The vector-parametric representation of a line is:
L(t) = p0 + tv
Note that, for a given line, there are many possible representations, so we can't check for this line simply by comparing functions of t.
Instead, we check if the student's and professor's lines have parallel direction vectors, and that the vector between the student's and professor's point is also parallel to the direction vectors.
Note that p0 can be obtained from L(0) and v from
L(1) - L(0).