Question

56. Let Li and L2 be the lines whose parametric equations are L]: x = 41, y = 1 -21, z = 2 + 21 L2: x = 1+1, y = 1-1, Z=-1+ 41 (a) Show that Li and L2 intersect at the point (2,0, 3). (b) Find, to the nearest degree, the acute angle between L and L2 at their intersection. c) Find parametric equations for the line that is perpen- dicular to L, and L2 and passes through their point of intersection.

Answer 1

Then > 56 4X=4t, 281-26, Z=2+2t Lid=1tt, yol-t, Z=-1+Ar Any point on line to is of the form (A4, 1-2t, , 2+24) for some t, ER Any point on line to y of the form (itta, 1-22, 11 Qt2) for some t, ER Now if the line 4, L2 intersects. At = 1+ta 1-2t, = 1-t₂ 2+2t, = -1 + 462 & 41,- 2 1 - - d = 27, 2 from (2) 41-21, 21 *) 4 t, = 12 so the point of intersection is (4. 12, 1-2. 12, 2+2. 4) that is (2,0, 3) by direction cosines of the line 4 are (4, -2, 2) direction cosing of the line L2 are (1,-1,4) so if a be the angle between two line Liste Azcost 4X1 + (-2)X(-1) + 1x4 ./4²+(-2)² + 22 XJ/² + (-1/2 +42 Then = Cost (14 1124 xv 18 co)." (W2x2x2x2x2X3X3 2 cs! 14 4x3√3 - Coy" (V) K 14-22 -14 2 = 47.65 6387' -22 - ;(-8+2)+j (2-16) + (-1+2) -6i-14j-2k sº (-6, -19,-) is te direction cosines of the perpendiculare line of both LOLL whose So equation of the line passes through (2,0, 3) and Srection cosines are (-6, -19,-2) is 1-2-67, 430-14+, Z=3-2t where ty parameter. 327 4:2 NI8+2, y = 6-8+, Z=10+ L :3 = 3+8t, y 5-3A, Z= 6+7 Any point on line an 4 is of the form (2+8+, 6-84, 1984) for some t, on line L₂ is of the form (37842,5-342, 6+tz) for some te If the line interjects then 2+8t, = 3 + 8 tz 6-871 25-3t₂ 10x, obrtz fromff 20+84723+8047 48 2 t 108,-6 W

so 8t = 1+8tz 6-(1+ 8 tz) 25-34₂ 422 104, -6 - v (from (1) 5-8t₂ = 5-34₂ 5420 *+2=0 de troba from (2) t = 3 but to 13 et 20 doo nor sritisfy (0) So Ligh, are not internects & Lule are skew line. 287 X=2-4, 4= 34. , Z21+ 2t. Any point on this line is of the form (er (2-4 , 34,-1+2+) ler at (2-1, 34,-1+2t), the line and the core 24 +32 = 6 interects. 2(34) + 3(-1+2+) = 6 6t, -3 +67, 26 So the point is (2-3, 3x - 1 + 2 x 2 ) (4,95 Then 9.3 12