Quick Intro to Geometric Algebra
Here are some concepts to bear in mind. This is a very brief introduction to geometric algebra; a longer one is here.
Bases and geometric products
Every dimension of space comes with a basis vector: an arrow of length 1 unit pointed towards the positive end of the axis.
In our 3-dimensional world, there are the basis vectors \(\hat x\), \(\hat y\), and \(\hat z\), which are pointed towards the positive ends of the \(x\), \(y\), and \(z\) axes respectively.
The fourth dimensional basis vector is \(\hat w\). In higher dimensions, usually all bases are numbered instead of lettered: the fifth dimensional basis vectors are \(\hat e_1\), \(\hat e_2\), \(\hat e_3\), \(\hat e_4\), and \(\hat e_5\).
The geometric product of two basis vectors is their simple multiplication - not the dot or cross product! The geometric product of \(\hat x\) and \(\hat y\) is simply \(\hat x \hat y\).
The geometric product of two basis vectors is a basis plane. \(\hat x \hat y\) is the basis plane of the \(xy\) plane. The other basis planes are \(\hat y \hat z\) and \(\hat x \hat z\).
The geometric product of three basis vectors is a basis volume. \(\hat x \hat y \hat z\) is the basis volume of 3D space, which only has one basis volume, but also one of the four basis volumes of 4D space.
The geometric product of a basis vector with itself is 1. That is, \(\hat x \hat x = \hat x^2 = \hat y \hat y = \hat y^2 = \hat z \hat z = \hat z^2 = 1\).
The geometric product of different basis vectors anticommutes: \(\hat x \hat y = - \hat y \hat x\) and \(\hat x \hat y \hat z = - \hat x \hat z \hat y = \hat z \hat x \hat y = - \hat z \hat y \hat x\).
Blades and multivectors
A blade is a scaled basis: a scalar (regular real number) multiplied by a basis. For example, \(3 \hat x \hat y\) is a blade. Note that this means all bases are blades scaled by 1.
A \(k\)-blade is a blade of grade \(k\): the geometric product of a scalar and \(k\) different basis vectors. \(3 \hat x \hat y\) has grade \(2\); it is a \(2\)-blade.
Scalars are \(0\)-blades - blades consisting of no basis vectors.
A multivector is a sum of multiple blades. For example, \(1 + 2 \hat x - 3 \hat y \hat z\) is a multivector.
The sum of multiple (and only) 1-blades is usually called a simple vector. For example, \(3 \hat x + 2 \hat y\) is a vector.
The sum of multiple (and only) 2-blades is a bivector. Basis planes are also known as basis bivectors. For example, \(3 \hat x \hat y\) is a bivector.
The rules of linearity, associativity and distributivity in multiplication apply, as long as order of arguments is maintained:
\((\hat x)(a \hat y) = a \hat x \hat y\) (linearity, for scalar \(a\))
\((\hat x \hat y)(\hat z) = \hat x (\hat y \hat z)\) (associativity)
\(\hat x (\hat y + \hat z) = \hat x \hat y + \hat x \hat z\) (distributivity)
\((\hat y + \hat z)(a \hat x) = a (\hat y + \hat z)(\hat x)\) (linearity) \(= a (\hat y \hat x + \hat z \hat x)\) (distributivity) \(= a (- \hat x \hat y - \hat x \hat z)\) (anticommutativity) \(= -a (\hat x \hat y + \hat x \hat z)\) (converse of distributivity)
However, some things which require commutativity break down, such as the binomial theorem.
The choose operator, inner (dot) and outer (wedge) products
\(\langle V \rangle_n\) chooses all \(n\)-blades from the multivector \(V\). For example, if \(V = 1 + 2 \hat x + 3 \hat y + 4 \hat x \hat y + 5 \hat y \hat z\), then \(\langle V \rangle_0 = 1\) and \(\langle V \rangle_1 = 2 \hat x + 3 \hat y\) and \(\langle V \rangle_2 = 4 \hat x \hat y + 5 \hat y \hat z\).
