Geometric Algebra HW 4 (Geometric Product in \(\mathbb{R}^3\))
MultiV 2021-22 / Dr. Kessner

1. Let \(w = e_1 + e_3\). Let \(w' = (e_2 e_1)w(e_1 e_2)\)

   Show that \(w' = -e_1 + e_3\).

   Draw \(w\) and \(w'\). Verify that \(w'\) is the result of reflecting \(w\) in \(e_1\), and then \(e_2\). Also verify that this equivalent to rotation by \(\pi\) in the \(e_1e_2\) plane.

2. Let \(w = e_1+e_3\). Let \(u = \frac{w}{\sqrt{2}}\) and \(v = e_1\). Note that \(u\) and \(v\) are unit vectors in the \(e_1e_3\) plane, and the angle between the two vectors is \(\frac{\pi}{4}\).

   Let \(w' = (vu)w(uv)\).

   Show that \(w' = e_1-e_3\).

   Draw \(w\) and \(w'\). Verify that \(w'\) is the result of rotating \(w\) by \(\frac{\pi}{2}\) in the \(e_1e_3\) plane .

3. Let \(w = e_1 + e_3\). Find two vectors \(u\) and \(v\) to represent rotation by \(-\frac{\pi}{4}\) in the \(e_2e_3\) plane. (Clockwise \(45^{\circ}\) if you’re on the positive \(e_1\) axis looking at the origin). Let \(w' = (vu)w(uv)\).

   Show that \(w' = e_1 + e_2\).