After four years of research we designed a 20-item multiple choice vector concept test (Test of Understanding of Vectors, TUV). In this article we analyze: 1) the reliability and discriminatory power of the test, and 2) students' understanding of the vector concepts evaluated in the test. The final version of the test was administered in English to 423 students who were finishing an Electricity and Magnetism course at a large private Mexican university. In the first part of the article, we show results indicating that the TUV is a reliable assessment tool. In the second part, we examine students' overall performance on the test and analyze the results of the five most difficult items for students: geometric interpretation of dot product, calculation of dot product of two vectors written in unit-vector notation, graphic representation of a unit vector, calculation of the direction of a vector written in unit-vector notation, and graphical subtraction of vector in 2D.