Multiply two vectors a different way and you get another vector, pointing at right angles to both of them. The hard part is which of the two perpendicular directions it points. The right-hand rule answers that, and here you actually do it — rotate a 3D hand until it lines up, curl the fingers, and see where the thumb ends up.
Three plane presets — xy, yz, xz — or drag the vector tips yourself for a custom pair. Then work through three steps. Point the fingers along A. Rotate so the curl sweeps toward B. Curl, and the answer appears. Each step unlocks only when the hand is within 10 degrees of correct, so you cannot skip ahead. Try curl previews the motion without giving anything away. After finishing, Try B × A runs it the other way round.
The two vectors, the angle between them, and once you finish the step, the cross product as coordinates and as a magnitude. Show parallelogram draws the shape whose area equals that magnitude.
No determinant or component formula here. No left-hand rule.
You should be able to place your own hand correctly and say which way A × B points before checking.