Cross product

The cross product of two vectors a and b in space is a vector, a × b. Its length is the area of the parallelogram they span, which for nonzero vectors is $|a|\,|b|\sin\theta$, where θ is the angle between them. When that area is not 0, a × b stands at right angles to both and points along the right thumb when the fingers curl from a toward b. Here you can turn a and b in a room and see the parallelogram’s shadows on the walls, which turn out to be the three components of a × b.

Drop the parallelogram straight down onto the floor. Its shadow is the parallelogram spanned by $(a_1, a_2)$ and $(b_1, b_2)$, and its area, with a sign, is the 2 × 2 determinant $a_1 b_2 - a_2 b_1$, which is the third component. The sign is + when the turn from the shadow of a to the shadow of b is counterclockwise seen from above. The two walls work the same way, each seen from the side its axis points to: the shadow on the yz wall gives the first component and the shadow on the zx wall the second. A shadow with a negative area turns clockwise, and the page hatches it.

Why do the shadows make a vector at right angles to the parallelogram? Take a parallelogram with nonzero area. Call its area A and let n be the unit vector at right angles to it, on the side the right-hand rule picks. Seen from a wall with unit normal m, the parallelogram is squashed in one direction only, by the factor $|n\cdot m|$. So its shadow on that wall has signed area $A\,(n\cdot m)$. Shadow on one wall draws this view with the current numbers. The three walls’ normals are the three axes, so the three shadows together are

This is a vector along n with length A. The coordinate formula and the geometric description are the same thing.

Swapping the order traces every shadow the other way round, so b × a = −(a × b): the cross product is not commutative. When a and b are parallel, or one of them is zero, the parallelogram collapses to a segment or a point, every shadow has area 0, and a × b = 0. When the area is not 0, Face looks straight along a × b, where the parallelogram shows its true shape, and Edge-on looks along the parallelogram, which becomes a line at right angles to a × b.

The right hand is a convention that matches the room’s right-handed axes, in which the turn from x to y is counterclockwise seen from +z.