The cross product

We define the cross product of two three-dimensional vectors a and b by the requirements:

  • a × b is a vector that is perpendicular to both a and b.
  • ||a × b|| is the area of the parallelogram spanned by a and b (i.e. the parallelogram whose adjacent sides are the vectors a and b).
  • The direction of a×b is determined by the right-hand rule. (This means that if we curl the fingers of the right hand from a to b, then the thumb points in the direction of a × b.)

From simple trigonometry, we can calculate that the area of the parallelogram spanned by a and b is

||a|| ||b|| sin θ,
where θ is the angle between a and b. (One can view the parallelogram as having a base of length ||b|| and perpendicular height ||a|| sin θ, as pictured below.)

PIC

This formula shows that the magnitude of the cross product is largest when a and b are perpendicular. On the other hand, if a and b are parallel or if either vector is the zero vector, then the cross product is the zero vector. (It is a good thing that we get the zero vector so that the above definition still makes sense. If the vectors are parallel or one vector is the zero vector, then there is not a unique line perpendicular to both a and b. But since there is only one vector of zero length, the definition still uniquely determines the cross product.)

Below is a Concept-Visualization Tool (CVT) that helps illustrate how the cross product works. The vector a is shown in blue, b is green, and the cross product c = a × b is red. (I’ve drawn vectors in a nonstandard way with balls on their ends. The parametrization of a line reading has a discussion on this.) I’ve also drawn the parallelogram formed by a and b.

Drag the balls at the ends of the vectors a and b to change these vectors. See how the cross product c and the parallelogram change in response.

Important note: The three-dimensional perspective of this graph is hard to perceive when the graph is still. If you keep the figure rotating, you’ll see it much better.

Although the CVT is admittedly hard to manipulate in a precise manner, you can see that the above properties are true. The area of the parallelogram (and hence the magnitude of the cross product) go to zero as a and b approach parallel (where the term “parallel” also includes what you might think as anti-parallel).

You can also verify with the CVT that b × a = -a × b and a × a = 0, which are important properties of the cross product.

If you like to see it with numbers, we also have some examples of calculating cross products and areas of parallelograms. These examples assume you know the formula for the cross product in terms of components.