For several years now the physics and chemistry of condensed matter attracts a great deal of attention, while also occupying an extremely important position in the research activity worldwide. Liquid crystalline materials, in particular, present numerous applications in the fields of science and technology. Since the development of the liquid crystals display (LCD) technology, a significant concern was devoted to the development and characterization of these fascinating mesophases. In the present paper we perform several Monte Carlo simulations, by using the Lebwohl-Lasher model, for investigating the molecular director configuration in a nematic liquid crystal cell having varying boundary anchoring conditions in asymmetric circumstances. For this geometry, we analyze the molecular spatial behaviour, while mapping the local order parameter distribution for a nematic phase temperature. We also characterize the shape of the transition regions which strongly depend on the distance to the boundaries, and analyze the neighboring spins behaviour throughout the bulk. Furthermore, by using the Mueller matrix approach, we simulated the transmission of light through the nematic cell at normal incidence under crossed polarizers condition. These investigations seek to enrich our scientific knowledge about the fascinating research topics homeotropic in condensed matter physics by exploring interesting issues related to the orientational and optical properties of liquid crystals in confined geometries.