Answer
$$\theta=\frac{\pi}{2}.$$
Work Step by Step
\begin{align*}
\cos \theta &=\frac{\langle 1,1,-1\rangle \cdot \langle1,-2,-1\rangle}{\|\langle 1,1,-1\rangle\|\|\langle 1,-2,-1\rangle\|}\\
&=\frac{1-2+1}{\sqrt{1+1+1}\sqrt{1+4+1}}\\
&=\frac{0}{\sqrt{3}\sqrt{6}}\\
&=0.
\end{align*}
That is $$\theta=\cos^{-1}0=\frac{\pi}{2}.$$
