Comparison of vector algebra and geometric algebra

In GA, vectors are not normally written boldface as the meaning is usually clear from the context.

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 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.

(right handed orthonormal frame) and so This yields a convenient definition for the cross product of traditional vector algebra: (this is antisymmetric).

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

Essentially, the geometric product of a bivector and the pseudoscalar of Euclidean 3-space provides a method of calculation of the Hodge dual.

As the pseudovectors/bivectors form a vector space, each pseudovector/bivector can be defined as the sum of three orthogonal components parallel to the standard basis pseudovectors/bivectors:

An alternative treatment is to axiomatically introduce the wedge product, and then demonstrate that this can be used directly to solve linear systems.

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 inversion (Cramer's rule) and determinants can be naturally expressed in terms of the wedge product.

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

For numerical problems row reduction with pivots and other methods are more stable and efficient.

parallelogram area and parallelepiped volumes (and higher-dimensional generalizations thereof) also comes as a nice side-effect.

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 result is then simple enough that it could be derived easily if required instead of having to remember or look up a rule.

factors of all the wedge products divide out, leaving the familiar determinants.

An alternate formulation is possible that puts the projection in a form that differs from the usual vector formulation Working backwards from the 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][usurped] ), the problem of orthogonal decomposition can be posed directly, Let

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.

, take the wedge product Having done this calculation with a vector projection, one can guess that this quantity equals

However, skipping ahead slightly, this to-be-proven fact allows for a nice closed form solution of the vector component outside of the plane: Notice the similarities between this planar rejection result and the vector rejection result.

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

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

If a vector is factored directly into projective and rejective terms using the geometric product

Namely, 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.

In order to justify the normal to a plane result above, a general examination of the product of a vector and bivector is required.

The properties of this generalized dot product remain to be explored, but first here is a summary of the notation

When the objective isn't comparing to the cross product, it's also notable that this unit vector derivative can be written