Orthomagnetic materials

 $$Z=e^{0\beta\mu_B B}+e^{\beta\mu_B B}+e^{-\beta\mu_B B}=1+2\cosh(\beta\mu_B B)$$

Periodic magnetization:

$$M=i\frac{N\mu_B}{V}\tan(\beta\mu_B B)$$

How to describe two vectors orthogonal whose magnitudes are proportional?

$$\textbf{v}\cdot\textbf{w}=vw e^{i\theta}$$