Consider two vectors and their cross product. The definition of the cross product is:
Since the angle is between the two vectors,
is
the altitude of the parallelogram and B is its base; therefore, the magnitude
of the cross product is the area of the parallelogram.
Since the vector C's magnitude
is equal to the area, C is used as the vector defining the
area. So an area vector is perpendicular to the plane that the area is
in, and has as a magnitude the geometrical value of that area.
Volume, on the other hand, is a scalar, since it is
where the vector D is the third
vector of a triad which defines a parallelopiped. To help you see this
better, A and B lie in the xy plane. B is dashed because
it is hidden. Vector C (not shown) is the cross product of A
and B, and points along the z axis. The cosine of the angle between
C
and D provides the altitude of the parallelopiped which, times the
base area (
),
yields the volume.
