Answer
$$\cos \theta= 0.$$
That is, $\theta=\frac{\pi}{2}$.
Work Step by Step
Let $u=(1,1,1),\quad v=(2,-2,2), \quad \langle u, v\rangle=u_1v_1+2u_2v_2+u_3v_3 $.
The angle $\theta$ between $u$ and $v$ is given by the formula
$$\cos \theta=\frac{ \langle u, v\rangle}{\|u\| \cdot\|v\|}=\frac{2-4+2}{\sqrt{ 1+2+1}\sqrt{4+8+4}}=0.$$
That is, $\theta=\frac{\pi}{2}$.
