Two Forms of the Dot Product

In ordinary Euclidean space, the dot product of two vectors is the product of the length of both vectors times the cosine of the angle between them.

realized that there were two radically different ways of looking at the instruction we were receiving, and that the two approaches were mutually incompatible. The first approach consists of treating mathematics as knowledge. Mathematical statements are information that you have to ingest and be able to recall. You have to learn the definitions, lea

When you see things that others don’t yet perceive, sharing your vision requires finding a way to get others to re-create those things in their own heads. A mathematical definition serves this purpose. It provides detailed instructions allowing others, starting with things they are already able to see, to mentally construct those new things.

the inner product, or “dot product,” between the two vectors: