In mathematical physics, a geometric algebra is a multilinear algebra described technically as a Clifford algebra over a real vector space equipped with a non-degenerate quadratic form. Informally, a geometric algebra is a Clifford algebra that includes a geometric product. This allows the theory and properties of the algebra to be built up in an intuitive, geometric way. The term is also used in a more general sense to describe the study and application of these algebras: so Geometric algebra is the study of geometric algebras.
Geometric algebra is useful in physics problems that involve rotations, phases or imaginary numbers. Proponents of geometric algebra argue it provides a more compact and intuitive description of classical and quantum mechanics, electromagnetic theory and relativity. Current applications of geometric algebra include computer vision, biomechanics and robotics, and spaceflight dynamics.
A geometric algebra is an algebra constructed over a vector space in which a geometric product is defined. The elements of geometric algebra are multivectors. The original vector space is constructed over the real numbers as scalars. From now on, a vector is something in itself. Vectors will be represented by boldface, small case letters (e.g. ), and multivectors by boldface, upper case letters (e.g. ).
The geometric product has the following properties, for all multivectors :
The distinctive point of this formulation is the natural correspondence between geometric entities and the elements of the associative algebra. This comes from the fact that the geometric product is defined in terms of the dot product and the wedge product of vectors as
The definition and the associativity of geometric product entails the concept of the inverse of a vector (or division by vector). Thus, one can easily set and solve vector algebra equations that otherwise would be cumbersome to handle. In addition, one gains a geometric meaning that would be difficult to retrieve, for instance, by using matrices. Although not all the elements of the algebra are invertible, the inversion concept can be extended to multivectors. Geometric algebra allows one to deal with subspaces directly, and manipulate them too. Furthermore, geometric algebra is a coordinate-free formalism.
Geometric objects like are called bivectors. A bivector can be pictured as a plane segment (a parallelogram, a circle etc.) endowed with orientation. One bivector represents all planar segments with the same magnitude and direction, no matter where they are in the space that contains them. However, once either the vector or is meant to depart from some preferred point (e.g. in problems of Physics), the oriented plane is determined unambiguously.
The outer product (the exterior product, or the wedge product ) "" is defined such that the graded algebra (exterior algebra of Hermann Grassmann) of multivectors is generated. Multivectors are thus the direct sum of grade k elements (k-vectors), where k ranges from 0 (scalars) to n, the dimension of the original vector space . Multivectors are represented here by boldface caps. Note that scalars and vectors become special cases of multivectors ("0-vectors" and "1-vectors", respectively).
Here are some comparisons between standard vector relations and their corresponding wedge and geometric product equivalents. All the wedge and geometric product equivalents here are good for more than three dimensions, and some also for two. In two dimensions the cross product is undefined even if what it describes (like torque) is perfectly well defined in a plane without introducing an arbitrary normal vector outside of the space.
Many of these relationships only require the introduction of the wedge product to generalize, but since that may not be familiar to somebody with only a traditional background in vector algebra and calculus, some examples are given.
They are both linear in the first operand
and in the second operand
In general, the cross product is not associative, while the wedge product is
Both the cross and wedge products of two identical vectors are zero:
is perpendicular to the plane containing and .
is an oriented representation of the same plane.
The norm (length) of a vector is defined in terms of the dot product
Using the geometric product this is also true, but this can be also be expressed more compactly as
This follows from the definition of the geometric product and the fact that a vector wedge product with itself is zero
In three dimensions the product of two vector lengths can be expressed in terms of the dot and cross products
The corresponding generalization expressed using the geometric product is
This follows from by expanding the geometric product of a pair of vectors with its reverse
Without justification or historical context, traditional linear algebra texts will often define the determinant as the first step of an elaborate sequence of definitions and theorems leading up to the solution of linear systems, Cramer's rule and matrix inversion.
An alternative treatment is to axiomatically introduce the wedge product, and then demonstrate that this can be used directly to solve linear systems. This is shown below, and does not require sophisticated math skills to understand.
It is then possible to define determinants as nothing more than the coefficients of the wedge product in terms of "unit k-vectors" ( terms) expansions as above.
When linear system solution is introduced via the wedge product, Cramer's rule follows as a side effect, and there is no need to lead up to the end results with definitions of minors, matrices, matrix invertibility, adjoints, cofactors, Laplace expansions, theorems on determinant multiplication and row column exchanges, and so forth.
