Let 2 Particles travel along space curve,

r1(t)= <t,t^2,t^3> and r2(t)= <1-3t, 1-5t, 1-7t>

Do their paths intersect?

set them equal to each other and see

Judging from my answer, the particles dont intersect because in the end, when I plug in (2/3)^3 = 1-7(1/9), they dont equal eachother. Is this the correct or did I mess up my math somewhere in the previous steps?

r1(t)= <t,t^2,t^3> and r2(t)= <1-3t, 1-5t, 1-7t>

t=1-3t
t=1/4

plug in 1/4

The question whether their paths intersect is not the same as the question whether the particles collides at a point.

Pluging in t = 1/4 answer the second question only (they share the same position at the same moment).

To answer the first question, you could solve for t1 and t2 :

t1 = 1 - 3t2
t1^2 = 1 - 5t2

and then plug in your results for t1 and t2 into the last coordinate.

wouldnt you have to define a plane for them to intersect in? otherwise, pretty much every 2 curve intersect depending on the point of view