Linear Algebra Done Right · Problem 6A.15

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · inner-products-and-norms

Problem

Suppose

u, v

are nonzero vectors in

\mathbf{R}^2

. Prove that

\langle u, v \rangle = \|u\|\, \|v\| \cos\theta,

where

\theta

is the angle between

u

and

v

(thinking of

u

and

v

as arrows with initial point at the origin). Hint: Use the law of cosines on the triangle formed by

u

v

, and

u - v