Triangle Inequality/Vectors in Euclidean Space

Theorem
Let $\mathbf{x}$,$\mathbf{y}$ be vectors in $\R^n$.

Let $\left\Vert{\cdot}\right\Vert$ denote vector length.

Then:


 * $\left \Vert {\mathbf{x} + \mathbf{y} }\right \Vert \le \left \Vert {\mathbf{x}}\right \Vert + \left \Vert { \mathbf{y} }\right \Vert$

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


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

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:


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