The question says that you have a body under circular motion, okay?
Now, that body is rotating around an axis. The axis is parallel to the vector <1, -2, 2>, and hence has this very direction.
The axis is perpendicular to the direction of motion of the body, right? Jcaron used a method called dot product. This is something that can be used to check whether two directional vectors are perpendicular or not, giving a result of 0 when the vectors are perpendicular.
So what we need, is to find the velocity vector, which also gives the direction of the body. So, if we find the dot product of the two vectors (velocity and axis), we should get the result 0 since they are perpendicular. This is where the work that jcaron did comes into play, that is, finding the dot product of the two vectors, one being the directional vector of the velocity, <-2, 3, 4> and the others in parts b to d and the other being the direction of the axis vector, <1, -2, 2>.
I hope this makes sense.