Linear Algebra Orthogonality Proof with scalar

Question

Let $u, v\in \mathbb{R}^n$. Show that $u⋅v \leq \| u \| ~ \|v \|$

Hint: expand $\| u−cv \|$ for $c \in \mathbb{R}$

I'm confused how to prove this theorem. From the hint, I got $(u-cv)(u-cv)$. Can someone please help understand this concept. Thank you.

Answer 1

This is the Cauchy-Schwarz inequality, take a look here for different proofs.