Geometric Algebra for Electrical Engineers

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

August 2, 2020 Geometric Algebra for Electrical Engineers No comments

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].

[Series intro] An introduction to geometric algebra.

July 25, 2020 Geometric Algebra for Electrical Engineers 3 comments , , , , , , , , , , , , , , , , ,

What’s in the pipe.

It’s been a while since I did any math or physics writing. This is the first post in a series where I plan to work my way systematically from an introduction of vectors, to the axioms of geometric algebra.  I plan to start with an introduction of vectors as directed “arrows”, building on that to discuss coordinates, tuples, and column matrix representations, and representation independent ideas. With those basics established, I’ll remind the reader about how generalized vector and dot product spaces are defined and give some examples. Finally, with the foundation of vectors and vector spaces in place, I’ll introduce the concept of a multivector space, and the geometric product, and start unpacking the implications of the axioms that follow naturally from this train of thought.

The applications that I plan to include in this series will be restricted to Euclidean spaces (i.e. where length is given by the Pythagorean law), primarily those of 2 and 3 dimensions.  However, it will be good to also lay the foundations for the non-Euclidean spaces that we encounter in relativistic electromagnetism (there is actually no other kind), and in computer graphics applications of geometric algebra, especially since we can do so nearly for free.  I plan to try to introduce the requisite ideas (i.e. the metric, which allows for a generalized dot product) by discussing Euclidean non-orthonormal bases.  Such bases have applications in condensed matter physics where there are useful for modelling crystal and lattice structure, and provide a hands conceptual bridge to a set of ideas that might otherwise seem abstract and without “real world” application.


Many introductions to geometric algebra start by first introducing the dot product, then bivectors and the wedge product, and eventually define the product of two vectors as the synthetic sum of the dot and wedge
\Bx \By = \Bx \cdot \By + \Bx \wedge \By.
It takes a fair amount of work to do this well. In the seminal work [4] a few pages are taken for each of the dot and wedge products, showing the similarities and building up ideas, before introducing the geometric product in this fashion. In [2] the authors take a phenomenal five chapters to build up the context required to introduce the geometric product.  I am not disparaging the authors for taking that long to build up the ideas, as their introduction of the subject is exceedingly clear and thorough, and they do a lot more than the minumum required to define the geometric product.

The strategy to introduce the geometric product as a sum of dot and wedge can result in considerable confusion, especially since the wedge product is often defined in terms of the geometric product
\Bx \wedge \By =
\inv{2} \lr{
\Bx \By – \By \Bx
The whole subject can appear like a chicken and egg problem. I personally found the subject very confusing initially, and had considerable difficulty understanding which of the many identities of geometric algebra were the most fundamental. For this reason, I found the axiomatic approach of [1] very refreshing. The cavaet with that work is that is is exceptionally terse, as they jammed a reformulation of most of physics using geometric algebra into that single book, and it would have been thousands of pages had they tried to make it readable by mere mortals.

When I wrote my own book on geometric algebra, I had the intuition that the way to introduce the subject ought to be like the vector space in abstract linear algebra. The construct of a vector space is a curious and indirect way to define a vector. Vectors are not defined as entities, but simply as members of a vector space, a space that is required to have a set of properties. I thought that the same approach would probably work with multivectors, which could be defined as members of a multivector space, a mathematical construction with a set of properties.

I did try this approach, but was not fully satisfied with what I wrote. I think that dissatisfaction was because I tried to define the multivector first. To define the multivector, I first introduced a whole set of prerequisite ideas (bivector, trivector, blade, k-vector, vector product, …), but that was also problematic, since the vector multiplication idea required for those concepts wasn’t fully defined until the multivector space itself was defined.

My approach shows some mathematical cowardness. Had I taken the approach of the vector space fully to heart, the multivector could have been defined as a member of a multivector space, and all the other ideas follow from that. In this multi-part series, I’m going to play with this approach anew, and see how it works out.  If it does work, I’ll see if I can incorporate this approach into a new version of my book.

Review and background.

In this series, I’m going to assume a reader interested in geometric algebra, is probably also familiar with a wide variety of concepts, including but not limited to

  • vectors,
  • coordinates,
  • matrices,
  • basis,
  • change of basis,
  • dot product,
  • real and complex numbers,
  • rotations and translations,
  • vector spaces, and
  • linear transformations.

