We create the cross-product to be a sort of ``right-hand rule'' operator. We might start by saying that we want the cross-product to have the following results:
Now assume the usual sorts of distributive properties, so that given
a 
and
b 
, we can compute
or
Note that the cross-product is a vector-product, by contrast with the dot product which produces a number (scalar). Furthermore, the direction of the cross-product is determined by using the right-hand rule.
This cross-product is only defined for three-dimensional vectors, by contrast with the dot product, which is defined for all vectors in all dimensions.
where 
is the angle between a and b. This
last property can be interpreted geometrically: the norm (length) of the cross
product 
is the area of the parallelogram constructed
using vectors a and b.
This can be generalized to the parallelpiped, constructed of three vectors, whose volume V is given by
If we discover that V is zero, then the vectors a, b and c must be coplanar (the parallelpiped is at most two-dimensional). Note: by symmetry,
Two non-zero vectors a and b are parallel if and only if
Thus the cross-product of two vectors produces a vector which is perpendicular to those two vectors, having magnitude the area of the parallelogram created by the two vectors. We can use this product as a means for determining when two vectors are parallel (they have a zero - vector - cross-product).