Usually, we say that the Lorentz force is a vector. However, in group theory, we make a distinction between vectors and pseudovectors based on how they transform under O(3). Vectors and pseudovectors transform similarly under rotations but not under reflections. For example, the magnetic field is a pseudovector, because it can be written in terms of a cross product, particularly as a curl:

$$\vec{B} = \nabla \times \vec{A}.$$

Any cross product is a pseudovector, because if both vectors reflect, then the cross product of those vectors do not, since the minus signs cancel out.

Now consider the Lorentz force:

$$\vec{F} = q(\vec{E} + \vec{v} \times \vec{B}).$$

The electric field is a vector. The velocity is a vector and the magnetic field is a pseudovector, so the cross product is a pseudovector as well. Because if the velocity reflects, B does not, but the cross product does since it picks up a minus sign. So is the total force then a vector or pseudovector? How does it transform under reflections?

Usually, we say that the Lorentz force is a vector. However, in group theory, we make a distinction between vectors and pseudovectors based on how they transform under $O(3)$. Vectors and pseudovectors transform similarly under rotations but not under reflections. For example, the magnetic field is a pseudovector, because it can be written in terms of a cross product, particularly as a curl:

$$\vec{B} = \nabla \times \vec{A}.$$

Any cross product is a pseudovector, because if both vectors reflect, then the cross product of those vectors do not, since the minus signs cancel out.

Now consider the Lorentz force:

$$\vec{F} = q(\vec{E} + \vec{v} \times \vec{B}).$$

The electric field is a vector. The velocity is a vector and the magnetic field is a pseudovector, so the cross product is a pseudovector as well. Because if the velocity reflects, $B$ does not, but the cross product does since it picks up a minus sign. So is the total force then a vector or pseudovector? How does it transform under reflections?