If and are any nonzero vectors in 3-space, which are not parallel to one another, write expressions for vectors representing the following:
The vector is defined to be the vector which is perpendicular to both and in the direction of the right hand rule. It has magnitude given by if is the angle between the two vectors.
Now, , but radians since the cross product of and is perpendicular to by part 2 above. Thus, .
Notice that we cannot distribute the dot product across the cross product like so: since the dot product of two vectors is a scalar and the cross product is only defined for vectors.