The geometric product

Given two vectors ${\displaystyle a}$ and ${\displaystyle b}$, if the geometric product ${\displaystyle ab}$ is^[10] anticommutative; they are perpendicular (top) because ${\displaystyle a\cdot b=0}$, if it is commutative; they are parallel (bottom) because ${\displaystyle a\wedge b=0}$.

Orientation defined by an ordered set of vectors.

Reversed orientation corresponds to negating the exterior product.

Geometric interpretation of grade-${\displaystyle n}$ elements in a real exterior algebra for ${\displaystyle n=0}$ (signed point), ${\displaystyle 1}$ (directed line segment, or vector), ${\displaystyle 2}$ (oriented plane element), ${\displaystyle 3}$ (oriented volume). The exterior product of ${\displaystyle n}$ vectors can be visualized as any ${\displaystyle n}$-dimensional shape (e.g. ${\displaystyle n}$-parallelotope, ${\displaystyle n}$-ellipsoid); with magnitude (hypervolume), and orientation defined by that on its ${\displaystyle (n-1)}$-dimensional boundary and on which side the interior is.^[11][12]

For vectors ${\displaystyle a}$ and ${\displaystyle b}$, we may write the geometric product of any two vectors ${\displaystyle a}$ and ${\displaystyle b}$ as the sum of a symmetric product and an antisymmetric product:

${\displaystyle ab={\frac {1}{2}}(ab+ba)+{\frac {1}{2}}(ab-ba)}$

Thus we can define the inner product^{[lower-alpha 3]} of vectors as

${\displaystyle a\cdot b:=g(a,b),}$

so that the symmetric product can be written as

${\displaystyle {\frac {1}{2}}(ab+ba)={\frac {1}{2}}\left((a+b)^{2}-a^{2}-b^{2}\right)=a\cdot b}$

Conversely, ${\displaystyle g}$ is completely determined by the algebra. The antisymmetric part is the exterior product of the two vectors, the product of the contained exterior algebra:

${\displaystyle a\wedge b:={\frac {1}{2}}(ab-ba)=-(b\wedge a)}$

Then by simple addition:

${\displaystyle ab=a\cdot b+a\wedge b}$ the ungeneralized or vector form of the geometric product.

The inner and exterior products are associated with familiar concepts from standard vector algebra. Geometrically, ${\displaystyle a}$ and ${\displaystyle b}$ are parallel if their geometric product is equal to their inner product, whereas ${\displaystyle a}$ and ${\displaystyle b}$ 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 ${\displaystyle 3}$ 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 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

${\displaystyle {\begin{aligned}1\wedge a_{i}&=a_{i}\wedge 1=a_{i}\\a_{1}\wedge a_{2}\wedge \cdots \wedge a_{r}&={\frac {1}{r!}}\sum _{\sigma \in {\mathfrak {S}}_{r}}\operatorname {sgn} (\sigma )a_{\sigma (1)}a_{\sigma (2)}\cdots a_{\sigma (r)},\end{aligned}}}$

where the sum is over all permutations of the indices, with ${\displaystyle \operatorname {sgn} (\sigma )}$ the sign of the permutation, and ${\displaystyle a_{i}}$ 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 ^[13][14]

${\displaystyle a_{1}a_{2}a_{3}\dots a_{n}=\sum _{i=0}^{[{\frac {n}{2}}]}\sum _{\mu \in {}{\mathcal {C}}}(-1)^{k}Pf(a_{\mu _{1}}\cdot a_{\mu _{2}},\dots ,a_{\mu _{2i-1}}\cdot a_{\mu _{2i}})a_{\mu _{2i+1}}\land \dots \land a_{\mu _{n}}}$

where ${\displaystyle Pf(A)}$ is the Pfaffian of A and ${\displaystyle {\mathcal {C}}={\binom {n}{2i}}}$ provides combinations, ${\displaystyle \mu }$, of n indicies divided into 2i and n-2i parts and k is the 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.^[15] The role the Pfaffian plays can be understood from a geometric viewpoint by developing Clifford algebra from simplices.^[16] This derivation provides a better connection between Pascal's triangle and simplices because it provides an interpretation of the first column of ones.

Blades, grades, and canonical basis

A multivector that is the exterior product of ${\displaystyle r}$ linearly independent vectors is called a blade, and is said to be of grade ${\displaystyle r}$.^{[lower-alpha 5]} A multivector that is the sum of blades of grade ${\displaystyle r}$ is called a (homogeneous) multivector of grade ${\displaystyle r}$. From the axioms, with closure, every multivector of the geometric algebra is a sum of blades.

Consider a set of ${\displaystyle r}$ linearly independent vectors ${\displaystyle \{a_{1},\ldots ,a_{r}\}}$ spanning an ${\displaystyle r}$-dimensional subspace of the vector space. With these, we can define a real symmetric matrix (in the same way as a Gramian matrix)

${\displaystyle [\mathbf {A} ]_{ij}=a_{i}\cdot a_{j}}$

By the spectral theorem, ${\displaystyle \mathbf {A} }$ can be diagonalized to diagonal matrix ${\displaystyle \mathbf {D} }$ by an orthogonal matrix ${\displaystyle \mathbf {O} }$ via

${\displaystyle \sum _{k,l}[\mathbf {O} ]_{ik}[\mathbf {A} ]_{kl}[\mathbf {O} ^{\mathrm {T} }]_{lj}=\sum _{k,l}[\mathbf {O} ]_{ik}[\mathbf {O} ]_{jl}[\mathbf {A} ]_{kl}=[\mathbf {D} ]_{ij}}$

