Editing Geometric algebra (section)

=== Geometric product ===
{{see also|Symmetric bilinear form|Exterior algebra}}
[[File:GA parallel and perpendicular vectors.svg|200px|right|thumb|Given two vectors <math>a</math> and {{tmath|1= b }}, if the geometric product <math>ab</math> is{{sfn|ps=|Hestenes|2005}} anticommutative; they are perpendicular (top) because {{tmath|1= a \cdot b = 0 }}, if it is commutative; they are parallel (bottom) because {{tmath|1= a \wedge b = 0 }}.]]
{{multiple image
 | left
 | footer = Geometric interpretation of grade-<math>n</math> elements in a real exterior algebra for <math>n = 0</math> (signed point), <math>1</math> (directed line segment, or vector), <math>2</math> (oriented plane element), <math>3</math> (oriented volume). The exterior product of <math>n</math> vectors can be visualized as any {{tmath|1= n }}-dimensional shape (e.g. {{tmath|1= n }}-[[Parallelepiped#Parallelotope|parallelotope]], {{tmath|1= n }}-[[ellipsoid]]); with magnitude ([[hypervolume]]), and [[Orientation (vector space)|orientation]] defined by that on its {{tmath|1= (n - 1) }}-dimensional boundary and on which side the interior is.{{sfn|ps=|Penrose|2007}}{{sfn|ps=|Wheeler|Misner|Thorne|1973|p=83}}
 | width1 = 220
 | image1 = N vector positive.svg
 | caption1 = Orientation defined by an ordered set of vectors.
 | width2 = 220
 | image2 = N vector negative.svg
 | caption2 = Reversed orientation corresponds to negating the exterior product.
}}
For vectors {{tmath|1= a }} and {{tmath|1= b }}, we may write the geometric product of any two vectors {{tmath|1= a }} and {{tmath|1= b }} as the sum of a symmetric product and an antisymmetric product:
: <math>ab = \frac{1}{2} (ab + ba) + \frac{1}{2} (ab - ba) .</math>

Thus we can define the ''inner product'' of vectors as
: <math>a \cdot b := g(a,b),</math>
so that the symmetric product can be written as
: <math>\frac{1}{2}(ab + ba) = \frac{1}{2} \left((a + b)^2 - a^2 - b^2\right) = a \cdot b .</math>

Conversely, {{tmath|1= g }} is completely determined by the algebra. The antisymmetric part is the exterior product of the two vectors, the product of the contained [[exterior algebra]]:
: <math>a \wedge b := \frac{1}{2}(ab - ba) = -(b \wedge a) .</math>

Then by simple addition:
: <math>ab=a \cdot b + a \wedge b </math> the ungeneralized or vector form of the geometric product.

The inner and exterior products are associated with familiar concepts from standard vector algebra. Geometrically, <math>a</math> and <math>b</math> are [[parallel (geometry)|parallel]] if their geometric product is equal to their inner product, whereas <math>a</math> and <math>b</math> are [[perpendicular]] if their geometric product is equal to their exterior product. In a geometric algebra for which the square of any nonzero vector is positive, the inner product of two vectors can be identified with the [[dot product]] of standard vector algebra. The exterior product of two vectors can be identified with the [[signed area]] enclosed by a [[parallelogram]] the sides of which are the vectors. The [[cross product]] of two vectors in <math>3</math> dimensions with positive-definite quadratic form is closely related to their exterior product.

Most instances of geometric algebras of interest have a nondegenerate quadratic form. If the quadratic form is fully [[nondegenerate quadratic form|degenerate]], the inner product of any two vectors is always zero, and the geometric algebra is then simply an exterior algebra. Unless otherwise stated, this article will treat only nondegenerate geometric algebras.

The exterior product is naturally extended as an associative bilinear binary operator between any two elements of the algebra, satisfying the identities
: <math>\begin{align}
                          1 \wedge a_i &= a_i \wedge 1 = a_i \\
  a_1 \wedge a_2\wedge\cdots\wedge a_r &= \frac{1}{r!}\sum_{\sigma\in\mathfrak{S}_r} \operatorname{sgn}(\sigma) a_{\sigma(1)}a_{\sigma(2)} \cdots a_{\sigma(r)},
\end{align}</math>

where the sum is over all permutations of the indices, with <math>\operatorname{sgn}(\sigma)</math> the [[parity of a permutation|sign of the permutation]], and <math>a_i</math> are vectors (not general elements of the algebra). Since every element of the algebra can be expressed as the sum of products of this form, this defines the exterior product for every pair of elements of the algebra. It follows from the definition that the exterior product forms an [[alternating algebra]].

The equivalent structure equation for Clifford algebra is{{sfn|ps=|Wilmot|1988a|p=2338}}{{sfn|ps=|Wilmot|1988b|p=2346}}
: <math> a_1 a_2 a_3 \dots a_n = \sum^{[\frac{n}2]}_{i=0} \sum_{\mu\in{}\mathcal{C}}
(-1)^k \operatorname{Pf}(a_{\mu_1}\cdot a_{\mu_2},\dots,a_{\mu_{2i-1}} \cdot a_{\mu_{2i}})
a_{\mu_{2i+1}}\land\dots\land a_{\mu_n}</math>
where <math>\operatorname{Pf}(A)</math> is the [[Pfaffian]] of {{tmath|1= A }} and <math display="inline">\mathcal{C} = \binom{n}{2i}</math> provides [[combination]]s, {{tmath|1= \mu }}, of {{tmath|1= n }} indices divided into {{tmath|1= 2i }} and {{tmath|1= n - 2i }} parts and {{tmath|1= k }} is the [[parity (mathematics)|parity]] of the [[combination]].

The Pfaffian provides a metric for the exterior algebra and, as pointed out by Claude Chevalley, Clifford algebra reduces to the exterior algebra with a zero quadratic form.{{sfn|ps=|Chevalley|1991}} The role the Pfaffian plays can be understood from a geometric viewpoint by developing Clifford algebra from [[simplex|simplices]].{{sfn|ps=|Wilmot|2023}} This derivation provides a better connection between [[Pascal's triangle]] and [[simplex|simplices]] because it provides an interpretation of the first column of ones.