What is the definition of an angle? How is the angle between two vectors defined in linear algebra textbooks? What motivates this definition and what are the results of this definition? How can we have a valid cosine function that works outside of axiomatic geometry before we can even define an angle in mathbbRⁿ via the cosine function? Can everything that is provable via Euclid's axioms of geometry be proven using only algebra in mathbbRⁿ without any of those axioms? If not, What other axioms must we include ________.