Define a new set of vectors ${\displaystyle \{e_{1},\ldots ,e_{r}\}}$, known as orthogonal basis vectors, to be those transformed by the orthogonal matrix:

${\displaystyle e_{i}=\sum _{j}[\mathbf {O} ]_{ij}a_{j}}$

Since orthogonal transformations preserve inner products, it follows that ${\displaystyle e_{i}\cdot e_{j}=[\mathbf {D} ]_{ij}}$ and thus the ${\displaystyle \{e_{1},\ldots ,e_{r}\}}$ are perpendicular. In other words, the geometric product of two distinct vectors ${\displaystyle e_{i}\neq e_{j}}$ is completely specified by their exterior product, or more generally

${\displaystyle {\begin{array}{rl}e_{1}e_{2}\cdots e_{r}&=e_{1}\wedge e_{2}\wedge \cdots \wedge e_{r}\\&=\left(\sum _{j}[\mathbf {O} ]_{1j}a_{j}\right)\wedge \left(\sum _{j}[\mathbf {O} ]_{2j}a_{j}\right)\wedge \cdots \wedge \left(\sum _{j}[\mathbf {O} ]_{rj}a_{j}\right)\\&=(\det \mathbf {O} )a_{1}\wedge a_{2}\wedge \cdots \wedge a_{r}\end{array}}}$

Therefore, every blade of grade ${\displaystyle r}$ can be written as the exterior product of ${\displaystyle r}$ vectors. More generally, if a degenerate geometric algebra is allowed, then the orthogonal matrix is replaced by a block matrix that is orthogonal in the nondegenerate block, and the diagonal matrix has zero-valued entries along the degenerate dimensions. If the new vectors of the nondegenerate subspace are normalized according to

${\displaystyle {\hat {e}}_{i}={\frac {1}{\sqrt {|e_{i}\cdot e_{i}|}}}e_{i},}$

then these normalized vectors must square to ${\displaystyle +1}$ or ${\displaystyle -1}$. By Sylvester's law of inertia, the total number of ${\displaystyle +1}$s and the total number of ${\displaystyle -1}$s along the diagonal matrix is invariant. By extension, the total number ${\displaystyle p}$ of these vectors that square to ${\displaystyle +1}$ and the total number ${\displaystyle q}$ that square to ${\displaystyle -1}$ is invariant. (The total number of basis vectors that square to zero is also invariant, and may be nonzero if the degenerate case is allowed.) We denote this algebra ${\displaystyle {\mathcal {G}}(p,q)}$. For example, ${\displaystyle {\mathcal {G}}(3,0)}$ models three-dimensional Euclidean space, ${\displaystyle {\mathcal {G}}(1,3)}$ relativistic spacetime and ${\displaystyle {\mathcal {G}}(4,1)}$ a conformal geometric algebra of a three-dimensional space.

The set of all possible products of ${\displaystyle n}$ orthogonal basis vectors with indices in increasing order, including ${\displaystyle 1}$ as the empty product, forms a basis for the entire geometric algebra (an analogue of the PBW theorem). For example, the following is a basis for the geometric algebra ${\displaystyle {\mathcal {G}}(3,0)}$:

${\displaystyle \{1,e_{1},e_{2},e_{3},e_{1}e_{2},e_{2}e_{3},e_{3}e_{1},e_{1}e_{2}e_{3}\}}$

A basis formed this way is called a canonical basis for the geometric algebra, and any other orthogonal basis for ${\displaystyle V}$ will produce another canonical basis. Each canonical basis consists of ${\displaystyle 2^{n}}$ elements. Every multivector of the geometric algebra can be expressed as a linear combination of the canonical basis elements. If the canonical basis elements are ${\displaystyle \{B_{i}\mid i\in S\}}$ with ${\displaystyle S}$ being an index set, then the geometric product of any two multivectors is

${\displaystyle \left(\sum _{i}\alpha _{i}B_{i}\right)\left(\sum _{j}\beta _{j}B_{j}\right)=\sum _{i,j}\alpha _{i}\beta _{j}B_{i}B_{j}.}$

The terminology "${\displaystyle k}$-vector" is often encountered to describe multivectors containing elements of only one grade. In higher dimensional space, some such multivectors are not blades (cannot be factored into the exterior product of ${\displaystyle k}$ vectors). By way of example, ${\displaystyle e_{1}\wedge e_{2}+e_{3}\wedge e_{4}}$ in ${\displaystyle {\mathcal {G}}(4,0)}$ cannot be factored; typically, however, such elements of the algebra do not yield to geometric interpretation as objects, although they may represent geometric quantities such as rotations. Only ${\displaystyle 0,1,(n-1)}$ and ${\displaystyle n}$-vectors are always blades in ${\displaystyle n}$-space.

Grade projection

Using an orthogonal basis, a graded vector space structure can be established. Elements of the geometric algebra that are scalar multiples of ${\displaystyle 1}$ are grade-${\displaystyle 0}$ blades and are called scalars. Multivectors that are in the span of ${\displaystyle \{e_{1},\ldots ,e_{n}\}}$ are grade-${\displaystyle 1}$ blades and are the ordinary vectors. Multivectors in the span of ${\displaystyle \{e_{i}e_{j}\mid 1\leq i<j\leq n\}}$ are grade-${\displaystyle 2}$ blades and are the bivectors. This terminology continues through to the last grade of ${\displaystyle n}$-vectors. Alternatively, grade-${\displaystyle n}$ blades are called pseudoscalars, grade-${\displaystyle (n-1)}$ blades pseudovectors, etc. Many of the elements of the algebra are not graded by this scheme since they are sums of elements of differing grade. Such elements are said to be of mixed grade. The grading of multivectors is independent of the basis chosen originally.

