Let events in the history of some particle be described by their phase angle $\theta$. And recall that the vectors $\hat{m} \equiv (1, 0, 0)$, $\hat{e} \equiv (0, 1, 0)$ and $\hat{z} \equiv (0, 0, 1)$ mark the magnetic, electric and polar axes. These algebraic entities can be used to construct another set of vectors. Definition: the axis of the **abscissa** is composed from all scalar multiples of

$\hat{x} \equiv \cos{\! 2\theta} \, \hat{m} + \sin{\! 2\theta} \, \hat{e}$

And similarily the **ordinate** axis is defined by

$\hat{y} \equiv - \sin{\! 2\theta} \, \hat{m} + \cos{\! 2\theta} \, \hat{e}$

These new vectors together with $\hat{z}$ are called a**Cartesian**basis after the work of René Descartes. Cartesian axes can be visualized by rotating the electric and magnetic axes by $2 \theta$ degrees around the polar axis. The factor of two means that $\hat{x}$ and $\hat{y}$ make two complete turns as $\theta$ goes through each cycle, one turn for quarks of each phase. These definitions can be rearranged to give $\hat{m} = \cos{\! 2\theta} \, \hat{x} - \sin{\! 2\theta} \, \hat{y}$ and $\hat{e} = \sin{\! 2\theta} \, \hat{x} + \cos{\! 2\theta} \, \hat{y}$.

Related WikiMechanics articles.