On the relativistic velocity composition paradox and the Thomas rotation

The non-commutativity and the non-associativity of the composition law of the non-colinear velocities lead to an apparent paradox, which in turn is solved by the Thomas rotation. A 3×3 parametric, unimodular and orthogonal matrix elaborated by Ungar is able to determine the Thomas rotation. However, the algebra involved in the derivation of the Thomas rotation matrix is overwhelming. The aim of this paper is to present a direct derivation of the Thomas angle as the angle between the composite vectors of the non-colinear velocities, thus obtaining a simplicity with which the rotation can be expressed. This allows the formulation of an alternative to the statement related to the necessity of the Thomas rotation of the Cartesian axes by the statement implying the necessity of the rotation of the direct (inverse) relativistic composite velocity to coincide with the inverse (direct) relativistic composite velocity.