This is a grading as a vector space, but not as an algebra. Because the product of an ${\displaystyle r}$-blade and an ${\displaystyle s}$-blade is contained in the span of ${\displaystyle 0}$ through ${\displaystyle r+s}$-blades, the geometric algebra is a filtered algebra.

A multivector ${\displaystyle A}$ may be decomposed with the grade-projection operator ${\displaystyle \langle A\rangle _{r}}$, which outputs the grade-${\displaystyle r}$ portion of ${\displaystyle A}$. As a result:

${\displaystyle A=\sum _{r=0}^{n}\langle A\rangle _{r}}$

As an example, the geometric product of two vectors ${\displaystyle ab=a\cdot b+a\wedge b=\langle ab\rangle _{0}+\langle ab\rangle _{2}}$ since ${\displaystyle \langle ab\rangle _{0}=a\cdot b}$ and ${\displaystyle \langle ab\rangle _{2}=a\wedge b}$ and ${\displaystyle \langle ab\rangle _{i}=0}$, for ${\displaystyle i}$ other than ${\displaystyle 0}$ and ${\displaystyle 2}$.

The decomposition of a multivector ${\displaystyle A}$ may also be split into those components that are even and those that are odd:

${\displaystyle A^{+}=\langle A\rangle _{0}+\langle A\rangle _{2}+\langle A\rangle _{4}+\cdots }$

${\displaystyle A^{-}=\langle A\rangle _{1}+\langle A\rangle _{3}+\langle A\rangle _{5}+\cdots }$

This is the result of forgetting structure from a ${\displaystyle \mathrm {Z} }$-graded vector space to ${\displaystyle \mathrm {Z} _{2}}$-graded vector space. The geometric product respects this coarser grading. Thus in addition to being a ${\displaystyle \mathrm {Z} _{2}}$-graded vector space, the geometric algebra is a ${\displaystyle \mathrm {Z} _{2}}$-graded algebra or superalgebra.

Restricting to the even part, the product of two even elements is also even. This means that the even multivectors defines an even subalgebra. The even subalgebra of an ${\displaystyle n}$-dimensional geometric algebra is isomorphic (without preserving either filtration or grading) to a full geometric algebra of ${\displaystyle (n-1)}$ dimensions. Examples include ${\displaystyle {\mathcal {G}}^{+}(2,0)\cong {\mathcal {G}}(0,1)}$ and ${\displaystyle {\mathcal {G}}^{+}(1,3)\cong {\mathcal {G}}(3,0)}$.

Representation of subspaces

Geometric algebra represents subspaces of ${\displaystyle V}$ as blades, and so they coexist in the same algebra with vectors from ${\displaystyle V}$. A ${\displaystyle k}$-dimensional subspace ${\displaystyle W}$ of ${\displaystyle V}$ is represented by taking an orthogonal basis ${\displaystyle \{b_{1},b_{2},\ldots ,b_{k}\}}$ and using the geometric product to form the blade ${\displaystyle D=b_{1}b_{2}\cdots b_{k}}$. There are multiple blades representing ${\displaystyle W}$; all those representing ${\displaystyle W}$ are scalar multiples of ${\displaystyle D}$. These blades can be separated into two sets: positive multiples of ${\displaystyle D}$ and negative multiples of ${\displaystyle D}$. The positive multiples of ${\displaystyle D}$ are said to have the same orientation as ${\displaystyle D}$, and the negative multiples the opposite orientation.

Blades are important since geometric operations such as projections, rotations and reflections depend on the factorability via the exterior product that (the restricted class of) ${\displaystyle n}$-blades provide but that (the generalized class of) grade-${\displaystyle n}$ multivectors do not when ${\displaystyle n\geq 4}$.

Unit pseudoscalars

Unit pseudoscalars are blades that play important roles in GA. A unit pseudoscalar for a non-degenerate subspace ${\displaystyle W}$ of ${\displaystyle V}$ is a blade that is the product of the members of an orthonormal basis for ${\displaystyle W}$. It can be shown that if ${\displaystyle I}$ and ${\displaystyle I'}$ are both unit pseudoscalars for ${\displaystyle W}$, then ${\displaystyle I=\pm I'}$ and ${\displaystyle I^{2}=\pm 1}$. If one doesn't choose an orthonormal basis for ${\displaystyle W}$, then the Plücker embedding gives a vector in the exterior algebra but only up to scaling. Using the vector space isomorphism between the geometric algebra and exterior algebra, this gives the equivalence class of ${\displaystyle \alpha I}$ for all ${\displaystyle \alpha \neq 0}$. Orthonormality gets rid of this ambiguity except for the signs above.

Suppose the geometric algebra ${\displaystyle {\mathcal {G}}(n,0)}$ with the familiar positive definite inner product on ${\displaystyle \mathbb {R} ^{n}}$ is formed. Given a plane (two-dimensional subspace) of ${\displaystyle \mathbb {R} ^{n}}$, one can find an orthonormal basis ${\displaystyle \{b_{1},b_{2}\}}$ spanning the plane, and thus find a unit pseudoscalar ${\displaystyle I=b_{1}b_{2}}$ representing this plane. The geometric product of any two vectors in the span of ${\displaystyle b_{1}}$ and ${\displaystyle b_{2}}$ lies in ${\displaystyle \{\alpha _{0}+\alpha _{1}I\mid \alpha _{i}\in \mathbb {R} \}}$, that is, it is the sum of a ${\displaystyle 0}$-vector and a ${\displaystyle 2}$-vector.

