Example:

Let's also look again at the vector-sum example, which is redrawn in Fig. 5.6. The norm of the vector sum $w=x+y$ is

$$\Vert w\Vert \isdef \Vert x+y\Vert \isdef \Vert(2, 3) + (4, 1)\Vert = \Vert(6, 4)\Vert = \sqrt{6^2 + 4^2} = \sqrt{52} = 2\sqrt{13}$$

while the norms of $x$ and $y$ are $\sqrt{13}$ and $\sqrt{17}$, respectively. We find that $\Vert x+y\Vert<\Vert x\Vert+\Vert y\Vert$ which is an example of the triangle inequality. (Equality occurs only when $x$ and $y$ are collinear, as can be seen geometrically from studying Fig. 5.6.)