Geometric problems are usually solved in the Euclidean space by using the standard vector algebra techniques. In this study, principles of the projective geometry and geometric algebra will be introduced via a novel method that significantly simplifies the solution of geometrical problems. Also, it supports the GPU parallel computation application. Besides that, an application of the principle of duality leads to a simple solution of the dual problems. We show that, the equivalence of the extended cross-product (outer product) and the solution of the system of linear equations. This gives a direct impact to scientific computation, solution of geometrical problems, robotics, computer graphics algorithms and virtual reality via fast computation through GPU parallel systems. Some numerical and graphical results are presented. © 2020. All rights reserved.