geometric algebra

Geometric algebra is a Clifford algebraMathworldPlanetmathPlanetmath ( which has been used with great success in the modeling of a wide varietyMathworldPlanetmathPlanetmath of physical phenomena. Clifford algebra is considered a more general algebraicMathworldPlanetmath framework than geometric algebra. The primaryMathworldPlanetmath distinction is that geometric algebra utilizes only real numbers as scalars and to represent magnitudes. The underlying philosophical justification for this is the interpretationMathworldPlanetmathPlanetmath that the unit imaginaryPlanetmathPlanetmath has geometric significance which naturally arises from the properties of the algebra and the interaction of its various subspacesPlanetmathPlanetmath.

Let 𝒱n be an n–dimensional vector spaceMathworldPlanetmath over the real numbers. As with traditional vector algebra, the vector space is spanned by a set of n linearly independentMathworldPlanetmath basis vectors. Any vector in this space may be represented by a linear combinationMathworldPlanetmath of the basis vectors. In the geometric algebra literature, such basis entities are also called blades.

Since vectors are one-dimensional directed quantities, they are assigned a grade of 1. Scalars are considered to be grade-0 entities. In geometric algebra, there exist higher dimensional analogues to vectors. Two-dimensional directed quantites are termed bivectors and they are grade-2 entites. In general a k-dimensional entity is known as a k-vector.

The geometric algebra 𝒒n=𝒒⁒(𝒱n) is a multi-graded algebra similarPlanetmathPlanetmath to Grassmann’s exterior algebraMathworldPlanetmath, except that the exterior product is replaced by a more fundamental multiplicationPlanetmathPlanetmath operationMathworldPlanetmath known as the geometric product. In general, the result of the geometric product is a multi-graded object called a multivector. A multivector is a linear combination of basis blades.

For vectors 𝐚,𝐛,πœβˆˆπ’±n and real scalars Ξ±,Ξ²βˆˆπ‘, the geometric product satisfies the following axioms:

associativity:𝐚⁒(π›πœ)=(πšπ›)⁒𝐜𝐚+(𝐛+𝐜)=(𝐚+𝐛)+𝐜commutativity:α⁒β=β⁒αα+Ξ²=Ξ²+αα⁒𝐛=𝐛⁒αα+𝐛=𝐛+Ξ±πšπ›=12⁒(πšπ›+π›πš)+12⁒(πšπ›-π›πš)𝐚+𝐛=𝐛+𝐚distributivity:𝐚⁒(𝐛+𝐜)=πšπ›+𝐚𝐜(𝐛+𝐜)⁒𝐚=π›πš+𝐜𝐚π₯𝐒𝐧𝐞𝐚𝐫𝐒𝐭𝐲α⁒(𝐛+𝐜)=α⁒𝐛+α⁒𝐜=(𝐛+𝐜)⁒αcontraction:𝐚2=𝐚𝐚=βˆ‘i=1nΟ΅i⁒|𝐚i|2=Ξ±where ⁒ϡi∈{-1,0,1}

Commutativity of scalar–scalar multiplication and vector–scalar multiplication is symmetricPlanetmathPlanetmathPlanetmathPlanetmath; however, in general, vector–vector multiplication is not commutativePlanetmathPlanetmath. The order of multiplication of vectors is significant. In particular, for parallel vectors:


and for orthogonal vectorsMathworldPlanetmath:


The parallelism of vectors is encoded as a symmetric property, while orthogonality of vectors is encoded as an antisymmetric property.

The contraction rule specifies that the square of any vector is a scalar equal to the sum of the square of the magnitudes of its componentsMathworldPlanetmathPlanetmath in each basis direction. Depending on the contraction rule for each of the basis directions, the magnitude of the vector may be positive, negative, or zero. A vector with a magnitude of zero is called a null vector.

The graded algebraMathworldPlanetmath 𝒒n generated from 𝒱n is defined over a 2n-dimensional linear space. This basis entities for this space can be generated by successive application of the geometric product to the basis vectors of 𝒱n until a closed set of basis entities is obtained. The basis entites for the space are known as blades. The following multiplication table illustrates the generation of basis blades from the basis vectors 𝐞1,𝐞2βˆˆπ’±n.


Here, Ο΅1 and Ο΅2 represent the contraction rule for 𝐞1 and 𝐞2 respectively. Note that the basis vectors of 𝒱n become blades themselves in addition to the multiplicative identityPlanetmathPlanetmath, Ο΅0≑1 and the new bivector 𝐞12β‰‘πž1⁒𝐞2. As the table demonstrates, this set of basis blades is closed under the geometric product.

The geometric product πšπ› is related to the inner product πšβ‹…π› and the exterior product πšβˆ§π› by


In the above example, the result of the inner (dot) product is a scalar (grade-0), while the result of the exterior (wedge) product is a bivector (grade-2).


  1. 1.

    David Hestenes, New Foundations for Classical Mechanics, Kluwer, Dordrecht, 1999

  2. 2.

    David Hestenes, Garret Sobczyk, Clifford Algebra to Geometric Calculus, Kluwer, Dordrecht, 1984

Title geometric algebra
Canonical name GeometricAlgebra
Date of creation 2013-03-22 13:17:03
Last modified on 2013-03-22 13:17:03
Owner PhysBrain (974)
Last modified by PhysBrain (974)
Numerical id 15
Author PhysBrain (974)
Entry type Definition
Classification msc 15A66
Classification msc 15A75
Synonym Clifford algebra
Related topic ExteriorAlgebra
Related topic CliffordAlgebra2
Related topic CCliffordAlgebra
Related topic SpinGroup
Defines geometric product