Deriving the law of cosines from the dot product by applying distributive and commutative properties

Deriving the law of cosines from the dot product by applying distributive and commutative properties.

First, we review some properties of the dot product: the dot product is both commutative and distributive, a vector difference points from the head of the second to the head of the first, and the dot product of a vector with itself yields the squared magnitude of the vector.

Then, we apply these ideas to a general triangle constructed of vectors for the sides. We dot the vector difference into itself and use the properties of the dot product to retrieve the law of cosines.