The vector cross product from an algebraic point of view (Q2773033)

The vector cross product of the three-dimensional vector space \(\mathbb{E}\) is studied from an algebraic point of view, considering the existence of a unique matrix \(T_a x= a\times z\), to derive the main properties of the cross product. This operator-theoretic point of view invites the use of the Moore-Penrose inverse of \(T_a\) and related matrices. The eigenvalues and eigenspaces are determined. Some proofs from analytical geometry can be simplified using the matrix oriented approach. The analysis of equations involving vector cross products can be facilitated applying the concept of the generalized matrix inverse.