By the properties of the geometric product, ${\displaystyle I^{2}=b_{1}b_{2}b_{1}b_{2}=-b_{1}b_{2}b_{2}b_{1}=-1}$. The resemblance to the imaginary unit is not incidental: the subspace ${\displaystyle \{\alpha _{0}+\alpha _{1}I\mid \alpha _{i}\in \mathbb {R} \}}$ is ${\displaystyle \mathbb {R} }$-algebra isomorphic to the complex numbers. In this way, a copy of the complex numbers is embedded in the geometric algebra for each two-dimensional subspace of ${\displaystyle V}$ on which the quadratic form is definite.

It is sometimes possible to identify the presence of an imaginary unit in a physical equation. Such units arise from one of the many quantities in the real algebra that square to ${\displaystyle -1}$, and these have geometric significance because of the properties of the algebra and the interaction of its various subspaces.

In ${\displaystyle {\mathcal {G}}(3,0)}$, a further familiar case occurs. Given a canonical basis consisting of orthonormal vectors ${\displaystyle e_{i}}$ of ${\displaystyle V}$, the set of all ${\displaystyle 2}$-vectors is spanned by

${\displaystyle \{e_{3}e_{2},e_{1}e_{3},e_{2}e_{1}\}.}$

Labelling these ${\displaystyle i}$, ${\displaystyle j}$ and ${\displaystyle k}$ (momentarily deviating from our uppercase convention), the subspace generated by ${\displaystyle 0}$-vectors and ${\displaystyle 2}$-vectors is exactly ${\displaystyle \{\alpha _{0}+i\alpha _{1}+j\alpha _{2}+k\alpha _{3}\mid \alpha _{i}\in \mathbb {R} \}}$. This set is seen to be the even subalgebra of ${\displaystyle {\mathcal {G}}(3,0)}$, and furthermore is isomorphic as an ${\displaystyle \mathbb {R} }$-algebra to the quaternions, another important algebraic system.

Extensions of the inner and exterior products

It is common practice to extend the exterior product on vectors to the entire algebra. This may be done through the use of the above mentioned grade projection operator:

${\displaystyle C\wedge D:=\sum _{r,s}\langle \langle C\rangle _{r}\langle D\rangle _{s}\rangle _{r+s}}$     (the exterior product)

This generalization is consistent with the above definition involving antisymmetrization. Another generalization related to the exterior product is the commutator product:

${\displaystyle C\times D:={\tfrac {1}{2}}(CD-DC)}$     (the commutator product)

The regressive product (usually referred to as the "meet") is the dual of the exterior product (or "join" in this context).^{[lower-alpha 6]} The dual specification of elements permits, for blades ${\displaystyle A}$ and ${\displaystyle B}$, the intersection (or meet) where the duality is to be taken relative to the smallest grade blade containing both ${\displaystyle A}$ and ${\displaystyle B}$ (the join).^[18]

${\displaystyle C\vee D:=((CI^{-1})\wedge (DI^{-1}))I}$

with ${\displaystyle I}$ the unit pseudoscalar of the algebra. The regressive product, like the exterior product, is associative.^[19]

The inner product on vectors can also be generalized, but in more than one non-equivalent way. The paper (Dorst 2002) gives a full treatment of several different inner products developed for geometric algebras and their interrelationships, and the notation is taken from there. Many authors use the same symbol as for the inner product of vectors for their chosen extension (e.g. Hestenes and Perwass). No consistent notation has emerged.

Among these several different generalizations of the inner product on vectors are:

${\displaystyle C\;\rfloor \;D:=\sum _{r,s}\langle \langle C\rangle _{r}\langle D\rangle _{s}\rangle _{s-r}}$   (the left contraction)

${\displaystyle C\;\lfloor \;D:=\sum _{r,s}\langle \langle C\rangle _{r}\langle D\rangle _{s}\rangle _{r-s}}$   (the right contraction)

${\displaystyle C*D:=\sum _{r,s}\langle \langle C\rangle _{r}\langle D\rangle _{s}\rangle _{0}}$   (the scalar product)

${\displaystyle C\bullet D:=\sum _{r,s}\langle \langle C\rangle _{r}\langle D\rangle _{s}\rangle _{|s-r|}}$   (the "(fat) dot" product)^{[lower-alpha 7]}

Dorst (2002) makes an argument for the use of contractions in preference to Hestenes's inner product; they are algebraically more regular and have cleaner geometric interpretations. A number of identities incorporating the contractions are valid without restriction of their inputs. For example,

${\displaystyle C\;\rfloor \;D=(C\wedge (DI^{-1}))I}$

${\displaystyle C\;\lfloor \;D=I((I^{-1}C)\wedge D)}$

${\displaystyle (A\wedge B)*C=A*(B\;\rfloor \;C)}$

${\displaystyle C*(B\wedge A)=(C\;\lfloor \;B)*A}$

${\displaystyle A\;\rfloor \;(B\;\rfloor \;C)=(A\wedge B)\;\rfloor \;C}$

${\displaystyle (A\;\rfloor \;B)\;\lfloor \;C=A\;\rfloor \;(B\;\lfloor \;C).}$

Benefits of using the left contraction as an extension of the inner product on vectors include that the identity ${\displaystyle ab=a\cdot b+a\wedge b}$ is extended to ${\displaystyle aB=a\;\rfloor \;B+a\wedge B}$ for any vector ${\displaystyle a}$ and multivector ${\displaystyle B}$, and that the projection operation ${\displaystyle {\mathcal {P}}_{b}(a)=(a\cdot b^{-1})b}$ is extended to ${\displaystyle {\mathcal {P}}_{B}(A)=(A\;\rfloor \;B^{-1})\;\rfloor \;B}$ for any blade ${\displaystyle B}$ and any multivector ${\displaystyle A}$ (with a minor modification to accommodate null ${\displaystyle B}$, given below).

