{pmatrix}\), is the conventional representation, effectively transforming vector components during a passive rotation. The alternative matrix, \(\begin{pmatrix} \sin(\theta) & \cos(\theta) \\ -\cos(\theta) & \sin(\theta) \end{pmatrix}\), is also valid as it maintains a determinant of 1 and its transpose equals its inverse. However, the latter representation lacks intuitive geometric significance, particularly in visualizing rotations.

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spaceofwaste
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The usual representation I see of an element of SO(2) is:

[tex]\left( \begin{array}{ c c } cos(\theta) & sin(\theta) \\ -sin(\theta) & cos(\theta) \end{array} \right)[/tex]

and it is easy to show that if you make a passive rotation of a cartesian frame by [tex]\theta[/tex] then this matrix will take the comps of an arbitrary vec to those in the new rotated frame.

However this matrix:

[tex]\left( \begin{array}{ c c } sin(\theta) & cos(\theta) \\ -cos(\theta) & sin(\theta) \end{array} \right)[/tex]

is also a valid representation of SO(2), since it has det=1, and transpose equal to inverse. I have played about with a few drawings but just don't see what this actually represents.
 
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In your second matrix, theta is the angle between the original y-axis and the transformed x axis, with positive theta denoting a clockwise rotation. Alternatively, it is the angle from the transformed x-axis to the original y axis, with positive theta denoting a counterclockwise rotation. Neither interpretation is particularly useful or intuitive, which is why it isn't used.
 

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