Arrow

the cross product leaps out of the plane. that’s the surprise.

drag two arrows on the floor -- swing them parallel and the third one, standing straight up, sinks away

Two vectors define a flat plane between them, and the cross product is the one direction that plane does not contain -- straight off it, with a length equal to the area of the parallelogram the two vectors span. Drag the arrows and watch the vertical one rise and fall as the parallelogram grows and shrinks. Swing the two arrows until they point the same way and the parallelogram flattens to a line with no area at all, which is exactly when the vertical arrow sinks back into the floor: nothing escapes a plane spanned by two parallel vectors, because there is no plane there to escape from.

Concepts: geometry, trigonometry.

An interactive you play with one thumb, on Wundr.