Probability theory plays a crucial role in the study of the geometry of Banach spaces. In the literature, notions from probability theory have been formulated and studied in the measure free setting of vector lattices. However, there is little evidence of these vector lattice techniques being used in the study of geometry of Banach spaces. In this thesis, we fill this niche. Using the l-tensor product of Chaney-Shaefer, we are able to extend the available vector lattice techniques and apply them to the Lebesgue-Bochner spaces. As a consequence, we obtain new characterizations of the Radon Nikod´ym property and the UMD property.