Definition of the Cross Product

The vector cross product takes two vector operands to produce a vector result. The result, like all geometric vectors, has two properties: length and orientation.

If u and v are vectors in three dimensional space (only), then u × v is a three dimensional vector, where:

Length:

|u × v |   =   | u | | v | sin θ,   where θ, is the angle between u and v.

Orientation:

u × v is perpendicular to both u and v. The choice (out of two) orientations perpendicular to u and v is made by the right hand rule.

To find a vector perpendicular to a particular plane, compute the cross product of two vectors in that plane. But there are two directions perpendicular to the plane. Which one does the cross product give you? That is determined by the right hand rule, which will be explained shortly.