Question

Let L1 be the line passing through the points Q1(−2, −5, −3) and Q2(2, −3, −1) and let L2 be the line passing through the point P1(11, 1, 4) with direction vector d=[3, 1, 2]T. Determine whether L1 and L2 intersect. If so, find the point of intersection Q.  

Answer 1

Remarcks Equation of the line passing through the point p(^4,8, ,2,) and with direction vector d=(lm,n) is arxa yoyi Z-21 e m Also, eqreation of line passing through A ca,, az, az), B (bo, bz, bg) is n-ay 2 y-ag bz-an z-az bz-az biroq Nore! dong y-w Z-ZI and 04 bi ne-dz y-y2 bz z-Z2 22 are coplanar ag if ng - -", 72-Z, G au bi C2 an 62

Now, we cheek whether o L, passing through a, (-2,-5,-3) and Q2 (2,-3,-1). so, the equation of his y +5 6 (using ) a+2 Z + 3 4 2 2 La passing through P, (11,1,4) and has direction veetor d= [3,1,2]. so, the equation of La is y-1 ② (using ) all . Z~4 3 aud are coplanar or not using 3 11+2 4+3 13 6 7 1+5 2. 4 T 2 2 2 2 3 2 3 - = 13:2-4.5+3(-2) = 0 then from ® , We can conclude that coplanar. and are

2 are - ན Now, direction vectors of ① and (4,2,2) and (3,1,2) then ② and non-parallel. this shows that ① and ② are intersecting Now, at2 4. Z+3 It 2 Y+5 2 3 = 4+-2, ya at-5, & z= 2t-3 Ler acat-2,,27-5,27– intersecting point of @ -3) is the and 2 then a also lies on ® so, from ® , we have all -1 3 4+ 'll 24-5-1 = 3 1 4t-13 64-18 9 t=5/ so, the point ois (8, 0, 2) thus, the point of intersection is @(8,0,2)