Triangle Inequality for Vectors and Distances

Theorem

Let $\mathbf{u},\mathbf{v},\mathbf{w}$ vectors in $R^n$

Then

• $||\mathbf{u}+\mathbf{v}\le ||\mathbf{u}||+||\mathbf{v}||$ (Triangle inequality for vectors)
• $d(\mathbf{u},\mathbf{v})\le d(\mathbf{u},\mathbf{w})+d(\mathbf{w},\mathbf{v})$ ( Triangle inequality for distances)