Table of contents An Introduction to Geometric Algebra over R^2 Geometric Algebra applied to Physics The Shoelace Formula via Geometric Algebra

You should be familiar with all the concepts presented in An Introduction to Geometric Algebra over R^2 before proceeding. Continuing our exploration of Geometric Algebra, let's look at calculating area. GA might have some advantages here since the exterior product of two vectors is the area of the parallelogram they define. $$ \boldsymbol{a} \wedge \boldsymbol{b} $$ We are being a little sloppy here as the exterior product really gives you a scalar times e12, but we'll ignore that for the rest of this article, presuming we'll just read off the scalar as the oriented area.

Geometric Algebra can be applied to Physics, and many of the introductions to GA online cover this, but they immediately jump to electromagnetic fields or quantum mechanics, which is unfortunate since GA can also greatly simplify 2D kinematics. One such example is uniform circular motion. You should be familiar with all the concepts presented in An Introduction to Geometric Algebra over R^2 before proceeding. If we have a vector p that moves at a constant rate of ω rad/s and has a starting position p0, then we can describe the vector p very easily: $$\boldsymbol{p} = \boldsymbol{p_0} e^{\omega t \boldsymbol{I}}$$ Start/Stop Let's figure out what the derivative of a Rotor looks like, by first recalling its definition: $$ e^{\theta \boldsymbol{I}} := \cos(\theta) + \sin(\theta)\boldsymbol{I}$$ We take the derivative with respect to θ: $$ \begin{align*} \frac{d}{d \theta} e^{\theta \boldsymbol{I}} &= \frac{d}{d \theta} (\cos(\theta) + \sin(\theta)\boldsymbol{I}) \\ &= -\sin(\theta) + \cos(\theta)\boldsymbol{I} \\ \end{align*} $$ At this point observe that cos and sin just changed places, along with a sign change, but we know of another operation that does the same thing, which is multiplication by I, so we get: $$ \begin{align*} \frac{d}{d \theta} e^{\theta \boldsymbol{I}} &= \frac{d}{d \theta} (\cos(\theta) + \sin(\theta)\boldsymbol{I}) \\ &= -\sin(\theta) + \cos(\theta)\boldsymbol{I} \\ &= \boldsymbol{I} (\cos(\theta) + \sin(\theta)\boldsymbol{I}) \\ &= \boldsymbol{I} e^{\theta \boldsymbol{I}} \\ \end{align*} $$ Not only does the derivative have a nice neat expression, we can read off from the formula what is happening, which is that the derivative is a vector that is rotated 90 degrees from the original vector.

Linear Algebra Geometric Algebra Applications Multiplying Vectors Rotors Double Angle Formula Complex Numbers Characterizing B Ratios Conjugates Geometric Algebra is fascinating, and I believe solves a large number of problems that arise from a more traditional approach to vectors, but I've been very disappointed with the quality of books and explanations I've found, most of them zooming off into abstract realms too quickly, or spending an inordinate amount of time building up a generalized theory before finally getting to something useful.