For the plane of all points through the plane passing through three independent points , , and , the normal form of the equation is
The equivalent wedge product equation is
For three dimensions the projective and rejective components of a vector with respect to an arbitrary non-zero unit vector, can be expressed in terms of the dot and cross product
For the general case the same result can be written in terms of the dot and wedge product and the geometric product of that and the unit vector
It's also worthwhile to point out that this result can also be expressed using right or left vector division as defined by the geometric product
If A is the area of the parallelogram defined by u and v, then
Note that this squared bivector is a geometric product.
The equivalent geometric product generalization is
Thus this derivative is the component of in the direction perpendicular to . In other words this is minus the projection of that vector onto .
This intuitively make sense (but a picture would help) since a unit vector is constrained to circular motion, and any change to a unit vector due to a change in its generating vector has to be in the direction of the rejection of from . That rejection has to be scaled by 1/|r| to get the final result.
When the objective isn't comparing to the cross product, it's also notable that this unit vector derivative can be written
Some fundamental geometric algebra manipulations will be provided below, showing how this vector product can be used in calculation of projections, area, and rotations. How some of these tie together and correlate concepts from other branches of mathematics, such as complex numbers, will also be shown.
In some cases these examples provide details used above in the cross product and geometric product comparisons.
One of the powerful properties of the Geometric product is that it provides the capability to express the inverse of a non-zero vector. This is expressed by:
Given a definition of the geometric product in terms of the dot and wedge products, adding and subtracting and demonstrates that the dot and wedge product of two vectors can also be defined in terms of the geometric product
This is the symmetric component of the geometric product. When two vectors are colinear the geometric and dot products of those vectors are equal.
As a motivation for the dot product it is normal to show that this quantity occurs in the solution of the length of a general triangle where the third side is the vector sum of the first and second sides .
The last sum is then given the name the dot product and other properties of this quantity are then shown (projection, angle between vectors, ...).
This can also be expressed using the geometric product
By comparison, the following equality exists
Without requiring expansion by components one can define the dot product exclusively in terms of the geometric product due to its properties of contraction, distribution and associativity. This is arguably a more natural way to define the geometric product, especially since the wedge product is not familiar to many people with traditional vector algebra background, and there is no immediate requirement to add two dissimilar terms (ie: scalar and bivector).
This is the antisymmetric component of the geometric product. When two vectors are orthogonal the geometric and wedge products of those vectors are equal.
Switching the order of the vectors negates this antisymmetric geometric product component, and contraction property shows that this is zero if the vectors are equal. These are the defining properties of the wedge product.
A generalization of the dot product that allows computation of the component of a vector "in the direction" of a plane (bivector), or other k-vectors can be found below. Since the signs change depending on the grades of the terms being multiplied, care is required with the formulas above to ensure that they are only used for a pair of vectors.
Reversing the order of multiplication of two vectors has the effect of the inverting the sign of just the wedge product term of the geometric product.
It is not a coincidence that this is a similar operation to the conjugate operation of complex numbers.
The reverse of a product is written in the following fashion
Thus, the dot product is
This is the symmetric component of the geometric product. When two vectors are colinear the geometric and dot products of those vectors are equal. The antisymmetric component is represented by the wedge product:
These symmetric and antisymmetric components extract the scalar and bivector components of a geometric product in the same fashion as the real and imaginary components of a complex number are extracted by its symmetric and antisymmetric components
This extraction of components also applies to higher order geometric product terms. For example
Using the Gram-Schmidt process a single vector can be decomposed into two components with respect to a reference vector, namely the projection onto a unit vector in a reference direction, and the difference between the vector and that projection.
With, , the projection of onto is
Orthogonal to that vector is the difference, designated the rejection,
The rejection can be expressed as a single geometric algebraic product in a few different ways
The similarity in form between the projection and the rejection is notable. The sum of these recovers the original vector
Here the projection is in its customary vector form. An alternate formulation is possible that puts the projection in a form that differs from the usual vector formulation
Working backwards from the end result, it can be observed that this orthogonal decomposition result can in fact follow more directly from the definition of the geometric product itself.
With this approach, the original geometrical consideration is not necessarily obvious, but it is a much quicker way to get at the same algebraic result.
However, the hint that one can work backwards, coupled with the knowledge that the wedge product can be used to solve sets of linear equations (see: ), the problem of orthogonal decomposition can be posed directly,
Let , where . To discard the portions of that are colinear with , take the wedge product
Here the geometric product can be employed
Because the geometric product is invertible, this can be solved for x
The same techniques can be applied to similar problems, such as calculation of the component of a vector in a plane and perpendicular to the plane.
The area of a parallelogram spanned between one vector and another equals the length of one of those vectors multiplied by the length of the rejection of that vector from the second.
The length of this vector is the area of the spanned parallelogram, and in the square is
There are a couple things of note here. One is that the area can easily be expressed in terms of the square of a bivector. The other is that the square of a bivector has the same property as a purely imaginary number, a negative square.
