We developed an instrument to systematically investigate student conceptual understanding of the relationships between the directions of net force, velocity, and acceleration in one dimension and report on data collected on the final version of the instrument from over 650 students. Unlike previous work, we simultaneously studied all six possible conditional relations between force, velocity, and acceleration in order to obtain a coherent picture of student understanding of the relations between all three concepts. We present a variety of evidence demonstrating the validity and reliability of the instrument. An analysis of student responses from three different course levels revealed three main findings. First, a significant fraction of students chose "partially correct" responses, and from pre- to post-test, many students moved from "misconception" to partially correct responses, or from partially correct to fully correct responses. Second, there were asymmetries in responding to conditional relations. For example, students answered questions of the form "Given the velocity, what can be inferred about the net force?" differently than converse questions "Given the net force, what can be inferred about the velocity?" Third, there was evidence of hierarchies in student responses, suggesting, for example, that understanding the relation between velocity and acceleration is necessary for understanding the relation between velocity and force, but the converse is not true. Finally, we briefly discuss how these findings might be applied to instruction. (Contains 4 figures and 12 tables.)