The simplest use of the cross product in physics today is in rotational motion. A rotating object results in a force directed toward the center of mass of the object. A force perpendicular to to the center of rotation creates a cross product force perpendicular to the plane of the other two forces.
Ships at sea carry a "gyrocompass", and very heavy rotating disc that results in a force direct toward its center. When the ship tries to "roll" the gyro and the roll result in a force in the direction of the ships motion - forward or backward.
Electromagnetic waves are also "users" of cross products. Electric waves and magnetic waves rotate perpendicular to each other. The resultant cross product "pushes" the linked waves forward in the direction of motion of the combined wave,