Irodov Problem 1.5

The displacement vector of the first particle as a function of time is s1 = r1 + t.v1 and that of the second will be s2 = r2+ t.v2. When and if the particles collide, s1 must be equal to s2. Thus,
r1 + t.v1 = r2 + t.v2. Thus, r1-r2 = t.(v2-v1). Since t is only a scalar (1 dimensional) while r and v are vector (more than 1 dimensional), for this condition to be true, r1-r2 must be aligned in the same direction as v2-v1. Thus, (r1-r2)/|r1-r2| = (v2-v1)/|v2-v1|.