Editing Geometric algebra (section)

== Examples and applications ==
=== 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.

=== Intersection of a line and a plane ===
[[File:LinePlaneIntersect.png|thumb|A line L defined by points T and P (which we seek) and a plane defined by a bivector B containing points P and Q.]]

We may define the line parametrically by {{tmath|1= p = t + \alpha \ v }}, where {{tmath|1= p }} and {{tmath|1= t }} are position vectors for points P and T and {{tmath|1= v }} is the direction vector for the line.

Then
: <math>B \wedge (p-q) = 0</math> and <math>B \wedge (t + \alpha v - q) = 0</math>
so
: <math>\alpha = \frac{B \wedge(q-t)}{B \wedge v} </math>
and
: <math>p = t + \left(\frac{B \wedge (q-t)}{B \wedge v}\right) v. </math>

=== Rotating systems ===
A rotational quantity such as [[torque]] or [[angular momentum]] is described in geometric algebra as a bivector.  Suppose a circular path in an arbitrary plane containing orthonormal vectors {{tmath|1= \widehat{u} }} and {{tmath|1= \widehat{\ \!v} }} is parameterized by angle.
: <math>\mathbf{r} = r(\widehat{u} \cos \theta + \widehat{\ \!v} \sin \theta) = r \widehat{u}(\cos \theta + \widehat{u} \widehat{\ \!v} \sin \theta)</math>

By designating the unit bivector of this plane as the imaginary number
: <math>{i} = \widehat{u} \widehat{\ \!v} = \widehat{u} \wedge \widehat{\ \!v}</math>
: <math>i^2 = -1 </math>
this path vector can be conveniently written in complex exponential form
: <math> \mathbf{r} = r \widehat{u} e^{i\theta} </math>
and the derivative with respect to angle is
: <math> \frac{d \mathbf{r}}{d\theta} = r \widehat{u} i e^{i\theta} = \mathbf{r} i .</math>

[[File:Exterior calc cross product.svg|right|thumb|The cross product in relation to the exterior product. In red are the unit normal vector, and the "parallel" unit bivector.]]

For example, torque is generally defined as the magnitude of the perpendicular force component times distance, or work per unit angle.  Thus the torque, the rate of change of work {{tmath|1= W }} with respect to angle, due to a force {{tmath|1= F }}, is
: <math> \tau = \frac{dW}{d\theta} = F \cdot \frac{dr}{d\theta} = F \cdot (\mathbf{r} i) .</math>

Rotational quantities are represented in [[vector calculus]] in three dimensions using the [[cross product]].  Together with a choice of an oriented volume form {{tmath|1= I }}, these can be related to the exterior product with its more natural geometric interpretation of such quantities as a bivectors by using the [[Hodge dual|dual]] relationship
: <math>a \times b = -I (a \wedge b) .</math>

Unlike the cross product description of torque, {{tmath|1= \tau = \mathbf{r} \times F }}, the geometric algebra description does not introduce a vector in the normal direction; a vector that does not exist in two and that is not unique in greater than three dimensions. The unit bivector describes the plane and the orientation of the rotation, and the sense of the rotation is relative to the angle between the vectors {{tmath|1= \widehat{u} }} and {{tmath|1= \widehat{\ \!v} }}.