If a vector is factored directly into projective and rejective terms using the geometric product , then it is not necessarily obvious that the rejection term, a product of vector and bivector is even a vector. Expansion of the vector bivector product in terms of the standard basis vectors has the following form
It can be shown that
(a result that can be shown more easily straight from ).
The rejective term is perpendicular to , since implies .
The magnitude of , is
So, the quantity
is the squared area of the parallelogram formed by and .
It is also noteworthy that the bivector can be expressed as
Thus is it natural, if one considers each term as a basis vector of the bivector space, to define the (squared) "length" of that bivector as the (squared) area.
Going back to the geometric product expression for the length of the rejection we see that the length of the quotient, a vector, is in this case is the "length" of the bivector divided by the length of the divisor.
This may not be a general result for the length of the product of two k-vectors, however it is a result that may help build some intuition about the significance of the algebraic operations. Namely,
Like vector projection and rejection, higher dimensional analogs of that calculation are also possible using the geometric product.
As an example, one can calculate the component of a vector perpendicular to a plane and the projection of that vector onto the plane.
Let , where . As above, to discard the portions of that are colinear with or , take the wedge product
Having done this calculation with a vector projection, one can guess that this quantity equals . One can also guess there is a vector and bivector dot product like quantity such that the allows the calculation of the component of a vector that is in the "direction of a plane". Both of these guesses are correct, and the validating these facts is worthwhile. However, skipping ahead slightly, this to be proved fact allows for a nice closed form solution of the vector component outside of the plane:
Notice the similarities between this planar rejection result a the vector rejection result. To calculation the component of a vector outside of a plane we take the volume spanned by three vectors (trivector) and "divide out" the plane.
Independent of any use of the geometric product it can be shown that this rejection in terms of the standard basis is
is the squared area of the parallelogram formed by , and .
The (squared) magnitude of is
Thus, the (squared) volume of the parallelopiped (base area times perpendicular height) is
Note the similarity in form to the w, u,v trivector itself
which, if you take the set of as a basis for the trivector space, suggests this is the natural way to define the length of a trivector. Loosely speaking the length of a vector is a length, length of a bivector is area, and the length of a trivector is volume.
In order to justify the normal to a plane result above, a general examination of the product of a vector and bivector is required. Namely,
This has two parts, the vector part where or , and the trivector parts where no indexes equal. After some index summation trickery, and grouping terms and so forth, this is
The trivector term is . Expansion of yields the same trivector term (it is the completely symmetric part), and the vector term is negated. Like the geometric product of two vectors, this geometric product can be grouped into symmetric and antisymmetric parts, one of which is a pure k-vector. In analogy the antisymmetric part of this product can be called a generalized dot product, and is roughly speaking the dot product of a "plane" (bivector), and a vector.
The properties of this generalized dot product remain to be explored, but first here is a summary of the notation
Let , where , and . Expressing and the , products in terms of these components is
With the conditions and definitions above, and some manipulation, it can be shown that the term , which then justifies the previous solution of the normal to a plane problem. Since the vector term of the vector bivector product the name dot product is zero when the vector is perpendicular to the plane (bivector), and this vector, bivector "dot product" selects only the components that are in the plane, so in analogy to the vector-vector dot product this name itself is justified by more than the fact this is the non-wedge product term of the geometric vector-bivector product.
While the cross product can only be defined in a three-dimensional space, the inner and outer products can be generalized to any dimensional .
Let be a vector and a homogeneous multivector of grade k, respectively. Their inner product is then
Writing, a vector in terms of its components, and left multiplying by the unit vector yields
The unit scalar and unit bivector pair can be considered an alternate basis for a two dimensional vector space. This alternate vector representation is closed with respect to the geometric product
This closure can be observed after calculation of the square of the unit bivector above, a quantity
that has the characteristics of the complex number .
This fact allows the simplification of the product above to
Thus what is traditionally the defining, and arguably arbitrary seeming, rule of complex number multiplication, is found to follow naturally from the higher order structure of the geometric product, once that is applied to a two dimensional vector space.
It is also informative to examine how the length of a vector can be represented in terms of a complex number. Taking the square of the length
This right multiplication of a vector with , is named the conjugate
And with that definition, the length of the original vector can be expressed as
This is also a natural definition of the length of a complex number, given the fact that the complex numbers can be considered an isomorphism with the two dimensional Euclidean vector space.
A point , of radius , located at an angle from the vector in the direction from to , can be expressed as
Writing , the square of this bivector has the property of the imaginary unit complex number.
