Geometric algebra is not to be confused with algebraic geometry.

Geometric algebra is also known as Clifford algebra which has many applications in physics and engineering. Algebraic geometry, on the other hand, is a branch of abstract mathematics.

Geometric algebra unifies projective geometry, complex numbers and quaternions in an intuitive framework. In geometric algebra lines, planes and volumes become the basic elements of the algebra. These elements are manipulated by algebraic operators that have intuitive geometric interpretations as well.

**Basis Elements**

Any point in a linear mathematical space can be expressed as a linear combination of the basis elements.

For example, a vector in the 2D Euclidean space known as can be described by the orthonormal basis elements and (see the perpendicular arrows in the figure below). The term “othonormal” means that and have unit length and they are orthogonal (90 degrees angle between them). In the most general sense, “orthogonal” means independent.

**Oriented Subspaces**

The product or represent oriented planes. The orientation of the plane is the opposite of the orientation of the plane (circular arrows in the figure above indicate the orientation).

Oriented subspaces can be used as basis elements too. We will see examples below.

**Bivector**

There are alternative names for subspaces. The oriented plane or is a bivector.

The term “bivector” is not very meaningful in .

**Trivector**

The alternative name for is trivector. There is no trivector in , obviously. Trivector represents an oriented volume. It is only meaningful in or higher dimensional Euclidean spaces.

To repeat, the terms “bivector” and “trivector” are meaningful in or higher dimensional Euclidean spaces.

**Vector space, multivector space and
**

When I introduced the basis elements above I was very careful to say “mathematical space” to keep it general. I avoided the term “vector space.” I left room for the introduction of the “multivector space.”

An Euclidean space is not technically a vector space but rather an affine space. The 3D Euclidean space is more general than a “vector space.” It is more general than a “multivector space” too.

Any 3D vector in can be written as

Any 3D multivector in can be written as

where , , , , , , , are coefficients. Since is a space of real numbers as the letter R indicates, the coefficients are real numbers.

**Multivector space = Clifford space**

The space containing all the 3D multivectors is called three-dimensional Clifford space . The basis elements of are , , , , , , .

The unit number can also be treated as a basis element associated with the coefficient .

In a 3D multivector there are 3 unit vectors (, , ) therefore the multiplicity of unit vectors in a general 3D multivector is 3.

In a 3D multivector there are 3 unit bivectors (, , ) therefore the multiplicity of unit bivectors in a general 3D multivector is 3.

In a 3D multivector there is 1 unit trivector () therefore the multiplicity of unit trivectors in a general 3D multivector is 1.

**n-dimensional Clifford space
**

The table [1] below shows the basis elements of n-dimensional Clifford space .

**Blades**

The basis elements (subspaces) of are known as *blades*.

Bivectors are blades of grade 2 (also known as 2-blades)

Trivectors are blades of grade 3 (also known as 3-blades)

Total Number of Blades in is .

Number of k-dimensional subspaces (k-blades) which is the multiplicity is given by

Multiplicity

**Is a trivector or pseudoscalar?
**

In the is a pseudoscalar as well as a trivector. Remember “trivector” is just an alternative name for . There is no mathematical significance in the name “trivector.”The name “pseudoscalar” has mathematical significance, however.

**What is a pseudoscalar?**

In physics pseudoscalar quantities change sign under parity operation. Parity operation is also known as space inversion. Space inversion means that the 3 spatial coordinates x,y,z change sign simultaneously.

The sign change of the trivector under parity operation is consistent with the multiplication table (axioms) of the basis elements of the 3D vector space. We will discuss this below. So, in this sense trivector is a pseudoscalar.

Parity operation is not well defined in 2D. So, if parity operation is defined for 3 space coordinates only then we should not generalize this definition of pseudoscalar. In my opinion, “pseudoscalar” terminology makes sense only in 3D. The term “pseudoscalar” has to be defined whenever we use it.

**Properties of the Basis Elements
**

A linear mathematical space is characterized by its basis elements. The characterization is not complete without defining the multiplication properties of the basis elements.

Any 3D vector in can be written as

Any quaternion (which is also embedded in in ) can be written as

where , , are the quaternion basis elements.

The vector and the quaternion has the same form. The similarity between the forms (as shown above) created a lot of confusion. People thought that a quaternion is some kind of vector. The confusion can be traced back to Gibbs-Heaviside who changed the definition of “vector.” In modern times we use the Gibbs-Heaviside definition of vector which is a directed line segment.

Vector and quaternion have the same form but their basis elements are completely different.

As a reminder, the multiplication rules for the basis elements are:

In the case of 3D vector basis elements the axioms are:

or, more generally

for

where the metric is

when and when

There are additional axioms for the trivector.

The axioms for 3D vector basis elements make sense geometrically if you remember that these basis vectors are orthonormal. They have unit length and they are perpendicular to each other. The geometric meanings of these basis elements and their products can be seen in this picture. Do not be bothered by the notation in the picture. Ignore for the time being. We will discuss it in the second installment of this tutorial series. For example, and is the same because by definition. In any case, we’ll discuss this new notation involving and in the second installment.

In the case of quaternion basis elements the axioms are:

Note the emphasis on the term “axiom” meaning that we start with these definitions. Everything else will be based on these axioms.

**Correspondence
**

Remember the multivector

The bivector part ( ) of the multivector is a quaternion.

To see this connection

we used the axioms (multiplication table) of the basis elements of the 3D vector space.

corresponds to quaternion basis element

corresponds to quaternion basis element

corresponds to quaternion basis element

**Observe
**

The bivectors and trivectors square to . Without this property the correspondence between bivector basis elements and quaternion basis elements cannot be established.

Some say that bivector and trivector (basis elements) are “imaginary” in the sense that they square to . I don’t emphasize this terminology.

**References**

[1] S. Franchini, G.Vassallo, F. Sorbello, “A Brief Introduction to Clifford Algebra“