Triangle Inequality/Vectors in Euclidean Space

Theorem
Let $\mathbf x, \mathbf y$ be vectors in the real Euclidean space $\R^n$.

Let $\norm {\, \cdot \,}$ denote vector length.

Then:


 * $\norm {\mathbf x + \mathbf y} \le \norm {\mathbf x} + \norm {\mathbf y}$

If the two vectors are scalar multiples where said scalar is non-negative, an equality holds:


 * $\exists \lambda \in \R, \lambda \ge 0: \mathbf x = \lambda \mathbf y \iff \norm {\mathbf x + \mathbf y} = \norm {\mathbf x} + \norm {\mathbf y}$

Proof
Let $\mathbf x, \mathbf y \in \R^n$.

We have:

From the Cauchy-Bunyakovsky-Schwarz Inequality:

To prove that the equality holds if the vectors are scalar multiples of each other, assume:


 * $\exists \lambda \in \R, \lambda \ge 0: \mathbf v = \lambda \mathbf w$