Mathematics — Linear Algebra · inner-products-and-norms
Problem
The angle between two vectors (thought of as arrows with initial point at the origin) in R2 or R3 can be defined geometrically. However, geometry is not as clear in Rn for n>3. Thus the angle between two nonzero vectors x,y∈Rn is defined to be
arccos∥x∥∥y∥⟨x,y⟩,
where the motivation for this definition comes from Exercise 15. Explain why the Cauchy--Schwarz inequality is needed to show that this definition makes sense.