This allows the point to be specified as a complex exponential
Complex numbers could be expressed in terms of the unit bivector . However this isomorphism really only requires a pair of linearly independent vectors in a plane (of arbitrary dimension).
Like complex numbers, quaternions may be written as a multivector with scalar and bivector components (a 0,2-multivector).
Where the complex number has one bivector component, and the quaternions have three.
One can describe quaternions as 0,2-multivectors where the basis for the bivector part is left-handed. There isn't really anything special about quaternion multiplication, or complex number multiplication, for that matter. Both are just a specific examples of a 0,2-multivector multiplication. Other quaternion operations can also be found to have natural multivector equivalents. The most important of which is likely the quaternion conjugate, since it implies the norm and the inverse. As a multivector, like complex numbers, the conjugate operation is reversal:
Thus . Note that this norm is a positive definite as expected since a bivector square is negative.
To be more specific about the left-handed basis property of quaternions one can note that the quaternion bivector basis is usually defined in terms of the following properties
The first two properties are satisfied by any set of orthogonal unit bivectors for the space. The last property, which could also be written , amounts to a choice for the orientation of this bivector basis of the 2-vector part of the quaternion.
As an example suppose one picks
Then the third bivector required to complete the basis set subject to the properties above is
Suppose that, instead of the above, one picked a slightly more natural bivector basis, the duals of the unit vectors obtained by multiplication with the pseudoscalar (). These bivectors are
A 0,2-multivector with this as the basis for the bivector part would have properties similar to the standard quaternions (anti-commutative unit quaternions, negation for unit quaternion square, same congugate, norm and inversion operations, ...), however the triple product would have the value , instead of .
The cross product of traditional vector algebra (on ) find its place in geometric algebra as a scaled outer product
(this is antisymmetric). Relevant is the distinction between axial and polar vectors in vector algebra, which is natural in geometric algebra as the mere distinction between vectors and bivectors (elements of grade two).
The here is a unit pseudoscalar of Euclidean 3-space, which establishes a duality between the vectors and the bivectors, and is named so because of the expected property
The equivalence of the cross product and the wedge product expression above can be confirmed by direct multiplication of with a determinant expansion of the wedge product
See also Cross product#Cross product as an exterior product. Essentially, the geometric product of a bivector and the pseudoscalar of Euclidean 3-space provides a method of calculation of the hodge dual.
Note that the division by a vector transforms the multivector into the sum of two vectors. Namely, -1 is the projection of on , and -1 is the rejection of from (i.e. the component of orthogonal to ). Note also that the structure of the solution does not depend on the chosen origin.
Torque is generally defined as the magnitude of the perpendicular force component times distance, or work per unit angle.
Suppose a circular path in an arbitrary plane containing orthonormal vectors and is parameterized by angle.
By designating the unit bivector of this plane as the imaginary number
this path vector can be conveniently written in complex exponential form
and the derivative with respect to angle is
So the torque, the rate of change of work W, due to a force F, is
Unlike the cross product description of torque, no vector in a normal direction had to be introduced, a normal that doesn't exist in two dimensions or 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 and .
At a glance this doesn't appear much like the familiar torque as a determinant or cross product, but this can be expanded to demonstrate its equivalence (the cross product is hiding there in the bivector ). Expanding the position vector in terms of the planar unit vectors
and expanding the force by components in the same direction plus the possible perpendicular remainder term
and then taking dot products yields is the torque
This determinant may be familiar from derivations with , and (See the Feynman lectures Volume I for example).
When the magnitude of the "rotational arm" is factored out, the torque can be written as
The vector is the unit vector perpendicular to the . Thus the torque can also be described as the product of the magnitude of the rotational arm times the component of the force that is in the direction of the rotation (ie: the work done rotating something depends on length of the lever, and the size of the useful part of the force pushing on it).
If the rotational arm that the force is applied to is not in the plane of rotation then only the components of the lever arm direction and the component of the force that are in the plane will contribute to the work done. The calculation above allowed for a force applied in an arbitrary direction, so to generalize this, a calculation that discards the component of the level arm direction not in the plane.
When is allowed to lie outside of the plane of rotation the component in the plane (bivector) can be described with the geometric product nicely
Thus, the vector with this magnitude that is perpendicular to this in the plane of the rotation is
and the total torque is thus
This makes sense when once considers that only the dot product part of contributes to the component of in the plane, and when the lever is in the rotational plane this wedge product component of is zero.
The use of the wedge product in the solution of linear equations can be quite useful for various geometric product calculations.
Traditionally, instead of using the wedge product, Cramer's rule is usually presented as a generic algorithm that can be used to solve linear equations of the form (or equivalently to invert a matrix). Namely