In this article we investigate students' understanding of: 1) vector components and, 2) vector products. We administered a test to 409 students completing introductory physics courses at a private Mexican university. In the first part, based on the work of Van Deventer , we analyze the understanding of components of a vector. We used multiple-choice questions asking for students' reasoning to elaborate on the misconceptions and difficulties of graphical representation of the x- and y-components of a vector. In the rest of this work, we analyze the understanding of the dot and cross products. We designed opened-ended questions to investigate the difficulties on the calculation and the misconceptions in the interpretation of these two products.