Question

Question 7 (10 points] Let Ly be the line passing through the points Q1-(3,-1,-4) and Q2=(5,-3,-2) and let La be the line passing through the point P4-(12,-4, 3) with direction vector a-(-6, -6, -21". Determine whether Ly and L2 intersect. If so, find the point of intersection Q. The lines intersect at the following point Q: Q=(0,0,0)

Answer 1

ая N MI (12,-4, 3) with direction guestion 72- Note 1: If two straight lines whose direction ratios L, M, N and L M N be parallel, then L M 고 N' Here, L, be the line passing through the points 8,2 =(3,-1,-4) and Qqz (5,-3,-2). So, the equation of the line L, is given by a-3 yti re Now, 2₂ be the the line passing through the point P vector d = [-6, -6, -27? Z +4 5-3 -3+1 -2 +4 X-3 yti Z + 4 2 - 2 2 1 So, the equation of the line L₂ is given by 2012 y +4 Z-3 -6 -6 2

line Now, the direction ratios of the line on are 2, -2,2 and the direction ratios of the line L₂ -6, -6, -2 Since 2 + - 2 + 2 2 -6 2 So, from Note ②, we can say that the lines L and L₂ intersect. we have to fina point of intersection Nows Q. let, for the Common point, x-3 yti Z +4 2 -2 2 Xriz Y+4 2-3 2 re -6 9– -2 so, the points (27, +3, -28,-1, 27, -4) and (-682 +12,6% -4, -2%,+3) are the Same. Hence, 27,+3 = - -6%2+12, -2% -112-602-4 L 3 28,-4= - 282 +3 © From ③ > 28, +682 29 From , - 2/0 +682 2-3 (adding) 1272 = 6

or 87 ܝܙܐܝ 12 So, from Boutting 82 2 2 we get 28,43 -6() +12 → 27, +39-3 +12 → 27.5 6. 8,03 Now, we see thats 8,23, &q= ½ satisfy 6. Therefore, the point of intersection of the two lines L&L₂ is (9 9,-7, 2) The lines intersect at the following : =(9,-7,2)