Answer
the vectors $u$ and $v$ satisfy the triangle inequality.
Work Step by Step
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.