\(U \cdot V = \langle UV \rangle_n\) where \(U\) is of grade \(r\), \(V\) is of grade \(s\), and \(n = |r - s|\). This is the inner or dot product.
The dot product associates and distributes the same way the geometric product does.
From this, for arbitrary vectors \(a \hat x + b \hat y\) and \(c \hat x + d \hat y\), we recover the typical meaning of the dot product:
\[\begin{split}& (a \hat x + b \hat y) \cdot (c \hat x + d \hat y) \\ &= ac (\hat x \cdot \hat x) + ad (\hat x \cdot \hat y) + bc (\hat y \cdot \hat x) + bd (\hat y \cdot \hat y) \\ &= ac \langle \hat x \hat x \rangle_0 + ad \langle \hat x \hat y \rangle_0 + bc \langle \hat y \hat x \rangle_0 + bd \langle \hat y \hat y \rangle_0 \\ &= ac \langle 1 \rangle_0 + ad (0) + bc (0) + bd \langle 1 \rangle_0 \\ &\text{(because }\hat x \hat y\text{ and }\hat y \hat z\text{ have no part with grade }0\text{)} \\ &= ac + bd\end{split}\]
\(U \wedge V = \langle UV \rangle_n\) where \(U\) is of grade \(r\), \(V\) is of grade \(s\), and \(n = r + s\). This is the outer or wedge product.
The outer product associates and distributes the same way the geometric product does.
From this, for arbitrary vectors \(a \hat x + b \hat y + c \hat z\) and \(d \hat x + e \hat y + f \hat z\), we recover something that looks very much like a cross product:
\[\begin{split}&(a \hat x + b \hat y + c \hat z) \wedge (d \hat x + e \hat y + f \hat z) \\ &= ad (\hat x \wedge \hat x) + bd (\hat y \wedge \hat x) + cd (\hat z \wedge \hat x) \\ &\quad + ae (\hat x \wedge \hat y) + be (\hat y \wedge \hat y) + ce (\hat z \wedge \hat y) \\ &\quad + af (\hat x \wedge \hat z) + bf (\hat y \wedge \hat z) + cf (\hat z \wedge \hat z) \\ &= ad \langle \hat x \hat x \rangle_2 + be \langle \hat y \hat y \rangle_2 + cf \langle \hat z \hat z \rangle_2 \\ &\quad + (ae - bd) \langle \hat x \hat y \rangle_2 + (af - cd) \langle \hat x \hat z \rangle_2 + (bf - ce) \langle \hat y \hat z \rangle_2 \\ &= 0 + 0 + 0 + \begin{vmatrix} a & b \\ d & e \end{vmatrix} (\hat x \hat y) + \begin{vmatrix} a & c \\ d & f \end{vmatrix} (\hat x \hat z) + \begin{vmatrix} b & c \\ e & f \end{vmatrix} (\hat y \hat z) \\ &\text{(because }\hat x \hat x\text{ etc.} = 1\text{, which has no grade }2\text{ part)} \\ &= \begin{vmatrix} \hat y \hat z & \hat x \hat z & \hat x \hat y \\ a & b & c \\ d & e & f \end{vmatrix}\end{split}\]
Euler’s formula applied to multivectors
\(e^{\theta B} = \cos \theta + B \sin \theta\) where \(\theta\) is a scalar in radians and \(B\) is a basis multivector.
Note
This formula only works when \(B^2 = -1\) (in the same way as the imaginary unit \(i\)), such as \((\hat x \hat y)^2 = \hat x \hat y \hat x \hat y = - \hat x \hat y \hat y \hat x = - \hat x \hat x = -1\).
For purposes of interest, the more general formula for \(e\) raised to a multivector power is the Taylor series:
\[e^V = \exp(V) = \sum_{n=0}^{\infty} \frac{V^n}{n!}\]
For reasons that are beyond my power to explain, the rotation of a multivector \(V\) by \(\theta\) through the plane \(B\) is \(e^{-\frac{\theta B}{2}} V e^{\frac{\theta B}{2}}\).