The definition u v = |u| |v| cos implies that |u v| |u||v|

The definition u • v = |u| |v| cos θ implies that |u • v| ≤ |u||v| (because |cos θ| ≤ 1). This inequality, known as the Cauchy–Schwarz Inequality, holds in any number of dimensions and has many consequences.

Show that

(u₁ + u₂ + u₃)² ≤ 3(u²₁ + u²₂ + u³₂),

for any real numbers u₁, u₂, and u₃. Use the Cauchy- Schwarz Inequality in three dimensions with u = (u₁, u₂, u₃) and choose v in the right way.)