Chapter 5 - Inner Product Spaces - 5.1 Length and Dot Product in Rn - 5.1 Exercises - Page 236: 58

the vectors $u$ and $v$ satisfy the triangle inequality.

Given the vectors ${u}=(1,1,1), {v}=(0,1,-2)$, then their lengths can be calculated as follows $$\|{u}\|=\sqrt{1+1+1}=\sqrt{3}, \quad \|{v}\|=\sqrt{0+1+4}=\sqrt{5}.$$ Also, $$\|{u+v}\|=(1,2,-1)=\sqrt{1+4+1}=\sqrt{6}.$$ One can see that, $$\|{u}+{v}\| \leq\|{u}\|+\|{v}\|.$$ Hence, the vectors $u$ and $v$ satisfy the triangle inequality.