Definition of the Cross Product

If u and v are vectors in three dimensional space, then u × v is a three dimensional vector. Unlike the other operations, its operands and its result must be 3D vectors. The operation is not defined for 2D.

Definition of the Cross product u × v

• u and v must be 3D vectors
• The result is a 3D vector with the following length and orientation:
  • 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.

The result is a vector perpendicular to the two operand vectors. But there are two directions that are perpendicular to both operands. Which one does the cross product give you? That is determined by the right hand rule, which is explained on the next page.