Almost exactly 200 years ago, Ørsted discovered that a compass needle pointed along circular paths around a current-carrying wire. Shortly afterwards, Ampère found that two wires carrying currents in the same direction attract each other. Whereas Ampère saw the phenomenon as action-at-a-distance between the wires, the notion of field-as-agent came to supersede action-at-a-distance. With this paradigm, it is clear that in the light of Ørsted's discovery, Ampère's observation implies a force on each current-carrying wire at right angles to the wire and to the field (from the other wire). This was confirmed and generalised by further experiments.

The names 'cross product' or 'vector product', and the vector algebra notation, to describe such a relationship came later. **The experimental findings helped to motivate the mathematical concept, not the other way round**. And, of course, it wasn't just electromagnetism, that gave rise to the cross product idea. Torque and angular momentum cry out for the treatment.