The definition u v = |u| |v| cos implies that |u v| |u||v| (because |cos | 1). This inequality, known

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.

Use the vectors u = (√a, √9) and v = (√b, √a) to show that √ab ≤ (a + b)/2, where a ≥ 0 and b ≥ 0.