The Vector Algebra War: A Historical Perspective

@article{Chappell2015TheVA,
  title={The Vector Algebra War: A Historical Perspective},
  author={James M. Chappell and Azhar Iqbal and John Gideon Hartnett and Derek Abbott},
  journal={IEEE Access},
  year={2015},
  volume={4},
  pages={1997-2004}
}
There are a wide variety of different vector formalisms currently utilized in engineering and physics. For example, Gibbs' three-vectors, Minkowski four-vectors, complex spinors in quantum mechanics, and quaternions used to describe rigid body rotations and vectors defined in Clifford geometric algebra. With such a range of vector formalisms in use, it thus appears that there is as yet no general agreement on a vector formalism suitable for science as a whole. This is surprising, in that, one… 

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References

SHOWING 1-10 OF 49 REFERENCES

Functions of Multivector Variables

A number of elementary functions extended to act over the skew field of Clifford multivectors, in both two and three dimensions are detailed, finding one key relationship that a complex number raised to a vector power produces a quaternion thus combining these systems within a single equation.

Quaternion and Clifford Fourier Transforms and Wavelets

Quaternion and Clifford Fourier and wavelet transformations generalize the classical theory to higher dimensions and are becoming increasingly important in diverse areas of mathematics, physics,

Relativity in Clifford's Geometric Algebras of Space and Spacetime

Of the various formalisms developed to treat relativistic phenomena, those based on Clifford's geometric algebra are especially well adapted for clear geometric interpretations and computational

Why Does the Geometric Product Simplify the Equations of Physics?

In the last decades it was observed that Clifford algebras and geometric product provide a model for different physical phenomena. We propose an explanation of this observation based on the theory of

Geometric Algebra for Physicists

Geometric algebra is a powerful mathematical language with applications across a range of subjects in physics and engineering. This book is a complete guide to the current state of the subject with

Geometric algebra for computer science - an object-oriented approach to geometry

An introduction to Geometric Algebra that will give a strong grasp of its relationship to linear algebra and its significance for 3D programming of geometry in graphics, vision, and robotics is found.

Geometry of Paravector Space with Applications to Relativistic Physics

Clifford’s geometric algebra, in particular the algebra of physical space (APS), lubricates the paradigm shifts from the Newtonian worldview to the post-Newtonian theories of relativity and quantum

Relativity in introductory physics

A century after its formulation by Einstein, it is time to incorporate special relativity early in the physics curriculum. The approach advocated here employs a simple algebraic extension of vector

LIGHT POLARIZATION : A GEOMETRIC-ALGEBRA APPROACH

The geometric algebra of three‐dimensional space (the ‘‘Pauli algebra’’) is known to provide an efficient geometric description of electromagnetic phenomena. Here, it is applied to the

Geometric computing with Clifford algebras: theoretical foundations and applications in computer vision and robotics

1. New Algebraic Tools for Classical Geometry.- 2. Generalized Homogeneous Coordinates for Computational Geometry.- 3. Spherical Conformai Geometry with Geometric Algebra.- 4. A Universal Model for