Wacker Art Clifford Algebra Wappen der Familie Wacker
Edersee - Fähre Scheid-Rehbach
Bild: "Edersee - Fähre Schein-Rehbach"

Prolog:

"... geometry, you know, is the gate of science, and the gate is so low and small that one can only enter it as a little child."

William Kingdon Clifford


In mathematics the name Clifford Algebra is used but Clifford himself named it Geometric Algebra. For an Overview on the subject visit my Geometric Algebra page. On this page it is assumed that the reader is familiar with the basic concepts of geometric algebra.

Edersee
Bild: "Edersee"

Vectors involved in the Polarisation Identity

Clifford Algebra

The Inner Product

We start with the geometric product of the two vectors u and v:

Geometric Product

Exchanging the vectors u and v gives the relation:

Geometric Product Commutated

By adding the two equations we get the scalar product:

Scalar Product

Or in a more general form:

Scalar Product general Form.

The relation of geometric products of the ortogonormal basic vectors, can be written in the form of a clifford product as follows:

Clifford Product

Or in a more general form with the metric tensor when the base vectors are not orthogonal.

Metric Tensor

If we square the sum of two vectors (using the geometric product) we get:

Sum of two vectors squared   (1)

Reordering the elements gives:

Clifford Product   (2)

If we square the difference of two vectors we get:

Difference of two vectors squared   (3)

Reordering the elements gives:

Clifford Product   (4)

Summing up equation (2) and (4) gives equation (5):

Clifford Product   (5)

We now have the following relations for the scalar product of two vectors:

Clifford Product   (2)

Clifford Product   (4)

Clifford Product   (5)

The Outer Product

The outer product can be defined in dependence from the geometric product as follows:

Outer Product

Using the following product relation:

Complex Length   (6)

Gives the following relation for the outer product:

Outer Product   (7)

Using the following product relation:

Complex Length   (8)

Gives a further relation for the outer product:

Outer Product   (9)

The diverens between the given two relations for the outer product results in the folowing expression:

Outer Product   (10)

We now have the following relations for the outer product of two vectors:

Outer Product

Outer Product

Outer Product

Geometric Product

When we sum up the results for the inner (5) and the outer product (10) we come back to the geometric product:

Geometric Product

Geometric Product

Geometric Product

Geometric Product

Edersee - Weißer Stein
Bild: "Edersee - Weißer Stein"

Double Products

Double Product

Double Product

Double Product

Double Product

Double Product

Double Product

Trible Products

Trible Product

Trible Product

Trible Product

Trible Product

Trible Product

Trible Product

The Vector Bivector Products

Vector Bivector Product

Vector Bivector Product  (1)

Vector Bivector Product

Vector Bivector Product  (2)

Vector Bivector Product

Vector Bivector Product

If we build the difference between (1) and (2) we come to the following expression:

Vector Bivector Product

Vector Bivector Product

Vector Bivector Product

Edersee
Bild: "Edersee"

Visit the Geometric Algebra page for an overview on Geometric Algebra.

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