# Identities of Vector Algebra as Associative Properties of Multiplicative Compositions of Quaternion Matrices

<- Back to II. Mechanical Engineering & Physics Vol. 8

Cite the paper

Victor Kravets, Tamila Kravets, Olexiy Burov (2017). Identities of Vector Algebra as Associative Properties of Multiplicative Compositions of Quaternion Matrices. Mechanics, Materials Science & Engineering, Vol 8. doi:10.2412/mmse.47.87.900

Authors: Victor Kravets, Tamila Kravets, Olexiy Burov

ABSTRACT. This paper is dedicated to the further development of matrix calculation in the sphere of quaternionic matrices. Mathematical description of transfer (displacement) and turn (rotation) in space are fundamental for the mechanics of rigid body. The transfer (displacement) in space is described by the vector (hodograph). The turn (rotation) in space is described by quaternion. Calculation quaternionic matrices generalizes vector algebra and is directly adapted for the computing experiment concerning nonlinear dynamics of discrete mechanical systems in spatial motion. It is proposed to examine the turn and transfer of the rigid body in space with four-dimensional orthonormal basis and corresponding matrices equivalent to quaternions or vectors.

The identities of vector algebra, including the Lagrange identity, Euler-Lagrange identity, Gram determinant and others, are found systematically. The associative products of conjugate quaternionic matrices are represented by the multiplicative compositions of vector algebra. The complex vector and scalar products are represented by the introduced matrices. With the use of associative property of conjugate quaternionic matrices’ products, the range of vector algebra identical equations is found, including the known ones, which serve, in particular, to justify the fidelity of the offered method. The method is being developed to represent associative products of conjugate and various quaternionic matrices by multiplicative compositions of vector algebra, containing scalar and vector products. The method is offered to represent complex (vector and scalar) vector algebra products as quaternionic matrices. This fundamental results constitute the first part of the study recommended for engineers, high school teachers and students who in their practical activity set and solve the problems of dynamic design of aeronautical engineering, rocket engineering, space engineering, land transport (railway and highway transport), robotics, etc. and exposed data to be able to contribute to the research area, to permit to enhance the intellectual performance, to provide the engineer with simple and efficient mathematical apparatus.

Keywords: quaternionic matrices, vector matrices, vector algebra identities, complex vector and scalar products, associative property of vector matrices products

References

[1] Bellman, R. Introduction to matrix analysis, Second edition, Book code: CL19, Series: Classics in Applied Mathematics, 1997, 403 p. DOI: http://dx.doi.org/10.1137/1.9781611971170.

[2] Mal’tsev, A.I. Osnovy linejnoj algebry [Fundamentals of linear algebra], Moscow, Nauka Publ., 1970, 400 p. (in Russian).

[3] Korenev, G.V. Tenzornoe ischislenie [Tensor calculus], MFTI Publ., 1995, 240 p. (in Russian).

[4] Kil’chevskij, N.A. Kurs teoreticheskoj mexaniki [Course of theoretical mechanics], Moscow, Nauka Publ., 1977, Part 1, 480 p., Part 2, 544 p. (in Russian).

[5] Elliot, J.P., Dawber, P.G. Symmetry in physics, Vol. 1: Principles and Simple Applications, Oxford University Press, 1985, 302 p., Vol. 2: Further Applications, Oxford University Press, 1985, 298 p.

[6] Berezin, A.V., Kurochkin, Yu.A., Tolkachev, E.A. Kvaterniony v relyativistskoj fizike [Quaternions in relativistic physics], Moscow, Editoreal Publ., 2003, 200 p. (in Russian).

[7] Branets, V.N., Shmyglevskij, I.P. Primenenie kvaternionov v zadachax orientatsii tverdogo tela [The use of quaternions in problems of solid-state orientation], Moscow, Nauka Publ., 1973, 320 p. (in Russian).

[8] Raushenbax, B.V., Tokar’, E.N. Upravlenie orientatsiej kosmicheskix apparatov [The orientation of the spacecraft management], Moscow, Nauka Publ., 1974, 600 p. (in Russian).

