- #1
spaceofwaste
- 3
- 0
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.
[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.