Editing Geometric algebra (section)

=== Hypervolume of a parallelotope spanned by vectors ===
For vectors {{tmath|1= a }} and {{tmath|1= b }} spanning a parallelogram we have
: <math> a \wedge b = ((a \wedge b) b^{-1}) b = a_{\perp b} b </math>
with the result that {{tmath|1= a \wedge b }} is linear in the product of the "altitude" and the "base" of the parallelogram, that is, its area.

Similar interpretations are true for any number of vectors spanning an {{tmath|1= n }}-dimensional [[Parallelepiped#Parallelotope|parallelotope]]; the exterior product of vectors {{tmath|1= a_1, a_2, \ldots , a_n }}, that is {{tmath|1= \textstyle \bigwedge_{i=1}^n a_i }}, has a magnitude equal to the volume of the {{tmath|1= n }}-parallelotope. An {{tmath|1= n }}-vector does not necessarily have a shape of a parallelotope – this is a convenient visualization. It could be any shape, although the volume equals that of the parallelotope.