Consider the lines L1 and L2, parametrized by ~v1(t) = t<2,1,7> + <1; 0; 1> and
ID: 3215721 • Letter: C
Question
Consider the lines L1 and L2, parametrized by ~v1(t) = t<2,1,7> + <1; 0; 1> and ~v2(t) = t<5; 3; 7> + <2; 1; 3> respectively. (a) Determine whether L1 and L2 are parallel, intersecting, or skew. Explain how you arrived at your conclusion (b) Find the plane P1 parallel to both L1 and L2 and containing L1. (c) Find the plane P2 parallel to both L1 and L2 and containing L2. (d) If L1 and L2 intersect, nd their point of intersection. Otherwise, find the shortest distance between them.Explanation / Answer
A) 2/5 is not equal to 1/3 so they are not parallel let suppose they intersect 2t + 1 = 5s + 2 t = 3s+ 1 solving for t and s t = -2 s = -1 for third co-ordinate 7 t + 1 = 7s + 3 putting values of t and s -14 + 1 = -7 + 3 ....which is not correct so lines do not intersect ...they are skew lines b) normal vector = (2,1,7) x ( 5,3,7) = (-14,-21,1) so plane is -14x -21y + z = k L1 lies so (1,0,1) satisfies -14 + 1 = k = -13 so eqn is 14x + 21y - z = 13 c) 14x + 21y - z = k (2,1,3) satisfies as L2 lies in it 28 +21 - 3 = 46 = k 14 x + 21y - z = 46 d) they do not intersect