The Cross Product

What to observe

1. Slide a and b apart and watch the patch between them open up. The answer grows with that patch and with nothing else: a cross product is an area wearing a direction, and the direction is just which way the patch faces.
2. Bring the two arrows into line. The patch collapses, the area goes to zero and so does the whole vector. Parallel vectors span nothing, which is the mirror image of perpendicular vectors having no shadow.
3. Now for the formula. Turn the wall shadows on. The patch throws a shadow on each of the three walls, and the signed area of each shadow is exactly one component of a × b. The bars on the right measure the shadows and compare them to the arithmetic, and they land on top of each other.
4. That is where a_xb_y − a_yb_xcomes from: it is the area of a parallelogram in the plane, base times height, written in coordinates. Every component is that same two dimensional formula, applied to a different pair of axes.
5. Look at the middle component in the panel: the pair is taken as z then x, not x then z. The three planes are visited in the cycle x → y → z → x, and the sign of every term is just which way round that cycle you went.
6. Press swap. Every component changes sign and the vector flips over, because the shadow you were measuring is now traced the other way round. The area never changed, only the sense of the loop around it.
7. Check the two bottom readouts: a · (a × b) and b · (a × b) are zero, always, to the last digit. The answer is perpendicular to both, which is forced, because a patch's facing direction cannot lie inside the patch.