Geometric Algebra for Electrical Engineers

Book update. Now includes recent work on best fit solutions.

October 1, 2023 Geometric Algebra for Electrical Engineers No comments , , , , , , , , ,


I’ve added a few new pages in the linear systems solution portion of my book, Geometric Algebra for Electrical Engineers.  This now includes the best fit content that was covered in my recent video and blog post on approximate solutions to linear systems.

The geometry that is associated with a Moore-Penrose or SVD-based pseudoinverse is not terribly obvious, and this result, providing the same answer, uses geometry exclusively.  I’ve included it in my book, since it’s a cool application, and not conceptually much trickier than the exact system solution.  This makes this section slightly more formal, as it now including an up front statement as a theorem — but that’s where formality ends, as I don’t formally prove the theorem.  I do, however, provide lots of examples and problems (with solutions), sufficient for the industrious to craft their own proof if desired.

The updated version of the book should be available on all amazon marketplaces within the next 3-5 days.  The free PDF version (and leanpub edition), both linked above, are already updated.


Updated figures in ‘Geometric Algebra for Electrical Engineers’

September 2, 2023 Geometric Algebra for Electrical Engineers , , ,

New version of the book is now published (online PDF and leanpub versions updated, with amazon updates in the approval pipeline)

  • V0.1.19-2 (Sep 2, 2023)
    • Reworked many of the Mathematica generated figures.  Now using the MaTeX[] extension to do the figure labelling (that was only done in a couple figures before this), as it looks much better, and is consistent with the fonts in the text.

      Each of these are individually very small changes, barely noticeable, but I think it makes a nice difference to overall quality.

      In many cases, I’ve generated new separate figures for the amazon paper editions of the book, using straight black instead of colors, so they don’t look as washed out, after conversion to black and white.

Here’s an example where just the captioning was changed:

The font is now whatever LaTeX uses for \\mathbf{n}, so it matches the text.

I think that the new Mathematica version (13.2) that I am using, also happens to render this 3D figure a bit nicer.

Here’s a comparison of one of the figures that now has a black and white specialization (old, new-color, new-bw):

In this particular case, I chose not to color the labels like I did previously, but I have retained that label color matching in some places.

Like I said, it’s a small difference, but the latex labelling just look better, period.  Notice that the numeric values at the tick marks on the border of the figure are not using a matching font (those are directly generated by Mathematica).  I’ll have to figure out how to make those use MaTeX too, and audit all the figures for that, but that’s a game for another day.

Video: Spherical basis vectors in geometric algebra

August 8, 2023 Geometric Algebra for Electrical Engineers , ,

I’ve made a new manim-based video with a geometric algebra application.

In the video, the geometric algebra form for the spherical unit vectors are derived, then unpacked to find the conventional vector algebra form. We will then use our new tools to find the expression for the kinetic energy of a particle in spherical coordinates.

Prerequisites: calculus (derivatives and chain rule), complex numbers (exponential polar form), and geometric algebra basics (single sided rotations, vector multiplication, vector commutation sign changes, …)

You can find the video on Google’s censorship-tube, or on odysee.

New revision of my book: Geometric Algebra for Electrical Engineers

May 16, 2022 Geometric Algebra for Electrical Engineers

A new version of my book is now available in free pdf form, in paperback and hardcover on amazon, on leanpub, and for the brave or crazy — as latex source files and a makefile.  All of these can be found here:

Geometric Algebra for Electrical Engineers.


Changes in this V0.1.16-13 (May 15, 2022) version, include:

  • Fix equations 2.81 2.82 — error found by Christopher
  • test drove build instructions (slight fix required.)
  • restore latexsym \Box as dAlembertian
  • gutter fixes.
  • perl -p -i -e ‘s/ *$//’ `cat spellcheckem.txt `
  • (massive change with no visible effects if done right): purge most dmath usage. MacTex upgrade has made these seemingly malfunction, and lines are getting split in all sorts of weird places.

[Part 1. Arrow representation of vectors] An introduction to geometric algebra.

August 2, 2020 Geometric Algebra for Electrical Engineers

This is a continuation of:

[Click here for a PDF of these posts with colored equations, and additional figures and commentary]


Cast yourself back in time, all the way to high school, where the first definition of vector that you would have encountered was probably very similar to the one made famous by the not very villainous Vector in Despicable Me [4].  His definition was not complete, but it is a good starting point:

Definition: Vector. A vector is a quantity represented by an arrow with both direction and magnitude.

All the operations that make vectors useful are missing from this definition, such as

  • a comparison operator,
  • a rescaling operation (i.e. a scalar multiplication operation that changes the length),
  • addition and subtraction operators,
  • an operator that provides the length of a vector,
  • multiplication or multiplication like operations.

