In this article we discuss the findings of our research on students' understanding of vector concepts in problems without physical context. First, we develop a complete taxonomy of the most frequent errors made by university students when learning vector concepts. This study is based on the results of several test administrations of open-ended problems in which a total of 2067 students participated. Using this taxonomy, we then designed a 20-item multiple-choice test [Test of understanding of vectors (TUV)] and administered it in English to 423 students who were completing the required sequence of introductory physics courses at a large private Mexican university. We evaluated the test's content validity, reliability, and discriminatory power. The results indicate that the TUV is a reliable assessment tool. We also conducted a detailed analysis of the students' understanding of the vector concepts evaluated in the test. The TUV is included in the Supplemental Material as a resource for other researchers studying vector learning, as well as instructors teaching the material.