Shifting magnetic field axis

Question

Considering the image below, I have magnetometer readings while my magnetometer is oriented along x' and y'. Can I convert these readings to get the equivalent reading if my magnetometer was oriented along x and y.

Answer 1

You have a $\mathbf{B}$ field, which we will write as $$\left[\begin{matrix}B\cos(\alpha)\\B\sin(\alpha)\end{matrix}\right],$$ where $B=\|\mathbf{B}\|,$ the magnitude of $\mathbf{B}.$ Here $\alpha$ is measured counterclockwise from the positive $x'$ axis. Now the $x,y$ axis is rotated from the $x',y'$ axis through a clockwise angle of $\theta,$ which is the same as if we had rotated the vector $\mathbf{B}$ counterclockwise through the same angle. You can go through the exercise of multiplying the vector above by the rotation matrix $$\left[\begin{matrix}\cos(\theta) &-\sin(\theta)\\\sin(\theta)&\cos(\theta)\end{matrix}\right],$$ but to beat to the punch, you will find that you can simplify the answer as $$\left[\begin{matrix}B\cos(\alpha+\theta)\\B\sin(\alpha+\theta)\end{matrix}\right].$$