Question

The journeys of two particles moving in space are described by the linear motions u(s) = (1, 2, 1) + s(6, -2,8) and v(t) = (-2,3,-3) +t(9, -3, 12) (a) Do the paths of the particles cross each other? (Hint: The paths cross each other exactly when there is a time s and a time t so that the location of the u-particle at time s is the location of the v-particle at time t; i.e. u(s) = v(t).) (b) Do the two particles collide? (Hint: The particles collide if they are at the same place at the same time; i.e. if there is a time t such that u(t) = v(t).)

Answer 1

I a The paths of the particles cross each other it us) = v(H) ih if 631+5 (6, -2, 8) = (-2, 3, 3) +(2, -3,12) 27 (it's, 2-25, 1+3) = (-2+9+, 3-373-3+12+) 27 S1+6 = –2+975 65-9t=-3 2-25 = 3-37 1-25+3t = 1 (1+88 = -3tiat (8542t =-4 sas-32= ~ nel-t et illa – 25 + 3t = but 25-342-1 I which is identical with 25-3t = -1 Let tok ER. Tums - - + 2K when tak,= 1+ 3, KER So the baths of the particles cross each other when there is a time tak, & sa - +3k ,KER. is for any case if kal, t = 1 2 3 = -1 +3:1 = ), then os ustales 2o; 1)+1. (6-2, 8) = (7,0, 9) 2 V (t = (-2, 3, -3) + 1 (9- 3,12)=(7,0, 9 So Uls) = V(t)

6 The two particles collide if there is a time t sit, uct) = V(A) we alsady have forkol, t=1& s=1 and then UL = (1,2,1)+1(6,28)=(7, 8, 9) 2 VD = (-2,3-3) + |(9,-3,13) = (7,0,0) e, uwavuliwa So it shows that there is a time +- (621) s.t. Ut = WH). so the two barices collide