Vector algebra and geometric algebra are alternative approaches to providing additional algebraic structures on vector spaces, with geometric interpretations, particularly vector fields in multivariable calculus and applications in mathematical physics.
Contents
- Basic concepts and operations
- Embellishments ad hoc techniques and tricks
- List of analogous formulas
- Algebraic and geometric properties of cross and exterior products
- Norm of a vector
- Lagrange identity
- Determinant expansion of cross and wedge products
- Matrix Related
- Equation of a plane
- Projection and rejection
- Area of the parallelogram defined by u and v
- Angle between two vectors
- Volume of the parallelopiped formed by three vectors
- Product of a vector and a bivector
- Derivative of a unit vector
- References
Vector algebra is specific to Euclidean 3-space, while geometric algebra uses multilinear algebra and applies in all dimensions and signatures, notably 3+1 spacetime as well as 2 dimensions. They are mathematically equivalent in 3 dimensions, though the approaches differ. Vector algebra is more widely used in elementary multivariable calculus, while geometric algebra is used in some more advanced treatments, and is proposed for elementary use as well. In advanced mathematics, particularly differential geometry, neither is widely used, with differential forms being far more widely used.
Basic concepts and operations
In vector algebra the basic objects are scalars and vectors, and the operations (beyond the vector space operations of scalar multiplication and vector addition) are the dot (or scalar) product and the cross product ×.
In geometric algebra the basic objects are multivectors (scalars are 0-vectors, vectors are 1-vectors, etc.), and the operations include the Clifford product (here called "geometric product") and the exterior product. The dot product/inner product/scalar product is defined on 1-vectors, and allows the geometric product to be expressed as the sum of the inner product and the exterior product when multiplying 1-vectors.
Vector algebra uses the cross product, while geometric algebra uses the exterior product (and the geometric product). More subtly, geometric algebra in 3-dimensional Euclidean space distinguishes 0-vectors, 1-vectors, 2-vectors, and 3-vectors, while elementary vector algebra identifies 1-vectors and 2-vectors (as vectors) and 0-vectors and 3-vectors (as scalars), though more advanced vector algebra distinguishes these as scalars, vectors, pseudovectors, and pseudoscalars respectively. Unlike vector algebra, geometric algebra includes sums of k-vectors of differing k.
The cross product does not generalize to dimensions other than 3 as a product of two vectors, yielding a unique third orthogonal vector, and in higher dimensions not all k-vectors can be identified with vectors or scalars. By contrast, the exterior product (and geometric product) is defined uniformly for all dimensions and signatures, and multivectors are closed under these operations.
Embellishments, ad hoc techniques, and tricks
More advanced treatments of vector algebra add embellishments to the initial picture – pseudoscalars and pseudovectors (in terms of geometric algebra in 3 dimensions, correspondingly 3-vectors and 2-vectors), while applications to other dimensions use ad hoc techniques and "tricks" rather than a general mathematical approach. By contrast, geometric algebra begins with a complete picture, and applies uniformly in all dimensions.
For example, applying vector calculus in 2 dimensions, such as to compute torque or curl, requires adding an artificial 3rd dimension and extending the vector field to be constant in that dimension, or alternately considering these to be scalars. The torque or curl is then a normal vector field in this 3rd dimension. By contrast, geometric algebra in 2 dimensions defines these as a pseudoscalar field (a 2-vector field), without requiring a 3rd dimension. Similarly, the scalar triple product is ad hoc, and can instead be expressed uniformly using the exterior product and the geometric product.
List of analogous formulas
Here are some comparisons between standard
Many of these relationships only require the introduction of the exterior product to generalize, but since that may not be familiar to somebody with only a background in vector algebra and calculus, some examples are given.
Algebraic and geometric properties of cross and exterior products
Cross and exterior products are both antisymmetric:
They are both linear in the first operand:
and in the second operand:
In general, the cross product is not associative, while the exterior product is:
Both the cross and exterior products of two identical vectors are zero:
The cross product of traditional vector algebra (on
(this is antisymmetric). Relevant is the distinction between axial and polar vectors in vector algebra, which is natural in geometric algebra as the distinction between vectors and bivectors (elements of grade two).
The
The equivalence of the
See also 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.
Norm of a vector
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 the exterior product of a vector wedge with itself is zero:
Lagrange identity
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 expanding the geometric product of a pair of vectors with its reverse
Determinant expansion of cross and wedge products
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" (
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.
Matrix Related
Matrix inversion (Cramer's rule) and determinants can be naturally expressed in terms of the wedge product.
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
This is a useful theoretic result. For numerical problems row reduction with pivots and other methods are more stable and efficient.
When the wedge product is coupled with the Clifford product and put into a natural geometric context, the fact that the determinants are used in the expression of
As is also shown below, results such as Cramer's rule also follow directly from the wedge product's selection of non-identical elements. The end result is then simple enough that it could be derived easily if required instead of having to remember or look up a rule.
Two variables example
Pre- and post-multiplying by
Provided
For
divide out.
Similarly, for three, or N variables, the same ideas hold
Again, for the three variable three equation case this is Cramer's rule since the
A numeric example with three equations and two unknowns: In case there are more equations than variables and the equations have a solution, then each of the k-vector quotients will be scalars.
To illustrate here is the solution of a simple example with three equations and two unknowns.
The right wedge product with
and a left wedge product with
Observe that both of these equations have the same factor, so one can compute this only once (if this was zero it would indicate the system of equations has no solution).
Collection of results for
Writing
Equation of a plane
For the plane of all points
The equivalent wedge product equation is
Projection and rejection
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,
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: [1] ), the problem of orthogonal decomposition can be posed directly,
Let
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.
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:
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
Having done this calculation with a vector projection, one can guess that this quantity equals
Notice the similarities between this planar rejection result and the vector rejection result. To calculate 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
where
is the squared area of the parallelogram formed by
The (squared) magnitude of
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
If a vector is factored directly into projective and rejective terms using the geometric product
It can be shown that
(a result that can be shown more easily straight from
The rejective term is perpendicular to
The magnitude of
So, the quantity
is the squared area of the parallelogram formed by
It is also noteworthy that the bivector can be expressed as
Thus is it natural, if one considers each term
Going back to the geometric product expression for the length of the rejection
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,
When a vector is divided out of the plane (parallelogram span) formed from it and another vector, what remains is the perpendicular component of the remaining vector, and its length is the planar area divided by the length of the vector that was divided out.Area of the parallelogram defined by u and v
If A is the area of the parallelogram defined by u and v, then
and
Note that this squared bivector is a geometric multiplication; this computation can alternatively be stated as the Gram determinant of the two vectors.
Angle between two vectors
Volume of the parallelopiped formed by three vectors
In vector algebra, the volume of a parallelopiped is given by the square root of the squared norm of the scalar triple product:
Product of a vector and a bivector
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
The trivector term is
The properties of this generalized dot product remain to be explored, but first here is a summary of the notation
Let
With the conditions and definitions above, and some manipulation, it can be shown that the term
Derivative of a unit vector
It can be shown that a unit vector derivative can be expressed using the cross product
The equivalent geometric product generalization is
Thus this derivative is the component of
This intuitively makes 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
When the objective isn't comparing to the cross product, it's also notable that this unit vector derivative can be written