The concept of vector, once supplemented with the operations above, will be useful since it models many directed physical quantities that we experience daily.  These include velocity, acceleration, forces, and electric and magnetic fields.

Vector comparison.

In fig. 1.1 (a), we have three vectors, labelled \( \Ba, \Bb, \Bc \), all with different directions and magnitudes, and in fig. 1.1 (b), those vectors have each been translated (moved without rotation or change of length) slightly. Two vectors are considered equal if they have the same direction and magnitude. That is, two vectors are equal if one is the image of the other after translation. In these figures \( \Ba \ne \Bb, \Bb \ne \Bc, \Bc \ne \Ba \), whereas any same colored vectors are equal.

Figure 1.1 (a): Three vectors

Figure 1.1 (a): Three vectors

Figure 1.1 (b): Example translations of three vectors.

Figure 1.1 (b): Example translations of three vectors.


Vector (scalar) multiplication.

We can multiply vectors by scalars by changing their lengths appropriately.

In this context a scalar is a real number (this is purposefully vague, as it will be useful to allow scalars to be complex valued later.)

Using the example vectors, some rescaled vectors include \( 2 \Ba, (-1) \Bb, \pi \Bc \), as illustrated in fig. 1.2.



Figure. 1.2 Scaled vectors.

Figure. 1.2 Scaled vectors.

Vector addition.

Scalar multiplication implicitly provides an algorithm for addition of vectors that have the same direction, as \( s \Bx + t \Bx = (s+t) \Bx \) for any scalars \( s, t \). This is illustrated in fig. 1.3 where \( 2 \Ba = \Ba + \Ba \) is formed in two equivalent forms. We see that the addition of two vectors that have the same direction requires lining up those vectors head to tail. The sum of two such vectors is the vector that can be formed from the first tail to the final head.

Figure 1.3. Twice a vector.

Figure 1.3. Twice a vector.


It turns out that this arrow daisy chaining procedure is an appropriate way of defining addition for any vectors.

Definition: Vector addition. The sum of two vectors can be found by connecting those two vectors head to tail in either order. The sum of the two vectors is the vector that can be formed by drawing an arrow from the initial tail to the final head. This can be generalized by chaining any number of vectors and joining the initial tail to the final head.

This addition procedure is illustrated in fig. 1.4, where \( \Bs = \Ba + \Bb + \Bc \) has been formed.

Figure 1.4. Addition of vectors.

Figure 1.4. Addition of vectors.

This definition of vector addition was inferred from the observation of the rules that must apply to addition of vectors that lay in the same direction (colinear vectors).  Is it a cheat to just declare that this rule for addition of colinear vectors also applies to arbitrary vectors?  Yes, it probably is, but it’s a cheat that works nicely, and one that models physical quantities that we experience daily (velocities, acceleration, force, …).  If you collect two friends you can demonstrate the workability of this inferred rule easily, by putting your arms out, and having your friends pull on them.  If you put your arms opposing to the sides, and have your friends pull with equal forces, you’ll see that the force that can be represented by the pulling of your friends add to zero.  If one of your friends is stronger, you’ll move more in that direction.  If you put your arms out at 45 degree angles, you’ll see that you move along the direction of the sum of the forces.  These scenarios are crudely sketched below in figure 1.x

Figure 1.x: Friends pulling on your arms.

Vector subtraction.

Since we can scale a vector by \( -1 \) and we can add vectors, it is clear how to define vector subtraction

Definition: Vector subtraction. The difference of vectors \( \Ba, \Bb \) is
\Ba – \Bb \equiv \Ba + ((-1)\Bb).

Graphically, subtracting a vector from another requires flipping the direction of the vector to be subtracted (scaling by \(-1\)), , and then adding both head to tail. This is illustrated in fig. 1.5.

Figure 1.5. Vector subtraction.

Figure 1.5. Vector subtraction.

Length and what’s to come.

It is easy to compute the length of a vector that has an arrow representation.
One simply lines a ruler of appropriate units along the vector and measures.

We actually want an algebraic way of computing length, but there is some baggage required, including

  • Coordinates.
  • Bases (plural of basis).
  • Linear dependence and independence.
  • Dot product.
  • Metric.

The next part of this series will cover these topics. Our end goal is geometric algebra, which allows for many coordinate free operations, but we still have to use coordinates, both to read the literature, and in practice. Coordinates and non-orthonormal bases are also a good way to introduce non-Euclidean metrics.


[4] Vector; supervillain extraordinaire (Despicable Me). A quantity represented by an arrow with direction and magnitude. Youtube. URL [Online; accessed 11-July-2020].

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