Geometric Algebra HW 2 (Geometric Product)
MultiV 2021-22 / Dr. Kessner

1. For each of the following vectors, find the inverse. Draw the unit circle on the plane, and draw each vector and its inverse.

   1. \(u = \begin{pmatrix} 2 \\ 0 \end{pmatrix}\)

   2. \(v = \begin{pmatrix} 0 \\ 3 \end{pmatrix}\)

   3. \(w = \begin{pmatrix} \frac{\sqrt{3}}{2} \\ \frac{1}{2} \end{pmatrix}\)

   4. \(x = \begin{pmatrix} \sqrt{3} \\ 1 \end{pmatrix}\)

Answers: \(u^{-1} = \begin{pmatrix} \frac{1}{2} \\ 0 \end{pmatrix}\), \(v^{-1}= \begin{pmatrix} 0 \\ \frac{1}{3} \end{pmatrix}\), \(w^{-1} = w = \begin{pmatrix} \frac{\sqrt{3}}{2} \\ \frac{1}{2} \end{pmatrix}\), \(x^{-1} = \begin{pmatrix} \frac{\sqrt{3}}{4} \\ \frac{1}{4} \end{pmatrix}\)

2. Let \(u = e_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\).
   Let \(v = (\cos\dfrac{\pi}{6})e_1+(\sin\dfrac{\pi}{6})e_2 = \begin{pmatrix} \frac{\sqrt{3}}{2} \\ \frac{1}{2} \end{pmatrix}\).

   Show the following:

   1. \(uv = (\cos\dfrac{\pi}{6})+(\sin\dfrac{\pi}{6})e_1 e_2 = \frac{\sqrt{3}}{2}+\frac{1}{2}e_1 e_2\)

      (\(uv\) is a rotor representing a rotation by \(\frac{\pi}{6}\).)

   2. \(vu = \frac{\sqrt{3}}{2}-\frac{1}{2}e_1 e_2\)

      (\(vu\) is a rotor representing a rotation by \(-\frac{\pi}{6}\).)

   3. \(vuv = v(uv) = (vu)v = \begin{pmatrix} \frac{1}{2} \\ \frac{\sqrt{3}}{2} \end{pmatrix}\)

      (applying \(uv\) on the right (or \(vu\) on the left) rotates \(v\) by \(\frac{\pi}{6}\))

   4. \(vvu = v(vu) = (uv)v = e_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\)

      (applying \(vu\) on the right (or \(uv\) on the left) rotates \(v\) by \(-\frac{\pi}{6}\))

   5. \(uvuv = \frac{1}{2} + \frac{\sqrt{3}}{2} e_1 e_2\)

      (\(uvuv = (uv)^2\) is a rotor representing rotation by \(\frac{\pi}{3}\))

   6. \((vu)(uv) = 1\)

   7. \(vuvuv = e_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}\)

   8. \(uvuvuv = e_1 e_2\)

      (\(uvuvuv = (uv)^3\) a rotor representing rotation by \(\frac{\pi}{2}\) (the unit bivector))