Mathematics

The key is understanding what the magnetic force law actually requires. A current-carrying wire in a magnetic field experiences a force given by $\vec{F} = I\vec{L} \times \vec{B}$—a cross product. When the wire is parallel to the field, these vectors point in the same (or opposite) directions, making their cross product zero. Parallel vectors can't "push" against each other, so no force arises.

The key is understanding what the magnetic force law actually requires. A current-carrying wire in a magnetic field experiences a force given by $\vec{F} = I\vec{L} \times \vec{B}$—a cross product. When the wire is parallel to the field, these vectors point in the same (or opposite) directions, making their cross product zero. Parallel vectors can't "push" against each other, so no force arises.