Dual basis

Let ${\displaystyle \{e_{1},\ldots ,e_{n}\}}$ be a basis of ${\displaystyle V}$, i.e. a set of ${\displaystyle n}$ linearly independent vectors that span the ${\displaystyle n}$-dimensional vector space ${\displaystyle V}$. The basis that is dual to ${\displaystyle \{e_{1},\ldots ,e_{n}\}}$ is the set of elements of the dual vector space ${\displaystyle V^{*}}$ that forms a biorthogonal system with this basis, thus being the elements denoted ${\displaystyle \{e^{1},\ldots ,e^{n}\}}$ satisfying

${\displaystyle e^{i}\cdot e_{j}=\delta ^{i}{}_{j},}$

where ${\displaystyle \delta }$ is the Kronecker delta.

Given a nondegenerate quadratic form on ${\displaystyle V}$, ${\displaystyle V^{*}}$ becomes naturally identified with ${\displaystyle V}$, and the dual basis may be regarded as elements of ${\displaystyle V}$, but are not in general the same set as the original basis.

Given further a GA of ${\displaystyle V}$, let

${\displaystyle I=e_{1}\wedge \cdots \wedge e_{n}}$

be the pseudoscalar (which does not necessarily square to ${\displaystyle \pm 1}$) formed from the basis ${\displaystyle \{e_{1},\ldots ,e_{n}\}}$. The dual basis vectors may be constructed as

${\displaystyle e^{i}=(-1)^{i-1}(e_{1}\wedge \cdots \wedge {\check {e}}_{i}\wedge \cdots \wedge e_{n})I^{-1},}$

where the ${\displaystyle {\check {e}}_{i}}$ denotes that the ${\displaystyle i}$th basis vector is omitted from the product.

A dual basis is also known as a reciprocal basis or reciprocal frame.

A major usage of a dual basis is to separate vectors into components. Given a vector ${\displaystyle a}$, scalar components ${\displaystyle a^{i}}$ can be defined as

${\displaystyle a^{i}=a\cdot e^{i}\ ,}$

in terms of which ${\displaystyle a}$ can be separated into vector components as

${\displaystyle a=\sum _{i}a^{i}e_{i}\ .}$

We can also define scalar components ${\displaystyle a_{i}}$ as

${\displaystyle a_{i}=a\cdot e_{i}\ ,}$

in terms of which ${\displaystyle a}$ can be separated into vector components in terms of the dual basis as

${\displaystyle a=\sum _{i}a_{i}e^{i}\ .}$

A dual basis as defined above for the vector subspace of a geometric algebra can be extended to cover the entire algebra.^[20] For compactness, we'll use a single capital letter to represent an ordered set of vector indices. I.e., writing

${\displaystyle J=(j_{1},\dots ,j_{n})\ ,}$

where ${\displaystyle j_{1}<j_{2}<\dots <j_{n},}$ we can write a basis blade as

${\displaystyle e_{J}=e_{j_{1}}\wedge e_{j_{2}}\wedge \cdots \wedge e_{j_{n}}\ .}$

The corresponding reciprocal blade has the indices in opposite order:

${\displaystyle e^{J}=e^{j_{n}}\wedge \cdots \wedge e^{j_{2}}\wedge e^{j_{1}}\ .}$

Similar to the case above with vectors, it can be shown that

${\displaystyle e^{J}*e_{K}=\delta _{K}^{J}\ ,}$

where ${\displaystyle *}$ is the scalar product.

With ${\displaystyle A}$ a multivector, we can define scalar components as ^[21]

${\displaystyle A^{ij\cdots k}=(e^{k}\wedge \cdots \wedge e^{j}\wedge e^{i})*A\ ,}$

in terms of which ${\displaystyle A}$ can be separated into component blades as

${\displaystyle A=\sum _{i<j<\cdots <k}A^{ij\cdots k}e_{i}\wedge e_{j}\wedge \cdots \wedge e_{k}\ .}$

We can alternatively define scalar components

${\displaystyle A_{ij\cdots k}=(e_{k}\wedge \cdots \wedge e_{j}\wedge e_{i})*A\ ,}$

in terms of which ${\displaystyle A}$ can be separated into component blades as

${\displaystyle A=\sum _{i<j<\cdots <k}A_{ij\cdots k}e^{i}\wedge e^{j}\wedge \cdots \wedge e^{k}\ .}$

Linear functions

Although a versor is easier to work with because it can be directly represented in the algebra as a multivector, versors are a subgroup of linear functions on multivectors, which can still be used when necessary. The geometric algebra of an ${\displaystyle n}$-dimensional vector space is spanned by a basis of ${\displaystyle 2^{n}}$ elements. If a multivector is represented by a ${\displaystyle 2^{n}\times 1}$ real column matrix of coefficients of a basis of the algebra, then all linear transformations of the multivector can be expressed as the matrix multiplication by a ${\displaystyle 2^{n}\times 2^{n}}$ real matrix. However, such a general linear transformation allows arbitrary exchanges among grades, such as a "rotation" of a scalar into a vector, which has no evident geometric interpretation.

