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

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.