My brief reading of this suggest it isn't mathematically sound in terms of its presentation.<p>On page 9 he discusses what he calls the geometric product. The problem is that the geometric product of two vectors is not a vector. This means that the space he is really working is larger than the vector space he started with. This isn't explained to the reader. What is this larger space?<p>Equation (7) shows that the geometric product of a and b is a.b + a wedge b. From this it's clear that he is working in the exterior algebra of the vector space.<p>Reading further shows that what he is really doing is giving geometric meanings to the operations in the exterior algebra where V is R^2 and R^3. This is useful and I think it has merit but I also think one should start with the proper setting.<p>Looking on page 9 going from (11) to (12) requires quite a leap. He says to square (11) but the right hand side of (11) is not a vector and the geometric product of this object hasn't been defined. It was only defined for vectors in V and not for other elements of the exterior algebra.
There are two very good introductory textbooks
on the subject by the same author:<p><a href="http://faculty.luther.edu/~macdonal/laga/" rel="nofollow">http://faculty.luther.edu/~macdonal/laga/</a>
<a href="http://faculty.luther.edu/~macdonal/vagc/index.html" rel="nofollow">http://faculty.luther.edu/~macdonal/vagc/index.html</a>
An article introducing Geometric Algebra (<i>A Unified Mathematical Language for Physics and Engineering</i>) was posted to HN (<a href="https://news.ycombinator.com/item?id=8192054" rel="nofollow">https://news.ycombinator.com/item?id=8192054</a>) and may be of interest.<p>It's authors also have other articles and teaching resources on their website: <a href="http://geometry.mrao.cam.ac.uk/" rel="nofollow">http://geometry.mrao.cam.ac.uk/</a>