The Cross Product

The cross product is simply a way to take two arrows in 3D and build a third arrow that sticks straight out, perpendicular to both. It's called "cross" because we write it with a times sign, $\vec a\times\vec b$. It's how you find the direction a flat surface faces, compute areas, and work out torque on a wrench. The component formula looks scary, but the meaning is in $\|\vec a\times\vec b\| = \|\vec a\|\,\|\vec b\|\sin\theta$, which is just $\|\vec a\|$ (a base) times $\|\vec b\|\sin\theta$ (a height), and base times height is an area. Bear with the bookkeeping; the idea is that simple.

The dot product turned two vectors into a number. The cross product, the second way to multiply them, gives back a vector, and it only works in 3D. Here's why you'd want it: in 3D, two directions that aren't parallel determine a whole plane, and there's exactly one direction (up to sign) sticking straight out of that plane. The cross product hands you that direction.

Computing it. The formula above is easy to garble, so most people get it by expanding the determinant

along the top row. For each component, cover that column, cross-multiply what's left, and remember the minus sign on the middle term. You can always check your answer by dotting it with $\vec a$ and $\vec b$: both should give $0$.

Direction. There are two directions perpendicular to the plane, so we need a rule to pick one. That's the right-hand rule: fingers along $\vec a$, curl toward $\vec b$, and your thumb points along $\vec a\times\vec b$. So $\vec i\times\vec j = \vec k$. Notice what that means for order: curl from $\vec b$ to $\vec a$ instead and your thumb flips, so $\vec b\times\vec a = -\vec a\times\vec b$.

Length. Compare $\|\vec a\times\vec b\| = \|\vec a\|\|\vec b\|\sin\theta$ with the dot product's $\cos\theta$. The dot product is biggest when the vectors line up; the cross product is biggest when they're perpendicular and zero when they're parallel. And $\|\vec b\|\sin\theta$ is the height of the parallelogram on base $\vec a$, so the length is base times height: the area.

What it's good for:

• A vector perpendicular to two given ones. This is how you'll get normal vectors to planes in the next unit.
• Area: the parallelogram is $\|\vec a\times\vec b\|$, and a triangle is half of that.
• Volume of a parallelepiped: $|\vec a\cdot(\vec b\times\vec c)|$. The cross product gives the base area times a perpendicular direction, and dotting with $\vec c$ picks up the height.
• Parallel test: $\vec a\times\vec b = \vec 0$ exactly when $\sin\theta = 0$.

Torque is $\vec r\times\vec F$, the magnetic force on a moving charge is $q\,\vec v\times\vec B$, and 3D graphics compute surface normals for lighting with cross products of triangle edges.