# Parametric curves intersection

I would like to ask how to solve following problem:

We have 2 "super heroes". Their path are given by parameterized function where t means time: $r_{1}(t)=$$\begin{bmatrix}t\\t^2\\t^3\end{bmatrix} and r_{2}(t)=$$\begin{bmatrix} 1+2t \\1+6t\\1+14t\end{bmatrix}$

The question is: Where do their path intersect (if they)? Are they ever going to meet if they start at the same time?

I don't know how to approach the problem. For first question I tried to set system of equations:

$t=1+2s$

$t^2=1+6s$

$t^3=1+14s$

But I wasn't able to solve it and I think similar approach is needed for next question

• Their paths definitely intersect at (1,1,1) though not at the same time. As the one curve is a line it seems unlikely that the two curves would encounter one another again. – Doug M Jun 6 '18 at 0:09
• $r_1(1) = r_2(0) = (1,1,1)$ – Doug M Jun 6 '18 at 0:16
• Are the $t$’s in the two parameterizations meant to be the same or not? – amd Jun 6 '18 at 1:52

We substitute the second equation with the first one to get $(1+2s)^2=1+6s$, and hence $s(2s-1)=0$. It follows from the two first equations that either $s=0, t=1$ or $s=1/2, t=2$.
Now, if they start at the same time, we see that they will never meet eachother, because for each intersection point, we do not have $t=s$.