Polar vectors and axial vectors

To establish the differential equations describing the motion of bodies in space, we choose, arbitrarily, frames of reference, or coordinate systems. The principle of relativity states that the laws of physics are independent of the choice of these inertial, or Galilean, frames, and in particular of their orientation.

Paradoxically, this is not true of the physical vectors that appear in these laws. Reversing the orientation of space reveals two types of vectors: polar vectors and axial vectors, often improperly called pseudovectors. The cross product, whose definition is intimately tied to the orientation of space, is the main culprit.

Vectors that multiply -------------------------------

The cross product, implicit in the work of Joseph-Louis Lagrange (1736‒1813), was not formally defined until 1878 by William Kingdon Clifford (1845‒1879), who drew on the quaternions of William Rowan Hamilton (1805‒1865) and the work of Hermann Günther Grassmann (1809‒1877). The physicist James Clerk Maxwell (1831‒1879) applied these ideas to physics, and Clifford himself refined his formalism to create vector analysis (see box). These theories spread quickly among physicists but were not accepted by mathematicians until much later, after a thorough reformulation.

Clifford defines the cross product of two vectors $\overrightarrow{u}\backslash overrightarrow\{u\}$ and $\overrightarrow{v}\backslash overrightarrow\{v\}$ as a vector $\overrightarrow{w}=\overrightarrow{u}\wedge \overrightarrow{v}\backslash overrightarrow\{w\}\; =\; \backslash overrightarrow\{u\}\; \backslash wedge\; \backslash overrightarrow\{v\}$ orthogonal to $\overrightarrow{u}\backslash overrightarrow\{u\}$ and $\overrightarrow{v}\backslash overrightarrow\{v\}$ and whose magnitude equals the area of the parallelogram formed by $\overrightarrow{u}\backslash overrightarrow\{u\}$ and $\overrightarrow{v}\backslash overrightarrow\{v\}$. This gives $\mathrm{\parallel }\overrightarrow{u}\wedge \overrightarrow{v}\mathrm{\parallel }=\mathrm{\parallel }\overrightarrow{u}\mathrm{\parallel }\mathrm{\parallel }\overrightarrow{v}\mathrm{\parallel }\sin (\theta )\mathrm{.}\backslash |\; \backslash overrightarrow\{u\}\; \backslash wedge\; \backslash overrightarrow\{v\}\; \backslash |\; =\; \backslash |\; \backslash overrightarrow\{u\}\; \backslash |\; \backslash |\; \backslash overrightarrow\{v\}\; \backslash |\; \backslash sin\; (\backslash theta).$