Despite those assumptions, as mentioned above, I’m going to attempt to build up the basics of vector representation and vector spaces in a systematic fashion, starting from a very elementary level.

My reasons for doing so are mainly to explore the logical sequencing of the ideas required.  I’ve always found well crafted pedagogical sequences rewarding, and will hopefully construct one here that is appreciated by anybody who chooses to follow along.

Next time.

As preparation for the next article in this series, the reader is asked to watch a short lesson from Vector, not so supervillain extraordinaire (Despicable Me).


[1] C. Doran and A.N. Lasenby. Geometric algebra for physicists. Cambridge University Press New York, Cambridge, UK, 1st edition, 2003.

[2] L. Dorst, D. Fontijne, and S. Mann. Geometric Algebra for Computer Science. Morgan Kaufmann, San Francisco, 2007.

[4] D. Hestenes. New Foundations for Classical Mechanics. Kluwer Academic Publishers, 1999.

A price increase for Geometric Algebra for Electrical Engineers

May 28, 2020 Geometric Algebra for Electrical Engineers No comments , , , ,

As of this week (end of May 2020), I raised the price of the black and white version of my Geometric Algebra book slightly (from $12 to $14.50 USD).  I say slightly, despite the 17% price increase, because the price is still pretty low from an absolute value perspective, as the markup I’d added to the minimum price was fairly small.  This price increase was an experiment in response to a reseller (SuperBookDeals) buying copies at $12 and then reselling them at higher prices.  For some reason amazon lists the higher price reseller copies before their own kindle-direct-publishing version, so a buyer had to go out of their way to find the lowest priced version.

I wouldn’t care if resellers undercut my list price, and then got a preferential listing from amazon.  The fact that this reseller doesn’t play this game with the color version of the book, which has a much higher printing cost (I haven’t changed my price for that, and am still selling it for $40 USD), suggests to me that I’d set the price too low for the black and white version.

If you are interested in a copy of the book, but don’t like the new higher price, please note that the (color) PDF version is still available for free.

I may drop the price back to the original $12 later, but for now I’m going to charge $14.50, and am curious to see how the pricing game plays out.

Note that a temporary side effect of me having changed the price is that SuperBookDeals appears to have dropped their price of one of their listings below $12 (my original price) to clear out their stock. Amazon also appears to be offering a couple copies at the old $12 price, which now lists as a sale price.


Finding the cheapest copy of my geometric algebra book on amazon

May 3, 2020 Geometric Algebra for Electrical Engineers 1 comment , , ,

My book, “Geometric Algebra for Electrical Engineers” is available as a free PDF here on my website, but also available in color ($40) and black-and-white ($12) formats on amazon.  Both versions are basically offered close to cost, should the reader be like me, preferring a print copy that can be marked up.  In fact, I made it available initially just so that I could get a cheap bound copy for my own use that I could mark up myself.

I noticed today that amazon now hides the cheapest version of my book, and seems shows the price of a reseller first.  For example, if you click the link to the $12 black-and-white version, it now appears that the book is selling for $13.01

but if you click on “Other Sellers”, the kindle-direct (print on demand) version that amazon offers itself hides further down in the list of sellers.  The version that I’m selling directly through is third on the list, despite it being the cheapest:

I guess that I’ve priced the black-and-white version of the book so low, that there are resellers that are willing to try to make some profit selling their own copies.  Do they depend on amazon giving them preferential listing order to make those sales?  I wonder how many of the people who have bought my book have ended up accidentally paying a higher price, using one of these resellers?

It does not appear that any resellers have played this game with the color version of the book, which has a higher price point.  I’m curious now to look at the sales stats for the two variations of the book to see how many of each version are selling (hardly any in either case, as the subject matter is too esoteric, but it was actually enough over the whole year that I did include the revenue on my income taxes.)

Amazon’s kindle-direct now has Canadian manufacturing

December 14, 2019 Geometric Algebra for Electrical Engineers No comments , ,

As a “kdp” author, I got an email about new Canadian manufacturing for kindle-direct orders (i.e. my Geometric Algebra book and various UofT physics and engineering class notes compilations.)

Here’s a fragment of that email:

“We’re excited to announce paperback manufacturing in Canada! This enables new features for KDP authors, including:

    • Faster shipping to your readers in Canada. Manufacturing in Canada enables FREE Two-Day Shipping for Prime Members.

Please note that, as of today, proof copies and author orders for authors in Canada will still be printed and shipped from the US.”

