Comparison of vector algebra and geometric algebra

Geometric algebra is an extension of vector algebra, providing additional algebraic structures on vector spaces, with geometric interpretations.
Vector algebra uses all dimensions and signatures, as does geometric algebra, notably 3+1 spacetime as well as 2 dimensions.
Basic concepts and operations.
Geometric algebra (GA) is an extension or completion of vector algebra (VA). The reader is herein assumed to be familiar with the basic concepts and operations of VA and this article will mainly concern itself with operations in formula_1 the GA of 3D space (nor is this article intended to be mathematically rigorous). In GA, vectors are not normally written boldface as the meaning is usually clear from the context.
The fundamental difference is that GA provides a new product of vectors called the "geometric product". Elements of GA are graded multivectors, scalars are grade 0, usual vectors are grade 1, bivectors are grade 2 and the highest grade (3 in the 3D case) is traditionally called the pseudoscalar and designated formula_2.
The ungeneralized 3D vector form of the geometric product is:
that is the sum of the usual dot (inner) product and the outer (exterior) product (this last is closely related to the cross product and will be explained below).
In VA, entities such as pseudovectors and pseudoscalars need to be bolted on, whereas in GA the equivalent bivector and pseudovector respectively exist naturally as subspaces of the algebra.
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 bivector), 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.
Translations between formalisms.
Here are some comparisons between standard formula_4 vector relations and their corresponding exterior product and geometric product equivalents. All the exterior 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 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.
Cross and exterior products.
formula_5 is perpendicular to the plane containing formula_6 and formula_7.
formula_8 is an oriented representation of the same plane.
We have the pseudoscalar formula_9 (right handed orthonormal frame) and so
This yields a convenient definition for the cross product of traditional vector algebra:
(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 formula_2 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 formula_16 cross product and the exterior product expression above can be confirmed by direct multiplication of formula_17 with a determinant expansion of the exterior product
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.
Cross and commutator products.
The pseudovector/bivector subalgebra of the geometric algebra of Euclidean 3-dimensional space form a 3-dimensional vector space themselves. Let the standard unit pseudovectors/bivectors of the subalgebra be formula_19, formula_20, and formula_21, and the anti-commutative commutator product be defined as formula_22, where formula_23 is the geometric product. The commutator product is distributive over addition and linear, as the geometric product is distributive over addition and linear. 
From the definition of the commutator product, formula_24, formula_25 and formula_26 satisfy the following equalities:
which imply, by the anti-commutativity of the commutator product, that 
The anti-commutativity of the commutator product also implies that
These equalities and properties are sufficient to determine the commutator product of any two pseudovectors/bivectors formula_34 and formula_35. 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: 
Their commutator product formula_38 can be expanded using its distributive property:
which is precisely the cross product in vector algebra for pseudovectors.
Norm of a vector.
Ordinarily:
Making use of the geometric product and the fact that the exterior product of a vector 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.
Linear algebra texts will often use the determinant for the solution of linear systems by Cramer's rule or for 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" (formula_49 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.
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 formula_56 (or equivalently to invert a matrix). Namely
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 formula_58 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 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 formula_60 and formula_61,
Provided formula_64 the solution is
For formula_66, this is Cramer's rule since the formula_67 factors of the wedge products
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 formula_71 factors of all the wedge products divide out, leaving the familiar determinants.
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 formula_73 solves for formula_74
and a left wedge product with formula_76 solves for formula_77
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
formula_74 and formula_77 yields a Cramer's rule-like form:
Writing formula_82, we have the end result:
Equation of a plane.
For the plane of all points formula_84 through the plane passing through three independent points formula_85, formula_86, and formula_87, the normal form of the equation is
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, formula_90, the projection of formula_91 onto formula_92 is
Orthogonal to that vector is the difference, designated the rejection,
Note the similarity in form to the "w", "u", "v" trivector itself
which, if you take the set of formula_96 as a basis for the trivector space, suggests this is the natural way to define the measure of a trivector. Loosely speaking, the measure of a vector is a length, the measure of a bivector is an area, and the measure of a trivector is a volume.
If a vector is factored directly into projective and rejective terms using the geometric product formula_97, 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
The trivector term is formula_99. Expansion of formula_100 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 formula_105, where formula_106, and formula_107. Expressing formula_108 and the formula_109, products in terms of these components is
With the conditions and definitions above, and some manipulation, it can be shown that the term formula_111, 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.
Derivative of a unit vector.
It can be shown that a unit vector derivative can be expressed using the cross product
\frac{d}{dt}\left(\frac{\mathbf r}{\Vert \mathbf r \Vert}\right)