1 The vector algebra war: a historical perspective James M. Chappell, Azhar Iqbal, John G. Hartnett, and Derek Abbott, Fellow, IEEE Abstract—There are a wide variety of different vector for- that this system indeed provides the most natural vector system malisms currently utilized in engineering and physics. For for three-dimensional space. example, Gibbs’ three-vectors, Minkowski four-vectors, complex spinors in quantum mechanics, quaternions used to describe rigid body rotations and vectors defined in Clifford geometric algebra. II. ANALYSIS With such a range of vector formalisms in use, it thus appears The concept of vectorial quantities actually appears to that there is as yet no general agreement on a vector formalism suitable for science as a whole. This is surprising, in that, one of be quite an ancient one, with the parallelogram law for the primary goals of nineteenth century science was to suitably the addition of vectors well known to Aristotelian science describe vectors in three-dimensional space. This situation has from the fourth century B.C.E. [2] and later repeated in also had the unfortunate consequence of fragmenting knowledge Newton’s Principia. Descartes, in 1637 however, proposed a across many disciplines, and requiring a significant amount of much more radical view of vectors as quantities such that time and effort in learning the various formalisms. We thus historically review the development of our various vector systems ‘Just as arithmetic consists of only four or five operations, and conclude that Clifford’s multivectors best fulfills the goal of namely, addition, subtraction, multiplication, division and the describing vectorial quantities in three dimensions and providing extraction of roots....so in geometry, to find required lines a unified vector system for science. it is merely necessary to add or subtract lines.’ [3] This Index Terms—Vectors, Gibbs, Hamilton, Clifford, Multivectors revolutionary idea was indeed successfully formulated alge- braically in the nineteenth century by Wessel, Argand and Gauss. It was achieved through the use of complex numbers I. INTRODUCTION z = a + ib, where a,b are real numbers and i = √ 1 ∈ ℜ − ENERALLY speaking, the concept of a vector has been is the unit imaginary. The number z was then interpreted as G an extremely useful one with nearly all branches of representing a point in the plane located at the coordinates physical science now described in the language of vectors [1]. [a,b]. This point could also be viewed as representing a Despite its great value as a concept there is nevertheless vector extending out from the origin to this point. The ability a plethora of different vector formalisms currently in use. to use a single number ‘z’ to describe a two-dimensional Listed in the approximate order of their creation are: complex planar point means that we now can undertake geometrical numbers (planar vectors), quaternionic vectors, Gibbs’ vectors, analysis in a coordinate free manner. It is also compatible Minkowski four-vectors, complex spinors, Dirac matrix four- with intuition in having a single number ‘z’ to refer to a vectors and finally vectors defined using Clifford algebra, as single point, rather then needing to always refer to two separate well as several other vector-type formalisms. coordinates. For example, if we have two vectors represented This fact is surprising as one of the main goals of nineteenth by the complex numbers z1 and z2 then vectorial addition is century science was to find a suitable vector system for simply z1 + z2. That is, if z1 = a1 + ib1 and z2 = a2 + ib2 three-dimensional Euclidean space. This objective was initially then z1 + z2 = (a1 + a2)+ i(b1 + b2) and so satisfies the led by Hamilton who produced the quaternionic vectors. parallelogram law for adding vectorial quantities, as required. Unfortunately Hamilton’s system failed to live up to the We thus have extended real numbers to a more general type arXiv:1509.00501v2 [physics.hist-ph] 22 Apr 2016 initial high expectations and following an intense debate over of number with the addition of the imaginary component with several years, it was replaced by the Gibbs vector system in all fundamental arithmetic operations essentially unchanged. mainstream use today. We firstly identify why the quaternions Indeed, complex numbers are a division algebra and so satisfy fail to produce a suitable description of Cartesian vectors but Descartes vision of vectorial quantities being amenable to also importantly identify what their natural role is. We then the four common arithmetic operations. This principle of show the serious failings of the Gibbs vector system before Descartes was also consistent with a later principle by Hankel demonstrating a reconciliation of these two rival systems for extending mathematical concepts, of the the principle of 3 the permanence of the rules of calculation [4]. within Clifford geometric algebra Cℓ( ). We then conclude ℜ One defect of this approach to representing a Cartesian James M. Chappell, Azhar Iqbal and Derek Abbott are with the School vector by a complex number is that we are setting up a real and of Electrical and Electronic Engineering, University of Adelaide, SA 5005, an imaginary axis for the plane that is clearly not isotropic1 Australia e-mail: [email protected]. John G. Hartnett is with the Institute for Photonics & Advanced Sensing and so somewhat inconsistent with the principles of relativity. (IPAS), and the School of Physical Sciences, University of Adelaide, SA 5005 Australia. 