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.

Consider the vectors u, v, and u + v (in any number of dimensions). Use the following steps to prove that |u + v| ≤ |u | + |v|. 

a. Show that |u + v|² = (u + v) • (u + v) = |u|² + 2u • v + |v|².

b. Use the Cauchy–Schwarz Inequality to show that |u + v|² ≤ (| u| + |v|)².

c. Conclude that |u + v| ≤ |u| + |v|.

d. Interpret the Triangle Inequality geometrically in R² or R³.