Answer
all the vectors spanned by the vectors $\{(1,0,0,-1),(0,1,0,2),(0,0,1,2)\}$ are orthogonal to $u$.
Work Step by Step
Let $u=(1-2,2,1)$ and $v=(a,b,c,d)$ such that $u$ is orthogonal to $v$, that is
$$u\cdot v=0 \Longrightarrow a-2b+2c+d=0.$$
Now, assume that, $a=r$ and $b=s$, $c=t$ then we get the solution,
$$a=r, \quad b=s, \quad c=t,\quad d=-r+2s-2t$$
that is, $v=(r,s,t,-r+2s-2t)=r(1,0,0,-1)+s(0,1,0,2)+t(0,0,1,2)$.
Hence, all the vectors spanned by the vectors $\{(1,0,0,-1),(0,1,0,2),(0,0,1,2)\}$ are orthogonal to $u$.
