The Cross Product

The cross product VXW of vectors V and W is a VECTOR that is perpendicular to the computer screen. The sketch shows you how its length and direction change as you vary V and W.

The rectangle and the parallelogram have the same area. By definition this area is also the length of the cross product VXW.

The height of the rectangle is |W| sin a, and its width cd = |V|, so |VXW| = |V||W| sin a. Thus we know the magnitude of VXW; what's its direction?

By definition, The cross product VXW
has the same length as this vector.

- VXW points straight out from the screen
if it is depicted as pointing upward.

- VXW points straight into the screen
if it is depicted as pointing downward.

This follows the famous "right-hand rule": curl the fingers of your right hand from V to W. Then your thumb points in the direction of VXW.

If you like calculations, see Projection: Numbers.