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.
:A one by one determinant is the coefficient of for an 1-vector.:A two-by-two determinant is the coefficient of for an bivector:A three-by-three determinant is the coefficient of for an trivector:...
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.
Equation of a plane
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:
Projective and rejective components of a vector
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
Area of the parallelogram defined by u and v
If A is the area of the parallelogram defined by u and v, then :
Note that this squared bivector is a geometric product.
Angle between two vectors
Volume of the parallelopiped formed by three vectors
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 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 properties and examples
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.
Inversion of a vector
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:
Dot and wedge products defined in terms of the geometric product
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
The dot 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).
The wedge product
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.
Note on symmetric and antisymmetric dot and wedge product formulas
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.
Dot and wedge products compared to the real and imaginary parts of a complex number
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
Orthogonal decomposition of a vector
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
A quicker way to the end result
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: [http://www.grassmannalgebra.info/grassmannalgebra/book/bookpdf/TheExteriorProduct.pdf] ), 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.
Area of parallelogram spanned by two vectors
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.
Expansion of a bivector and a vector rejection in terms of the standard basis
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: 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 .
Cross product as outer product
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