Posted on Mon 01 October 2018 in Thesis • Tagged with thesis, quaternions, linear algebra
We're nearly ready to give the geometric interpretation of quaternionic multiplication that we've been working toward ever since we first defined the quaternions. Instead of considering the four-dimensional quaternions, we've spent the last few posts considering three-dimensional vectors, specifically discussing a product operation called the dot product that we can …
Posted on Wed 12 September 2018 in Thesis • Tagged with thesis, quaternions, linear algebra
Before the hiatus, we began discussing the geometry of vectors in three dimensions. At the end of that post, we discussed why it was impossible to define a multiplication operation on \(\mathbb{R}^3\) that also has an inverse (division) operation. However, there are still two useful "product" operations involving …
Posted on Mon 30 April 2018 in Thesis • Tagged with thesis, quaternions, linear algebra
The previous posts in this series discussed two-dimensional vector geometry and how that geometry can be connected to the algebra of complex numbers. The main result is that the multiplication of complex numbers corresponds geometrically to rotations and dilations of the 2-dimensional plane.
As mentioned in an earlier post, this …