With the low price that I set my book prices at, paying just the US shipping for an “author proof” has been about the same as ordering a normal copy, so now there will really be no point to ordering proofs anymore.

“2nd” edition of “Geometric Algebra for Electrical Engineers”

May 4, 2019 Geometric Algebra for Electrical Engineers No comments

I’ve refreshed my Geometric Algebra for Electrical Engineers book, which could be considered a 2nd edition of sorts. The amazon color and black-and-white versions have been updated, as well as the pdf and the leanpub version (all of those are in available in the previous link.)


V0.1.15-6 (May 2, 2019)

  • Update figures (thicker lines, remove some ticks, …) and link them to the mathematica link anchors.
  • “in figure fig.” -> “in fig”.
  • Extend my hacks of the classic thesis template to use 6×9 with smaller than default margins. Now have the preface page numbers not in the bleed area of the page.
  • Split colorlablebox into separate .sty (for phy452 notes.)
  • Fix pdfbookmarks for contents and list of figures (so that they don’t show up under the preface)
  • Index quaternion (Bruce Gould)
  • GAelectrodynamics.tex: Want scrheadings starting before contents otherwise page numbers are out of bounds (and the page headings are MIA)
  • Bruce: “May I suggest that the proofs should have the end-of-proof symbol at the end?” Used the amsthm proof environment to do this.
  • Theorem 1.2: turn the converse into a footnote, to be seen later. (Bruce)
  • Added Bruce Gould to the thanks.


Why to study electromagnetism with geometric algebra.

February 3, 2019 Geometric Algebra for Electrical Engineers No comments

The current draft of my book really ought to have some motivation in the preface. This is what I was thinking of.

Why you want to read this book.

When you first learned vector algebra you learned how to add and subtract vectors, and probably asked your instructor if it was possible to multiply vectors. Had you done so, you would have been told either “No”, or a qualified “No, but we can do multiplication like operations, the dot and cross products.” This book is based on a different answer, “Yes.” A set of rules that define a coherent multiplication operation are provided.

Were you ever bothered by the fact that the cross product was only defined in three dimensions, or had a nagging intuition that the dot and cross products were related somehow? The dot product and cross product seem to be complimentary, with the dot product encoding a projection operation (how much of a vector lies in the direction of another), and the magnitude of the cross product providing a rejection operation (how much of a vector is perpendicular to the direction of another). These projection and rejection operations should be perfectly well defined in 2, 4, or N dimemsions, not just 3. In this book you will see how to generalize the cross product to N dimensions, and how this more general product (the wedge product) is useful even in the two and three dimensional problems that are of interest for physical problems (like electromagnetism.) You will also see how the dot, cross (and wedge) products are all related to the vector multiplication operation of geometric algebra.

When you studied vector calculus, did the collection of Stokes’s, Green’s and Divergence operations available seem too random, like there ought to be a higher level structure that described all these similar operations? It turns out that such structure is available in the both the language of differential forms, and that of tensor calculus. We’d like a toolbox that doesn’t require expressing vectors as differentials, or resorting to coordinates. Not only does geometric calculus provides such a toolbox, it also provides the tools required to operate on functions of vector products, which has profound applications to electromagnetism.

Were you offended by the crazy mix of signs, dots and cross products in Maxwell’s equations? The geometric algebra form of Maxwells’s equation resolves that crazy mix, expressing Maxwell’s equations as a single equation. The formalism of tensor algebra and differential forms also provide simpler ways of expressing Maxwell’s equations, but are arguably harder to relate to the vector algebra formalism so familiar to electric engineers and physics practitioners. In this book, you will see how to work with the geometric algebra form of Maxwell’s equation, and how to relate these new techniques to familiar methods.

My book (Geometric Algebra for Electrical Engineers) now available in paper.

January 29, 2019 Geometric Algebra for Electrical Engineers No comments

Edition 0.1.14 of my first book, Geometric Algebra for Electrical Engineers is now available, in a variety of pricing options:

Both paper versions are softcover, and have a 6×9″ format, whereas the PDF is formatted as letter size.  The leanpub version was made when I had the erroneous impression that it was a print on demand service like kindle-direct-publishing (aka createspace.) — it’s not, but the set your own price aspect of their service is kind of neat, so I’ve left it up.

If you download the free PDF or buy the black and white version, and feel undercharged, feel free to send some bitcoin my way.