Hi I need help with this problem:
Given $$r(t)=(e^t,e^t\cos t, e^t \sin t),\quad t\in[0,2\pi]$$ It represents the trajectory of a particle $P$.
- Draw the curve.
Reparametrize $r$ such that it represents a particle $Q$ moving in the opposite direction.
If $P$ starts at one end of the curve and $Q$ starts at the other end (at the same time), find the point of the trajectory when they both crash.
1.- I first tried to draw the curve, but I don't know how to exactly do it with those $e^t$ in there. I tried by factoring them out like $e^t(1, \cos t, \sin t)$, so I think that the that it would be like an spiral, because of $e^t$, but I really don't know how to draw it.
2.- Then I tried to reparametrize r, so I put $-t$ instead of $t$ on $r$, but I don't know if I'm correct.
3.- I don't know how to find the point when both particles collide, I was thinking of setting both parametrizations as equal and then finding the time $t$ that satisfies that, but I'm not sure.