what you have to realize is that the particles might not travel at the same rate.

So you can't just equate the components.

What you can do is let

$r_1(t)=<t, t^2, t^3>$

and

$r_2(s) = < 1+2s, 1+6s, 1+14s>$

and see if the 3 equations (each component) in two unknowns, $s$ and $t$, have a solution.

As it turns out there are 2 solutions, or points where the space curves intersect.

Colliding on the other hand requires that the two particles' paths intersect at a common time.

In this case you would just equate the components and as you saw there was no collision solution.

See if you can solve for the intersections.