hmm... well the book did give one lousy example:

find a vector function that represents the curve of intersection of the cylinder x² + y² = 1 and the plane y + z = 2

answer: the projection of C (curve of intersection) onto the xy plane is the circle x² + y² = 1, z=0.  so we know that we can write
                                     x=cost  y=sint       0≤t≤2pi
from the equation of the plane we have z = 2 - y = 2 - sint

so we can write parametric equations for C as
       x=cost  y=sint  z= 2 - sint     0≤t≤2pi
the corresponding vector equation is
       r(t) = costi + sintj + (2-sint)k     0≤t≤2pi

i hope someone can make sense out of it.