A general linear transformation from vectors to vectors is of interest. With the natural restriction to preserving the induced exterior algebra, the outermorphism of the linear transformation is the unique^{[lower-alpha 8]} extension of the versor. If ${\displaystyle f}$ is a linear function that maps vectors to vectors, then its outermorphism is the function that obeys the rule

${\displaystyle {\underline {\mathsf {f}}}(a_{1}\wedge a_{2}\wedge \cdots \wedge a_{r})=f(a_{1})\wedge f(a_{2})\wedge \cdots \wedge f(a_{r})}$

for a blade, extended to the whole algebra through linearity.

Vector space model

${\displaystyle {\mathcal {G}}(3,0)}$ may be considered as an extension or completion of vector algebra.

The even subalgebra of ${\displaystyle {\mathcal {G}}(2,0)}$ is isomorphic to the complex numbers, as may be seen by writing a vector ${\displaystyle P}$ in terms of its components in an orthonormal basis and left multiplying by the basis vector ${\displaystyle e_{1}}$, yielding

${\displaystyle Z=e_{1}P=e_{1}(xe_{1}+ye_{2})=x(1)+y(e_{1}e_{2}),}$

where we identify ${\displaystyle i\mapsto e_{1}e_{2}}$ since

${\displaystyle (e_{1}e_{2})^{2}=e_{1}e_{2}e_{1}e_{2}=-e_{1}e_{1}e_{2}e_{2}=-1.}$

Similarly, the even subalgebra of ${\displaystyle {\mathcal {G}}(3,0)}$ with basis ${\displaystyle \{1,e_{2}e_{3},e_{3}e_{1},e_{1}e_{2}\}}$ is isomorphic to the quaternions as may be seen by identifying ${\displaystyle i\mapsto -e_{2}e_{3}}$, ${\displaystyle j\mapsto -e_{3}e_{1}}$ and ${\displaystyle k\mapsto -e_{1}e_{2}}$.

Every associative algebra has a matrix representation; replacing the three Cartesian basis vectors by the Pauli matrices gives a representation of ${\displaystyle {\mathcal {G}}(3,0)}$:

${\displaystyle {\begin{aligned}e_{1}=\sigma _{1}=\sigma _{x}&={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\\e_{2}=\sigma _{2}=\sigma _{y}&={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}\\e_{3}=\sigma _{3}=\sigma _{z}&={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}\,.\end{aligned}}}$

Dotting the "Pauli vector" (a dyad):

${\displaystyle \sigma =\sigma _{1}e_{1}+\sigma _{2}e_{2}+\sigma _{3}e_{3}}$ with arbitrary vectors ${\displaystyle a}$ and ${\displaystyle b}$ and multiplying through gives:

${\displaystyle (\sigma \cdot a)(\sigma \cdot b)=a\cdot b+a\wedge b}$ (Equivalently, by inspection, ${\displaystyle a\cdot b+i\sigma \cdot (a\times b)}$)

Spacetime model

In physics, the main applications are the geometric algebra of Minkowski 3+1 spacetime, ${\displaystyle {\mathcal {G}}(1,3)}$, called spacetime algebra (STA),^[3] or less commonly, ${\displaystyle {\mathcal {G}}(3,0)}$, interpreted the algebra of physical space (APS).

While in STA, points of spacetime are represented simply by vectors, in APS, points of ${\displaystyle (3+1)}$-dimensional spacetime are instead represented by paravectors, a three-dimensional vector (space) plus a one-dimensional scalar (time).

In spacetime algebra the electromagnetic field tensor has a bivector representation ${\displaystyle {F}=({E}+ic{B})\gamma _{0}}$.^[23] Here, the ${\displaystyle i=\gamma _{0}\gamma _{1}\gamma _{2}\gamma _{3}}$ is the unit pseudoscalar (or four-dimensional volume element), ${\displaystyle \gamma _{0}}$ is the unit vector in time direction, and ${\displaystyle E}$ and ${\displaystyle B}$ are the classic electric and magnetic field vectors (with a zero time component). Using the four-current ${\displaystyle {J}}$, Maxwell's equations then become

Formulation	Homogeneous equations	Non-homogeneous equations
Fields	${\displaystyle DF=\mu _{0}J}$
	${\displaystyle D\wedge F=0}$	${\displaystyle D~\rfloor ~F=\mu _{0}J}$
Potentials (any gauge)	${\displaystyle F=D\wedge A}$	${\displaystyle D~\rfloor ~(D\wedge A)=\mu _{0}J}$
Potentials (Lorenz gauge)	${\displaystyle F=DA}$ ${\displaystyle D~\rfloor ~A=0}$	${\displaystyle D^{2}A=\mu _{0}J}$

