The Question:

Two particles travel along the space curves:

$\displaystyle r1(t) = < t , t^2, t^3 >$

$\displaystyle r2(t) = < 1 + 2t, 1 + 6t, 1 + 14t >$

(a) Do the particles collide?

(b) Do their paths intersect?

My Attempt:

(a)

For the two particles to collide, there must be a value of t such that r1(t) = r2(t). So, I set the components of the positions functions equal to each other:

t = 1 + 2t => t = -1

t^2 = 1 + 6t => t = 3 + or - sqrt(10)

I can see already that because t is equal to different quantities in the first two components, the particles do not collide.

(b)

I'm not sure how I might find if the space curves ever intersect...Other than using the same method I used above, which just can't apply for both parts.

Thanks for any responses!