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*Derivation and Generalization of the Law of Cosines onto the nth Dimension  (2021)

S. Yoon.

international Baccalaureate (IB) Extended Essay

While the Law of Cosines is one of the most well-known and straightforward theorems in mathematics, would it also hold true in higher dimensions? While much research has been conducted on the generalization of Cosine or operators such as the cross product to any arbitrary dimension, there is a lack of research on bringing the Law of Cosines to a higher dimension. It must be noted that a couple of research has successfully generalized the theorem, but this report will prove the theorem without the use of the Divergence theorem yet with greater mathematical rigour. By using a basic property of determinants and the exterior product on a n dimensional simplex, we were able to successfully prove the Law of Cosines in any finite dimension. Finally, we were able to confirm that the law is true by applying it to a 4 dimensional simplex.

An implication of this theorem is the multidimensional Pythagorean rule for special cases of the simplex. While the theorem might not have practical applicability due to the vast amount of information about the simplex needed to apply the theorem to a simplex, introducing the exterior product is new in research on this theorem. Further research may be conducted to generalize the theorem to a hyperbolic simplex, combining the hyperbolic law of cosines and the multidimensional law of cosines investigated in this report.

Keywords: Simplex, Inner product, Exterior product, Law of Cosines, Determinant

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