Magnitude and File:GeometricProduct201.svg: Difference between pages

@@ Line 1: / Line 1: @@
-A ''magnitude'' is a quantity that represents a concrete distance of some kind. In rigid geometric algebra, a magnitude $$\mathbf z$$ is composed of two components, a scalar and an antiscalar, as follows:
-:$$\mathbf z = x\mathbf 1 + y {\large\unicode{x1d7d9}}$$
-Magnitudes are homogeneous just like everything else in a projective geometric algebra. This means it has both a [[bulk]] and a [[weight]], and it is [[unitized]] by making the magnitude of its weight one.
-=== Examples ===
-* The [[geometric norm]] produces a magnitude that gives the perpendicular distance between an object and the origin. This is also half the distance that the origin is moved by an object used as an operator.
-* [[Euclidean distances]] between objects are expressed as magnitudes given by the sum of the [[bulk norms]] and [[weight norms]] of [[commutators]].
-* Exponentiating the magnitude $$d\mathbf 1 + \phi {\large\unicode{x1d7d9}}$$ produces a [[motor]] for which $$d/\phi$$ is the pitch of the screw transformation.
-== See Also ==
-* [[Geometric norm]]
-* [[Unitization]]

Revision as of 00:43, 26 August 2023