Counterexample of triangle inequality property for a Real Euclidean Space

Question

According to Apostol's Calculus Vol. II, Equality holds for the triangle inequality in the following scenarios:

Let $\mathbb{x}$ and $\mathbb{y}$ be two vectors. Then equality holds if

• $\mathbb{x} = c\mathbb{y}$ for $c \geq 0$.

However, it does not state that this is a necessary condition, and neither does ProofWiki. My question is: is there a case where equality holds in the triangle inequality where this condition is not met? Thank you.