Answer
$$\theta=\cos^{-1}\frac{1}{\sqrt{3}}.$$
Work Step by Step
\begin{align*}
\cos \theta &=\frac{\langle 1,1,1\rangle \cdot \langle 2,-1,2\rangle}{\|\langle 1,1,1\rangle\|\|\langle 2,-1,2\rangle\|}\\
&=\frac{2-1+2}{\sqrt{1+1+1}\sqrt{4+1+4}}\\
&=\frac{3}{\sqrt{3}\sqrt{9}}\\
&=\frac{1}{\sqrt{3}}
\end{align*}
That is $$\theta=\cos^{-1}\frac{1}{\sqrt{3}}.$$
