On the beginning of this academic year, my math teacher gave us a piece of homework to practice mathematical writing: to write a short proof of the law of cosines using overleaf. While doing some elementary research if there is a simple or elegant proof, I found that there was a 3-dimensional equivalent of the law of cosines, or an extension on a 3 dimensional surface with constant curvature. I had also learnt linear algebra through MIT OpenCourseWare and the Berkeley summer camp in the previous year. I had also recently begun to find enjoyment in writing math papers, thanks to KSEF! With these, I, newly exposed to the field of linear algebra, **became intrigued whether the law of cosines could be extended to any finite dimension greater than 1. **

To answer this question, I searched up if there are previous works that have already been on this area. I first discovered some relevant concepts, such as a simplex and hypervolume. The rigorous definition of a hypervolume seems complex than I expected, so I will begin reading on it, or find a way to complete the derivation of the multidimensional law of cosines without a rigorous definition. I found a research done by another high school student on generalizing the shoelace theorem to a tetrahedron, which may be some help for this research. I also found two works that are closely related to this topic--__one by M A Murray-Lasso__ and __another by Ding__.

I was quite surprised that this theorem has not been investigated that much. In Murray-Lasso’s work, although very resourceful, a few errors in notations were noticeable, and was slightly lacking in mathematical rigour. Ding’s proof used a different method, which used the Divergence Theorem. However, neither thoroughly defined the multidimensional cosine or how the a n-dimensional simplex’s “face” (which would be a n-1 dimensional simplex) could be represented as a vector. After some research, I found the exterior product to be a suitable explanation, but I will have to study much more about the exterior product to maintain mathematical rigour.

My objective for this research is to follow Murray-Lasso’s method to prove the theorem, but with greater mathematical rigour and an introduction of concepts such as the exterior product.