Chapter 13 - Vector Geometry - 13.3 Dot Product and the Angle Between Two Vectors - Exercises - Page 666: 13

The angle between the vectors is obtuse.

Since $$\langle 1,1,1\rangle \cdot \langle 1,-2,-2\rangle =1-2-2=-3\neq 0$$ then the vectors are not orthogonal. The angle between the vectors is obtuse since their dot product is negative (negative cosine of the dot product indicates that we are in the 2nd quadrant).