Spatial Axes
Cartesian axes compared to the electric and magnetic axes.
Cartesian axes compared to the electric and magnetic axes.

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 DescartesXlink.png. 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 $\hat{e} = \sin{\! 2\theta} \, \hat{x} + \cos{\! 2\theta} \, \hat{y}$.
//Echinus,// Jean Baptiste Lamarck. Tableau Encyclopédique et Méthodique des Trois Regnes de la Nature, Paris 1791-1798. Photograph by D Dunlop.
Echinus, Jean Baptiste Lamarck. Tableau Encyclopédique et Méthodique des Trois Regnes de la Nature, Paris 1791-1798. Photograph by D Dunlop.
Right.png Next step: spatial orientation.
Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-ShareAlike 3.0 License