Dot product: Component vs. Geometric definition

The goal of this post is to answer a simple question: why are the following two definitions of the vector dot product in Euclidean space [1] equivalent for vectors and : Component definition: Geometric definition: , where is the magnitude of and is the angle between the vectors’ directions Here’s a graphical depiction of our vectors (focusing on for clarity, though this applies to any-dimensional vectors). It shows both the components of the vectors and the angle between them. The length of the arrow for is . We’ll show two proofs of the equivalence here, the geometric proof and the projection proof. The Appendix describes some properties of dot products that facilitate these proofs. Geometric proof We’ll be using this diagram of our vectors and , as well as the vector : Using the law of cosines [2] on the triangle formed by the three vectors: Since for any vector , we have (see Appendix), let’s rewrite this equation as: But and the dot product obeys the distributive property (see…