In geometric calculus, juxtaposition of vectors such as in ${\displaystyle DF}$ indicate the geometric product and can be decomposed into parts as ${\displaystyle DF=D~\rfloor ~F+D\wedge F}$. Here ${\displaystyle D}$ is the covector derivative in any spacetime and reduces to ${\displaystyle \nabla }$ in flat spacetime. Where ${\displaystyle \bigtriangledown }$ plays a role in Minkowski ${\displaystyle 4}$-spacetime which is synonymous to the role of ${\displaystyle \nabla }$ in Euclidean ${\displaystyle 3}$-space and is related to the d'Alembertian by ${\displaystyle \Box =\bigtriangledown ^{2}}$. Indeed, given an observer represented by a future pointing timelike vector ${\displaystyle \gamma _{0}}$ we have

${\displaystyle \gamma _{0}\cdot \bigtriangledown ={\frac {1}{c}}{\frac {\partial }{\partial t}}}$

${\displaystyle \gamma _{0}\wedge \bigtriangledown =\nabla }$

Boosts in this Lorentzian metric space have the same expression ${\displaystyle e^{\beta }}$ as rotation in Euclidean space, where ${\displaystyle {\beta }}$ is the bivector generated by the time and the space directions involved, whereas in the Euclidean case it is the bivector generated by the two space directions, strengthening the "analogy" to almost identity.

The Dirac matrices are a representation of ${\displaystyle {\mathcal {G}}(1,3)}$, showing the equivalence with matrix representations used by physicists.

Homogeneous models

Homogeneous models generally refer to a projective representation in which the elements of the one-dimensional subspaces of a vector space represent points of a geometry.

In a geometric algebra of a space of ${\displaystyle n}$ dimensions, the rotors represent a set of transformations with ${\displaystyle n(n-1)/2}$ degrees of freedom, corresponding to rotations – for example, ${\displaystyle 3}$ when ${\displaystyle n=3}$ and ${\displaystyle 6}$ when ${\displaystyle n=4.}$ Geometric algebra is often used to model a projective space, i.e. as a homogeneous model: a point, line, plane, etc. is represented by an equivalence class of elements of the algebra that differ by an invertible scalar factor.

The rotors in a space of dimension ${\displaystyle n+1}$ have $n(n-1)/2+n$ degrees of freedom, the same as the number of degrees of freedom in the rotations and translations combined for an ${\displaystyle n}$-dimensional space.

This is the case in Projective Geometric Algebra (PGA), which is used^[24][25][26] to represent Euclidean isometries in Euclidean geometry (thereby covering the large majority of engineering applications of geometry). In this model, a degenerate dimension is added to the three Euclidean dimensions to form the algebra ${\displaystyle {\mathcal {G}}(3,0,1).}$ With a suitable identification of subspaces to represent points, lines and planes, the versors of this algebra represent all proper Euclidean isometries, which are always screw motions in 3-dimensional space, along with all improper Euclidean isometries, which includes reflections, rotoreflections, transflections, and point reflections.

PGA combines ${\displaystyle {\mathcal {G}}(3,0,1)}$ with a dual operator to obtain meet, join, distance, and angle formulae. Depending on the author,^[27][28] this could mean the Hodge star or the projective dual, though both result in identical equations being derived, albeit with different notation. In effect, the dual switches basis vectors that are present and absent in the expression of each term of the algebraic representation. For example, in the PGA or 3-dimensional space, the dual of the line ${\displaystyle {\boldsymbol {e}}_{12}}$ is the line ${\displaystyle {\boldsymbol {e}}_{03}}$, because ${\displaystyle {\boldsymbol {e}}_{0}}$ and ${\displaystyle {\boldsymbol {e}}_{3}}$ are basis elements that are not contained in ${\displaystyle {\boldsymbol {e}}_{12}}$ but are contained in ${\displaystyle {\boldsymbol {e}}_{03}}$. In the PGA of 2-dimensional space, the dual of ${\displaystyle {\boldsymbol {e}}_{12}}$ is ${\displaystyle {\boldsymbol {e}}_{0}}$, since there is no ${\displaystyle {\boldsymbol {e}}_{3}}$ element.

PGA is a widely used system that combines geometric algebra with homogeneous representations in geometry, but there exist several other such systems. The conformal model discussed below is homogeneous, as is "Conic Geometric Algebra",^[29] and see Plane-based geometric algebra for discussion of homogeneous models of elliptic and hyperbolic geometry compared with the euclidean geometry derived from PGA.

Conformal model

Working within GA, Euclidean space ${\displaystyle {\mathcal {E}}^{3}}$ (along with a conformal point at infinity) is embedded projectively in the CGA ${\displaystyle {\mathcal {G}}(4,1)}$ via the identification of Euclidean points with 1D subspaces in the 4D null cone of the 5D CGA vector subspace. This allows all conformal transformations to be performed as rotations and reflections and is covariant, extending incidence relations of projective geometry to circles and spheres.

Specifically, we add orthogonal basis vectors ${\displaystyle e_{+}}$ and ${\displaystyle e_{-}}$ such that ${\displaystyle e_{+}^{2}=+1}$ and ${\displaystyle e_{-}^{2}=-1}$ to the basis of the vector space that generates ${\displaystyle {\mathcal {G}}(3,0)}$ and identify null vectors

${\displaystyle n_{\infty }=e_{-}+e_{+}}$ as a conformal point at infinity (see Compactification) and

${\displaystyle n_{\text{o}}={\tfrac {1}{2}}(e_{-}-e_{+})}$ as the point at the origin, giving

${\displaystyle n_{\infty }\cdot n_{\text{o}}=-1}$.

This procedure has some similarities to the procedure for working with homogeneous coordinates in projective geometry and in this case allows the modeling of Euclidean transformations of ${\displaystyle \mathbf {R} ^{3}}$ as orthogonal transformations of a subset of ${\displaystyle \mathbf {R} ^{4,1}}$.

