Question 1:

Two particles travel along the space curves

r1(t)=<t, t2, t3>, r2(t)=<1+2t, 1+6t, 1+14t>

Do the particles collide? Do their paths intersect?

I know how to find out how the particles collide, just set the corresponding components of both vectors equal to one another and look for a t-value that works for all of them (which there isn't). My question is how do you determine if their paths intersect?

Question 2:

Use the parametric equations

x=(2+cos(1.5t))cos(t)

y=(2+cos(1.5t))sin(t)

z=sin(1.5t)

to sketch the curve by hand as viewed from above, with gaps indicating where the curve passes over itself. Start by showing that the projection of the curve onto the xy-plane has polar coordinates r=2+cos(1.5t) and Θ=t, so r varies between 1 and 3. Then show that z has maximum and minimum values when the projection is halfway between r=1 and r=3.