1Isotropy implies that for an isolated physical system experimental out- Manuscript received November 13, 2015; revised: XX YY, 2015. comes are independent of its orientation in space. 2 Complex numbers actually describe the algebra of rotations It thus appeared at first that quaternions may indeed be for the plane, which is two dimensional. Hence this has the the ideal algebra for three-dimensional physical space that same dimension as a two-dimensional vector space. This is had been sought. Unfortunately there was one cloud on the why complex numbers can doubly serve as rotation operators horizon, the fact that a vector quaternion squares to the nega- as well as vectors for the plane. That is, a rotation of a vector tive Pythagorean length. Indeed, Maxwell commented on this given by a complex number z can be written unusual fact, noting that the kinetic energy, which involves the ′ square of the velocity vector, would therefore be negative [9]. z =eiθz. (1) Maxwell, despite these reservations, formulated the equations In this case both the rotation operator eiθ and the vector z are of electromagnetism in quaternionic form. Maxwell, however, represented by complex numbers. backed away from a complete endorsement of the quaternions, However, following the generally successful use of complex recommending in his treatise on electricity and magnetism ‘the numbers in describing vectors in the plane, researchers of the introduction of the ideas, as distinguished from the operations nineteenth century then turned their attention to the generaliza- and methods of Quaternions’ [10]. tion of complex numbers to three-dimensional space in order The difficulties with quaternions led to a breakaway for- to naturally describe these vectors. malism of the Gibbs vector system. Gibbs considered that the Hamilton led this program, and in 1843 he succeeded in most useful function of the quaternions was in their forming generalizing the complex number algebra to the quaternion of the dot product and the cross product operations. That algebra [5]. A quaternion can be written is, for two vector quaternions v = v1i + v2j + v3k and w = w1i+w2j+w3k we find through expanding the brackets, q = a + v1i + v2j + v3k, (2) that where the three basis vectors have a negative square i2 = 2 2 vw i j k i j k j = k = 1 and are anticommuting with each other. = (v1 + v2 + v3 )(w1 + w2 + w3 ) (4) − Hamilton’s quaternions form a four-dimensional associative = v1w1 v2w2 v3w3 + (v2w3 v3w2)i H − − − − division algebra over the real numbers represented by . Once +(v3w1 v1w3)j + (v1w2 v2w1)k − − again being a division algebra, like complex numbers, they = v w + v w, are amenable to all the common arithmetic operations. The − · × sequence of algebras , C and H are constructed to be division using the negative square and the anticommuting properties of algebras, that is, theyℜ are closed with an inverse operation. the basis vectors. We can also see how the vector quaternion Indeed, the required algebraic rules for quaternions follow squared v2 = v v is thus the negative of the Pythagorean from this closure property [6]. These properties also, in fact, length as noted− previously.· We can see though how indeed make them naturally suited to describe rotations in space as the dot and cross products naturally arise from the product of they also have the closure property. two vector quaternions. Gibbs then considered that adopting Quaternions are also isotropic, as required, with a three the separate operations of the dot and cross products acting dimensional ‘vector’ represented as v = v1i + v2j + v3k. on three-vectors could thus form the basis for a more efficient Hamilton then claimed that being a generalization of complex and straightforward vector system. numbers to three dimensions it would therefore logically be This led to an intense and lengthy debate over several years the appropriate algebra to describe three-dimensional space. between the followers of Gibbs and the followers of Hamilton, Indeed he proposed, many years before Einstein or Minkowski, beginning in 1890, over the most efficient vectorial system to that if the scalar a, in Eq. (2), was identified with time then be adopted in mathematical physics [1]. The supporters of the four-dimensional quaternion can be a representation for Hamilton were able to claim that quaternions being general- a unified spacetime framework [7].