[9] Mejo, R.A. Perexodnaya matritsa dlya vychisleniya otnositel’nyx kvaternionov [The transition matrix to calculate the relative quaternion], Raketnaya texnika i kosmonavtika [Rocketry and Astronautics], 17 (3), 1979, p. 184-189. (in Russian).

[10] Lur’e, A.I. Analiticheskaya mexanika [Analytical mechanics], Moscow, Fizmatgiz Publ., 1961, 824 p. (in Russian).

[11] Plotnikov, P. K., Chelnokov, Yu. N. Primenenie kvaternionnyx matrits v teorii konechnogo povorota tverdogo tela [Application of quaternion matrices in the final turn in solid state theory], Sbornik nauchn.-metod. statej po teoret. mexanike. − Moscow, Vysshaya shkola Publ., Vol. 11, 1981, p. 122 − 128. (in Russian).

[12] Ishlinskij, A.Yu. Orientatsiya, giroskopy i inertsial’naya navigatsiya [Orientation, gyroscopes and inertial navigation], Moscow, Nauka Publ., 1976, 672 p. (in Russian).

[13] Onischenko, S.M. Primenenie giperkompleksnyx chisel v teorii inertsial’noj navigatsii. Avtonomnye sistemy [The use of hyper complex numbers in the inertial navigation theory. Stand-alone systems], Kyiv, Naukova dumka Publ., 1983, 208 p. (in Russian).

[14] Pars, L.A. A Treatise on analytical dynamics, Ox Bow Pr. Publ., 1981, 641 p.

[15] Chelnokov, Yu.N. Kvaternionnye i bikvaternionnye modeli i metody mexaniki tverdogo tela i ix prilozheniya. Geometriya i kinematika dvizheniya [Quaternion and biquaternions models and methods of solid mechanics and their applications. The geometry and kinematics motion], Moscow, Fizmatgiz Publ., 2006, 512 p. (in Russian).

[16] Kravets, T.V. Based on Gibbs vector solution of spatial angular stabilization of solid problem, Avtomatika 2001: Sbornik nauchnyx trudov konferentsii [Automation 2001: Proceedings of the Conference], Odessa Publ., 2001, Volume 2, p. 20.

[17] Kravets, V.V., Kravets, T.V. On the nonlinear dynamics of elastically interacting asymmetric rigid bodies, Int. Appl. Mech., 2006, 42(1), p. 110- 114.

[18] Kravets, V.V., Kravets, T.V., Kharchenko, A.V. Using quaternion matrices to describe the kinematics and nonlinear dynamics of an asymmetric rigid body, Int. Appl. Mech., 2009, 45 (223), DOI: 10.1007/s10778-009-0171-1.

[19] Pivnyak, G.G., Kravets, V.V., Bas, K.M., Kravets, T.V., Tokar, L.A. Elements of calculus quaternionic matrices and some applications in vector algebra and kinematics, MMSE Journal, 3, March 2016, p.p. 46-56. ISSN 2412-5954, Open access www.mmse.xyz, DOI 10.13140/RG.2.1.1165.0329.

[20] Kravets, V.V., Bass, K.M., Kravets, T.V., Tokar L.A. Dynamic design of ground transport with the help of computational experiment, MMSE Journal, 1, October 2015, p.p. 105 – 111. ISSN 2412-5954, Open access www.mmse.xyz, DOI 10.13140/RG.2.1.2466.6643.

[21] Kravets, V., Kravets, T., Burov, O. Monomial (1, 0, -1)-matrices-(4х4). Part 1. Application to the transfer in space. Lap Lambert Academic Publishing, Omni Scriptum GmbH&Co. KG., 2016, 137 p. ISBN: 978-3-330-01784-9

[22] Korn, G., Korn, T. Spravochnik po matematike dlya nauchnyh rabotnikov i inzhenerov [Mathematical Handbook for Scientists and Engineers], Moscow, Nauka Publ., 1984, 832 p. (in Russian).