Let L1 be the line passing through the points Q1=(−2, −3, −4) and Q2=(−5, −2, −3) and let L2 be the line passing through the point P1=(−27, 31, −25) with direction vector →d=[−6, 9, −6]T. Determine whether L1 and L2 intersect. If so, find the point of intersection Q.

Having a hard time with this one..

3 answers

direction of L1 is [-3, 1, 1]
equation of L1
L1 = (-2,-3,-4) + s(-3,1,1)

equation for L2
L2 = (−27, 31, −25) + t(−6, 9, −6)

so : -2 - 3s = -27 - 6t ---> 6t - 3s = -25
and -3 + s = 31 + 9t ---> s = 9t + 34
using substitution :
6t - 3(9t+34) = -25
6t - 27t - 102 = -25
-21t = 77
t = -11/3, then s = 9(-11/3) + 34 = 1

check in third equation:
-4 + s = -25 - 6t
LS = -4 + 1 = -3
RS = -25 - 6(1) = -31
LS ≠ RS
They do not intersect.
Check my arithmetic, I did not write it down first.
The method is correct.

thank you so much!

not at all any help