The relativistic Faddeev equation for three-nucleon scattering is formulated in momentum space and directly solved in terms of momentum vectors without employing a partial wave decomposition. Relativistic invariance is achieved by constructing a dynamical unitary representation of the Poincaré group on the three-nucleon Hilbert space. The exclusive breakup reaction at 508 MeV is calculated based on a Malfliet-Tjon type of two-body interaction and the cross sections are compared to measured cross sections at this energy. We find that the magnitude of the relativistic effects can be quite large and depends on the configurations considered. In spite of the simple nature of the model interaction, the experimental cross sections are in surprisingly good agreement with the predictions of the relativistic calculations. We also find that although for specific configurations the multiple scattering series converges rapidly, this is in general not the case.Key words: Relativistic Quantum Mechanics, Faddeev Equation, The Quantum Mechanical 24.10.Jv, Breakup reactions in the proton-deuteron (pd) system at intermediate energies have been studied experimentally quite intensively in recent decades. A prominent set of data can be found in the comprehensive overview of the experiments completed at Saturne-2 [1]. However, the theoretical interpretation faced and still faces serious challenges. At those energies pion production channels are open and nuclear resonances play a role. In contrast to the energy regime below the pion threshold, where high precision nucleonnucleon (NN) forces are established [2,3,4], and nuclear forces based on effective chiral dynamics are being developed [5] [8]. In addition, the standard partial wave decomposition, successfully applied below the pion-production threshold [9], is no longer an adequate numerical scheme due to the proliferation of the number of partial waves. Thus, the intermediate energy regime is a new territory for few-body calculations, which waits to be explored.The aim of this article is to address two aspects in that list of challenges: exact Poincaré invariance and calculations using vector variables instead of partial waves. In a series of publications [10] the technique of solving the nonrelativistic momentum-space Faddeev equation without partial waves has been mastered, for bound as well as scattering states. The Faddeev equation, based on a Poincaré invariant mass operator, has been formulated in detail in [11]. The resulting Faddeev equation has both kinematical and dynamical differences with respect to the corresponding nonrelativistic equation.The formulation of the theory is given in a representation of Poincaré invariant quantum mechanics where the interactions are invariant with respect to kinematic translations and rotations [12]. The model Hilbert space is a three-nucleon Hilbert space (thus not allowing for absorptive processes). The method used to introduce the NN interactions in the unitary representation of the Poincaré group allows to input high-precisi...