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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:

$$\mathbf z = x + y {\large\unicode{x1d7d9}}$$

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.


  • 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 $$d + \phi {\large\unicode{x1d7d9}}$$ produces a motor for which $$d/\phi$$ is the pitch of the screw transformation.

See Also