A fast changing and fluid area of GA, CGA is also being investigated for applications to relativistic physics.

Models for projective transformation

Two potential candidates are currently under investigation as the foundation for affine and projective geometry in three dimensions ${\displaystyle {\mathcal {R}}(3,3)}$^[30]and ${\displaystyle {\mathcal {R}}(4,4)}$^[31] which includes representations for shears and non-uniform scaling, as well as quadric surfaces and conic sections.

A new research model, Quadric Conformal Geometric Algebra (QCGA) ${\displaystyle {\mathcal {R}}(9,6)}$ is an extension of CGA, dedicated to quadric surfaces. The idea is to represent the objects in low dimensional subspaces of the algebra. QCGA is capable of constructing quadric surfaces either using control points or implicit equations. Moreover, QCGA can compute the intersection of quadric surfaces as well as the surface tangent and normal vectors at a point that lies in the quadric surface.^[32]

Projection and rejection

In 3D space, a bivector ${\displaystyle a\land b}$ defines a 2D plane subspace (light blue, extends infinitely in indicated directions). Any vector ${\displaystyle c}$ in 3D space can be decomposed into its projection ${\displaystyle c_{\|}}$ onto a plane and its rejection ${\displaystyle c_{\perp }}$ from this plane.

For any vector ${\displaystyle a}$ and any invertible vector ${\displaystyle m}$,

${\displaystyle a=amm^{-1}=(a\cdot m+a\wedge m)m^{-1}=a_{\|m}+a_{\perp m},}$

where the projection of ${\displaystyle a}$ onto ${\displaystyle m}$ (or the parallel part) is

${\displaystyle a_{\|m}=(a\cdot m)m^{-1}}$

and the rejection of ${\displaystyle a}$ from ${\displaystyle m}$ (or the orthogonal part) is

${\displaystyle a_{\perp m}=a-a_{\|m}=(a\wedge m)m^{-1}.}$

Using the concept of a ${\displaystyle k}$-blade ${\displaystyle B}$ as representing a subspace of ${\displaystyle V}$ and every multivector ultimately being expressed in terms of vectors, this generalizes to projection of a general multivector onto any invertible ${\displaystyle k}$-blade ${\displaystyle B}$ as^{[lower-alpha 9]}

${\displaystyle {\mathcal {P}}_{B}(A)=(A\;\rfloor \;B)\;\rfloor \;B^{-1},}$

with the rejection being defined as

${\displaystyle {\mathcal {P}}_{B}^{\perp }(A)=A-{\mathcal {P}}_{B}(A).}$

The projection and rejection generalize to null blades ${\displaystyle B}$ by replacing the inverse ${\displaystyle B^{-1}}$ with the pseudoinverse ${\displaystyle B^{+}}$ with respect to the contractive product.^{[lower-alpha 10]} The outcome of the projection coincides in both cases for non-null blades.^[33][34] For null blades ${\displaystyle B}$, the definition of the projection given here with the first contraction rather than the second being onto the pseudoinverse should be used,^{[lower-alpha 11]} as only then is the result necessarily in the subspace represented by ${\displaystyle B}$.^[33] The projection generalizes through linearity to general multivectors ${\displaystyle A}$.^{[lower-alpha 12]} The projection is not linear in ${\displaystyle B}$ and does not generalize to objects ${\displaystyle B}$ that are not blades.

Reflection

Simple reflections in a hyperplane are readily expressed in the algebra through conjugation with a single vector. These serve to generate the group of general rotoreflections and rotations.

Reflection of vector ${\displaystyle c}$ along a vector ${\displaystyle m}$. Only the component of ${\displaystyle c}$ parallel to ${\displaystyle m}$ is negated.

The reflection ${\displaystyle c'}$ of a vector ${\displaystyle c}$ along a vector ${\displaystyle m}$, or equivalently in the hyperplane orthogonal to ${\displaystyle m}$, is the same as negating the component of a vector parallel to ${\displaystyle m}$. The result of the reflection will be

${\displaystyle c'={-c_{\|m}+c_{\perp m}}={-(c\cdot m)m^{-1}+(c\wedge m)m^{-1}}={(-m\cdot c-m\wedge c)m^{-1}}=-mcm^{-1}}$

This is not the most general operation that may be regarded as a reflection when the dimension ${\displaystyle n\geq 4}$. A general reflection may be expressed as the composite of any odd number of single-axis reflections. Thus, a general reflection ${\displaystyle a'}$ of a vector ${\displaystyle a}$ may be written

${\displaystyle a\mapsto a'=-MaM^{-1},}$

where

${\displaystyle M=pq\cdots r}$ and ${\displaystyle M^{-1}=(pq\cdots r)^{-1}=r^{-1}\cdots q^{-1}p^{-1}.}$

If we define the reflection along a non-null vector ${\displaystyle m}$ of the product of vectors as the reflection of every vector in the product along the same vector, we get for any product of an odd number of vectors that, by way of example,

${\displaystyle (abc)'=a'b'c'=(-mam^{-1})(-mbm^{-1})(-mcm^{-1})=-ma(m^{-1}m)b(m^{-1}m)cm^{-1}=-mabcm^{-1}\,}$

and for the product of an even number of vectors that

${\displaystyle (abcd)'=a'b'c'd'=(-mam^{-1})(-mbm^{-1})(-mcm^{-1})(-mdm^{-1})=mabcdm^{-1}.}$