Geometric Algebra Classwork (Reflection)
MultiV 2021-22 / Dr. Kessner

Let \(u=e_1\) and \(v = \dfrac{1}{\sqrt{2}}(e_1+e_2)\).

Define the transformations \(R_u(w) = uwu\), \(R_v(w) = vwv\), and \(R_{uv}(w) = (vu)w(uv)\).

1. Show that \(R_u(e_1+e_3) = e_1-e_3\) and \(R_u(u) = u\).

2. For a general \(w = w_xe_1+w_ye_2+w_ze_3\), show that \(R_u(w) = w_xe_1-w_ye_2-w_ze_3\). In other words, the \(y\) and \(z\) coordinates are negated. What is the transformation \(R_u\)?

3. Show that \(R_v(e_1)=e_2\), \(R_v(v) = v\) and \(R_v(e_1+e_3)=e_2-e_3\). What is the transformation \(R_v\)?

4. Show that \(R_{uv}(e_1) = e_2\), \(R_{uv}(e_3) = e_3\), and \(R_{uv}(e_1+e_3) = e_2+e_3\). What is the transformation \(R_{uv}\)? Note that \(R_{uv}=R_vR_u\).

5. Define the transformation \(M_x(w) = -R_u(w)\). Calculate \(M_x(w)\) for a general \(w = w_xe_1+w_ye_2+w_ze_3\). What is this transformation? Describe how the transformation changes the coordinates.

6. Define transformations \(M_y\) and \(M_z\) and show that they act as expected on a general \(w = w_xe_1+w_ye_2+w_ze_3\).