How do I rotate a linear function around the z-axis?

[Niles]

Homework Statement

I have two coordinate system (x, y), (x', y') that differ by a rotation around the z-axis by an angle \alpha. In the coordinate system (x', y') I have a function f(x', y') = C, where C is a constant.

I would like to express f in the coordinate system (x,y), where it is a linear function x\nabla +y_0. The gradient \nabla of this function is 1/\tan(\alpha).

The Attempt at a Solution

I need to find the shift in x now. I get that this is C/\cos(\alpha). Is there a way for me to test that this function indeed is constant in the coordinate system (x', y')?

 

[SteamKing]

Draw a picture of f(x',y') = C in your x'-y' coordinate system. Now imagine that this coordinate system is rotated about the origin by an angle alpha. What happens to the line f(x',y') = C? Don't you need more than one parameter to express this line in the (x,y) system?

 

[HallsofIvy]

Rotation about the z-axis through an angle \alpha is given by the matrix
\begin{bmatrix}cos(\alpha) & -sin(\alpha) & 0 \\ sin(\alpha) & cos(\alpha) & 0 \\ 0 & 0 & 1\end{bmatrix}