We investigated the influence of context on students' understanding of cross products of vectors using three isomorphic multiple-choice tests asking for the direction of a cross product in different geometrical settings. One version of the test involved the Lorentz force, the second version involved the torque on an electric dipole, and the third version was without physics context. We administered the tests to 216 first-year pre-med students at a Belgian (Flemish) university. We found that students perform significantly better in the context of the Lorentz force. Students more often chose the incorrect alternative corresponding to the vector sum in the test versions involving an electric dipole or without physics context when the vectors are not orthogonal. For orthogonal vectors, a sign error--i.e., inverting the direction of the resulting vector--was the most common mistake in both tests with physics context, while without physics context selecting the alternative corresponding to the sum remained the most common mistake. Prior familiarity with a right-hand rule in a specific context seems to be able to explain improved scores in the test version concerning the Lorentz force. Instructors and curriculum developers can benefit from adopting an integrated approach in which the mathematical aspects of the cross product are treated together with multiple examples in physics, allowing students to transition from using specific rules to determine a cross product, to a more integrated understanding of it.