Polar and axial vectors versus quaternions

Silva, Cibelle Celestino; Martins, Roberto de Andrade

doi:10.1119/1.1475326

Cited by 16 publications

(13 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Given this duality of four-dimensional vectors and quaternions one may consider (w, x, y, z) as either representing a vector in four dimensions or as representing a rotation and a scaling operation. This duality property was known to Hamilton and was emphasized by his successor Tait, who considered it a key property of quaternions (Silva and Martins, 2002). It is in the sense of this duality that one may consider quaternions to be the natural extension of complex numbers.…”

Section: Quaternionsmentioning

confidence: 93%

Understanding quaternions and the Dirac belt trick

Staley

2010

Eur. J. Phys.

View full text Add to dashboard Cite

The Dirac belt trick is often employed in physics classrooms to show that a 2π rotation is not topologically equivalent to the absence of rotation whereas a 4π rotation is, mirroring a key property of quaternions and their isomorphic cousins, spinors. The belt trick can leave the student wondering if a real understanding of quaternions and spinors has been achieved, or if the trick is just an amusing analogy. The goal of this paper is to demystify the belt trick and to show that it suggests an underlying four-dimensional parameter space for rotations that is simply connected. An investigation into the geometry of this four-dimensional space leads directly to the system of quaternions, and to an interpretation of three-dimensional vectors as the generators of rotations in this larger four-dimensional world. The paper also shows why quaternions are the natural extension of complex numbers to four dimensions. The level of the paper is suitable for undergraduate students of physics.

show abstract

Section: Quaternionsmentioning

confidence: 93%

Understanding quaternions and the Dirac belt trick

Staley

2010

Eur. J. Phys.

View full text Add to dashboard Cite

show abstract

“…Na seção 4, comparamos e contrastamos os produtos de quatérnions com os produtos escalar e vetorial, destacando as "adaptações" básicas efetuadas por Gibbs e Heaviside. Na seção 5 reunimos discussões sobre essas adaptações, os aspectos matemáticos envolvidos, o debate histórico a respeito das diferentes interpretações e comentamos algumas referências didáticas básicas [8][9][10][11][12] e recentes [13][14][15][16][17][18], nas quais o assunto pode ser aprofundado. Nossas conclusões são apresentadas na seção 6.…”

Section: Introductionunclassified

Sobre as origens das definições dos produtos escalar e vetorial

Menon

2009

Rev. Bras. Ensino Fís.

View full text Add to dashboard Cite

Nos livros-texto de física e de matemática utilizados em cursos básicos universitários, as operações de multiplicação de dois vetores (produtos escalar e vetorial) são introduzidas apenas como definições, sem nenhuma referência ou discussão a respeito das razões formais e/ou motivações que levaram ao estabelecimento de tais estruturas. Neste trabalho, apresentamos uma breve revisão didática sobre as origens dessas definições, discutindo os resultados pertinentes, formais, estabelecidos por Hamilton no contexto da álgebra de quatérnions e certas "adaptações" feitas por Gibbs e Heaviside, as quais deram origem ao ramo da matemática que hoje é popularmente conhecido como "álgebra vetorial". Comentamos algumas desvantagens decorrentes dessas "adaptações", fazendo referência a outros sistemas algébricos práticos e formalmente bem fundamentados (álgebras de Grassmann e Clifford). Indicamos e comentamos alguns artigos e trabalhos, básicos e também recentes, nos quais o assunto pode ser aprofundado.

show abstract

“…It is known that Hamilton's pure quaternions are not equivalent to vectors [3]. For ordinary Euclidean vectors, basis elements are three orthogonal unit polar vectors, while for Hamilton's quaternions they are unit 'versors' (imaginary units) and with respect to coordinate transformations behave similar to axial vectors.…”

Section: Introductionmentioning

confidence: 99%

Octonionic geometry

Gogberashvili¹

2005

AACA

View full text Add to dashboard Cite

We extend vector formalism by including it in the algebra of split octonions, which we treat as the universal algebra to describe physical signals. The new geometrical interpretation of the products of octonionic basis units is presented. Eight real parameters of octonions are interpreted as the space-time coordinates, momentum and energy. In our approach the two fundamental constants, c and , have the geometrical meaning and appear from the condition of positive definiteness of the octonion norm. We connect the property of non-associativity with the time irreversibility and fundamental probabilities in physics.

show abstract

Polar and axial vectors versus quaternions

Cited by 16 publications

References 11 publications

Understanding quaternions and the Dirac belt trick

Understanding quaternions and the Dirac belt trick

Sobre as origens das definições dos produtos escalar e vetorial

Octonionic geometry

Contact Info

Product

Resources

About