Rotations | Geometric algebra episode 13

#geometricalgebra #reflections #rotations #sandwichproduct We can easily compose two reflections into a rotation. This simple idea is expressed in a sandwich product. But we already had a simpler one-sided multiplication that performs rotations for us. So why do we even need the sandwich product? Find out in this video. You can support us on Patreon, where you can already watch all the Geometric Algebra videos and get access to exclusive content. Unfortunately, we won't be publishing any new videos after Geometric Algebra, at least not in the near future. Your support is still more than welcome of course: https://www.patreon.com/user?u=86649007 If you want to learn more about rotations or the sandwich product, here are some interesting links: [MOM 1]    • Geometric Algebra  - 3D Rotations and Rotors   When you rotate a vector that sticks out of the plane of rotation, the sandwich product automatically does exactly what we want. This video gives an elegant and easy-to-follow proof. [MOM 2]    • Geometric Algebra in 2D - Two Reflections ...   Combining two reflections to get a rotation. [AMD 1]    • Geometric Algebra 3   Alan MacDonald explains where the rotation formulas come from. He shows why we need a two-sided sandwich product and not just a one-sided traditional product. The details of the proof are less clear than in the Mathoma video. 0:00 Composing two reflections into a rotation 3:56 Complex multiplication rotates inside a plane 5:15 The sandwich product rotates objects that don't live in the plane 9:08 Treating parallel and orthogonal vectors differently 10:37 Turning the one-sided product into a sandwich 11:41 Conclusion This video is published under a CC Attribution license ( https://creativecommons.org/licenses/... )