Question

One particle travels along the space curve rı(t) = (t,t?, t) and another particle travels along the space curve rz(t) = (1 + 2t, 1 + 61,1 + 14t). Answer the following two questions: 1. Do the particles collide? 2. Do their paths intersect?

Answer 1

(i) for the collisions we need that two particles will be at the same point (wi4, 2) for the same value of P, that is we need three equations to be satisfied by the same t. 1 + 2 + 72 1+6t t3 =1+144 The First equation is satisfied for t=-1, But this value doesn't satisfy the second equation, which means that value of 4 to satisfy all 3 egns. there is no common There is no collision (2) For the path interesection we need to checks 8f there same points on trajectories, we need to three othe equations. are the Sole to = It a tz U t2 = totz 3 til = It 14tz substitute te in ea 9 int ③ 2 (1+2+) totz 4 tz* 4t₂² 6tz te=0 or 1

If t₂=0 it follows t = 142.co) =1 and these valves satisfy third equation 132 1+ 14:0 = 2 If t, - qöt follows that to = 1+ 2. and these values satisfy the 3rd eqn. 23 It 14. Ź -8 Finally, we have that there are two anteresection points , 4, 2) = (ti, t, 2, 7,3)=(Lill) (for tr = ) 201412) = (t,, t, 2, +13) - (2,4,89 (tz = 2