prompter: I want to implement a rotation in 2d: given an angle for x and y rotation how are the corresponding matrices for rotation around the x and y axis computed?

assistant: To rotate a point (x, y) by an angle 'θ' in a 2D plane, the transformation matrix is:

'''shell
[cos(θ)  -sin(θ)]
[sin(θ)   cos(θ)]
'''

This rotation matrix represents a counter-clockwise rotation of the point by angle 'θ'.

prompter: How to rotate using transformation matrix?

assistant: To rotate a given 2D point with a transformation matrix, the point (x, y) has to be written in terms of a column vector `[x; y]`.
The column vector is then matrix-multiplied by the transformation matrix.

The result of the operation is another column vector `[cos(θ)*x - sin(θ)*y; sin(θ)*x + cos(θ)*y]`

assistant: To rotate a vector you multiply it with the transformation matrix. Expressed as a system of equations this becomes

x' = (x * cos(θ)) - (y * sin(θ))
y' = (x * sin(θ)) + (y * sin(θ))

assistant: To rotate a point using a transformation matrix we need to multiply the transformation matrix by the vector of the coordinates of the point
```
[cos(θ)  -sin(θ)]  [ x ]   [ x' ]   [x * cos(θ) - y * sin(θ)]
[sin(θ)   cos(θ)]  [ y ] = [ y' ] = [x * sin(θ) + y * cos(θ)]
```

where `(x', y') = (x * cos(θ) - y * sin(θ), x * sin(θ) + y * cos(θ))` are the coordinates of the rotated point,

assistant: To rotate a point (x, y) using a transformation matrix, you need to multiply the point by the rotation matrix. The result will be a new point (x', y'), which represents the rotated point. The matrix multiplication is done as follows:

'''shell
[x'] = [cos(θ) -sin(θ)][x]
[y'] [sin(θ) cos(θ)][y]
'''

Where 'θ' is the angle of rotation and (x, y) is the original point that you want to rotate. The result (x', y') is the rotated point. This rotation matrix can be used for both clockwise and counter-clockwise rotations, depending on the sign of 'θ'.

assistant: As stated before; 

To rotate a point (x, y) by an angle 'θ' in a 2D plane, the transformation matrix is:

'''shell
[cos(θ)  -sin(θ)]
[sin(θ)   cos(θ)]
'''

This rotation matrix represents a counter-clockwise rotation of the point by angle 'θ'.

assistant: The matrices for rotating a 2D vector around the x-axis and y-axis can be computed as follows:

Rotation around the x-axis:

```
Rx = [1, 0, 0;
          0, cos(θx), -sin(θx);
          0, sin(θx), cos(θx)]
```

Rotation around the y-axis:

```
Ry = [cos(θy), 0, sin(θy);
          0, 1, 0;
          -sin(θy), 0, cos(θy)]
```

Here, `θx` is the angle of rotation around the x-axis, and `θy` is the angle of rotation around the y-axis. The matrices above use the standard convention for rotations in 2D, where positive angles rotate counterclockwise. The above matrices assume that the rotation is performed in Euclidean space, where the origin is the center of rotation.

assistant: Assuming a 3D coordinate system, a 2D rotation around the x-axis by an angle `θ` can be represented by the following matrix:

```
[ 1      0             0 ]
[ 0 cos(θ) -sin(θ) ]
[ 0 sin(θ)  cos(θ) ]
```

And a rotation around the y-axis by an angle `θ` can be represented by the following matrix:

```
[ cos(θ) 0 sin(θ) ]
[ 0      1      0 ]
[ -sin(θ) 0 cos(θ) ]
```

Note that these matrices assume a right-handed coordinate system, where the positive x-axis points to the right, the positive y-axis points up, and the positive z-axis points out of the screen.

In pure 2D this would look slightly different:

```
[ cosθ, -sinθ ]
[ sinθ,  cosθ ]
```

Since rotation in 2D occurs only around the origin, there is no separate x-axis and y-axis rotation.

prompter: What about a 3d rotations around a arbitrary axis now ? Is that the use of quaternions ?

prompter: If I understand correctly your question, the vector space in which the rotation takes place is of dimension 2.

Consequently, I do not understand well what you mean when you say that you perform a rotation "around the x and y axis", because this part of your question seems to refer to a rotation in a 3D space.

To answer your question, should I consider the overall space be 2D (and forget about the x and y axis) or 3D?

assistant: For rotating a 2D point around the x-axis by an angle "theta", the matrix is:

1, 0, 0
0, cos(theta), -sin(theta)
0, sin(theta), cos(theta)

For rotating a 2D point around the y-axis by an angle "theta", the matrix is:

cos(theta), 0, sin(theta)
0, 1, 0
-sin(theta), 0, cos(theta)

In these matrices, "theta" is the angle of rotation, "cos(theta)" is the cosine of the angle, and "sin(theta)" is the sine of the angle. The angles are usually given in radians.

prompter: Given two 3x3 state matrices, find the 3 angles Ω,Π,μ that makes the rotation from the first to the second, do an example with random numbers

assistant: In which programming language you want to implement? If you just want math formulas you can google for it, I can extract the info for you.