The Geometric Product
The geometric product is a fundamental operation of geometric algebra. In this section the basic operations are given.
The geometric product of two vectors is the sum of the scalar product and the outer product:
So the result of the geometric product is a multi vector with two parts: a scalar and a bivector.
Commutating the two vectors in the geometric product gives:
Expressing the inner product with the geometric product gives the following formular:
Expressing the outer product with the geometric product gives the following formular:
The inner and outer products are the symmetric and antisymmetric parts of the
geometric product. Rearranging the two equations above gives the following two formulas:
Dependens of the geometric product of two vectors on the dot product of these vectors.
Dependens of the geometric product of two vectors on the wedge product of these vectors.
Application of the Geometric Product on Base Vectors
Vector product of base vectors that are equal has the result 1.
Vector product of base vectors that are orthogonal produces a bivector. (i is not equal to j)
Canonic Form in two Dimensions
Having u and v expressed by their canonic base vectors in two dimensions gives:
The geometric product of u and v can then be expressed as follows:
The result of the geometric product of two vectors can be interpretated as a complex number.
The Geometric Product and Eulers Formular
Inverse of a Vector
Square of a vector:
For the geometric product the inverse of a vector that is not zero is defined (which is not the case for the dot and the cross product!):
Definition of a Blade
The outer product of k linear independent vectors is called a k-blade.
Reversion of a Blade
The reversion of a k-blade is defined by the reversion of the order of the defining vectors.
The ˜ tilde symbol is used to mark a reversed k-blade.
Inversion of a Blade
The invers of a blade is defined as follows
Inverse of the Unit Pseudo Scalar
The inverse of the unit pseudo scalar is equal to the reverse of the unit pseudo scalar.
In 3 Dimensions we have the following Example:
Dual Vector Space
The dual Multivector A* of a given Multivector A of is defined as follows:
I is the unit pseudoscalar of the geometric algebra.