Maths - 3D Clifford Algebra - Arithmetic

There is a 3D mulivector calculator at the end of this page.

The arithmatic on this page asumes the operands are a general case of a 3D multivector:

a.e + a.e1 e₁ + a.e2 e₂ + a.e3 e₃ + a.e12 e₁^ e₂ + a.e31 e₃^ e₁ + a.e23 e₂^ e₃ + a.e123 e₁^ e₂^ e₃

and a second multivector:

b.e + b.e1 e₁ + b.e2 e₂ + b.e3 e₃ + b.e12 e₁^ e₂ + b.e31 e₃^ e₁ + b.e23 e₂^ e₃ + b.e123 e₁^ e₂^ e₃

Adding multi vectors numbers

Just add each component independently as follows:

e = a.e + b.e
e1 = a.e1 + b.e1
e2 = a.e2 + b.e2
e3 = a.e3 + b.e3
e12 = a.e12 + b.e12
e31 = a.e31 + b.e31
e23 = a.e23 + b.e23
e123 = a.e123 + b.e123

This operation will be coded in the multi3d class (see this class here).

When adding blades of different grade then we cant reduce it further and we leave the + in the number.

For example:

3 + 4 e1 + 5 e12

added to

5 + 4 e2 + 3 e12

gives

8 + 4 e1 + 4 e2 + 8 e12

Subtracting multi vectors numbers

Just subtract each component independently as follows:

e = a.e - b.e
ex = a.ex - b.ex
ey = a.ey - b.ey
ez = a.ez - b.ez
exy = a.exy - b.exy
ezx = a.ezx - b.ezx
eyz = a.eyz - b.eyz
exyz = a.exyz - b.exyz

This operation will be coded in the multi3d class (see this class here).

Multiplying 3D multivectors numbers (geometric product)

The main type of multiplication, which is described here, is geometric multiplication.

So if we have a general case of a 3D multivector:

a.e + a.e1 e₁ + a.e2 e₂ + a.e3 e₃ + a.e12 e₁^ e₂ + a.e31 e₃^ e₁ + a.e23 e₂^ e₃ + a.e123 e₁^ e₂^ e₃

and a second multivector:

b.e + b.e1 e₁ + b.e2 e₂ + b.e3 e₃ + b.e12 e₁^ e₂ + b.e31 e₃^ e₁ + b.e23 e₂^ e₃ + b.e123 e₁^ e₂^ e₃

Since * is distributive over + we can multiply out the terms:

(a.e + a.e1 e₁ + a.e2 e₂ + a.e3 e₃ + a.e12 e₁^ e₂ + a.e31 e₃^ e₁ + a.e23 e₂^ e₃ + a.e123 e₁^ e₂^ e₃) * (b.e + b.e1 e₁ + b.e2 e₂ + b.e3 e₃ + b.e12 e₁^ e₂ + b.e31 e₃^ e₁ + b.e23 e₂^ e₃ + b.e123 e₁^ e₂^ e₃)

this gives the result:

This is derived from the multipication table:

This multipication table is derived on this page.

alternative method

Some programs implement multiplication by using Eddington Basis

Eddington Basis for aMonad
E Number 1 : 0, 0, 0, *0
E Number 2 : 0, 0, 1, *1
E Number 3 : 0, 0, 2, *2
E Number 4 : 0, 0, 3, *3
E Number 5 : 0, 1, 2, *6
E Number 6 : 0, 1, 3, *7
E Number 7 : 0, 2, 3, *11
E Number 8 : 1, 2, 3, *27


Basis products for aMonad
E Number 1 : 1, 2, 3, 4, 5, 6, 7, 8, *0
E Number 2 : 2, 1, 5, 6, 3, 4, 8, 7, *1
E Number 3 : 3, -5, 1, 7, -2, -8, 4, -6, *2
E Number 4 : 4, -6, -7, 1, 8, -2, -3, 5, *3
E Number 5 : 5, -3, 2, 8, -1, -7, 6, -4, *6
E Number 6 : 6, -4, -8, 2, 7, -1, -5, 3, *7
E Number 7 : 7, 8, -4, 3, -6, 5, -1, -2, *11
E Number 8 : 8, 7, -6, 5, -4, 3, -2, -1, *27


Basis products for aMonad
E Number 1 : 1, 2, 3, 4, 5, 6, 7, 8, *0
E Number 2 : 2, -1, 5, 6, -3, -4, 8, -7, *1
E Number 3 : 3, -5, -1, 7, 2, -8, -4, 6, *2
E Number 4 : 4, -6, -7, -1, 8, 2, 3, -5, *3
E Number 5 : 5, 3, -2, 8, -1, 7, -6, -4, *6
E Number 6 : 6, 4, -8, -2, -7, -1, 5, 3, *7
E Number 7 : 7, 8, 4, -3, 6, -5, -1, -2, *11
E Number 8 : 8, -7, 6, -5, -4, 3, -2, 1, *27

Inner and Outer products

In addition to the geometric product there are two more types of multiplication used in Geometric Algebra. These extend and generalise the 'dot' and 'cross' products used in 3D vector algebra.

Inner product by a vector reduces the grade of a multivector. It is related to the dot product.

Outer product by a vector increases the grade of a multivector. It is related to the cross product.

Outer product

We want the outer product to be an extension of the vector cross product:

x = Ay * Bz - By * Az
y = Az * Bx - Bz * Ax
z = Ax * By - Bx * Ay

This gives a bivector which is mutually perpendicular to the vectors being multiplied. But how do we extend this so that we can multiply non-vectors? There are different definitions for the outer product so we need to be careful, if you use other websites or books you may find that they use different definitions. Here is the full 3D Outer multiplication table:

This is explained here.

Inner product.

We want the outer product to be an extension of the vector dot product:

A • B = Ax * Bx + Ay * By + Az * Bz

This gives a scalar which depends on the angle between the vectors being multiplied, it is zero if the vectors are perpendicular and zero if the vectors are parallel. But how do we extend this so that we can multiply non-vectors? There are different definitions for the inner product so we need to be careful, if you use other websites or books you may find that they use different definitions. Here we will use the 'semi-commutative inner product' which has the following 3D Inner multiplication table:

This is explained here.

Division

We don't tend to use the notation for division, since multivector multiplication is not commutative we need to be able to distinguish between [a][b]^-1 and [b]^-1[a]. So instead of a divide operation we tend to multiply by the inverse.

So the problem is, how to calculate the inverse of a multivector, this is discussed here.

Rules of Mathematical Operations

Geometric Algebras Categories