Let L1 be the line passing through the points Q1=(4, −1, −2) and Q2=(1, 0, −1) and let L2 be the line passing through the point P1=(−6, 21, −8) with direction vector →d=[−3, 9, −3]T. Determine whether L1 and L2 intersect. If so, find the point of intersection Q.

Answer 1

for L1, direction vector is (3,-1,-1)
and an equation is
x = 1 + 3t
y = -t
z = -1 - t

for L2, a possible equation is
x = -6 - 3k
y = 21 + 9k
z = -8 - 3k

if they intersect,
1+3t = -6-3k AND -t = 21+9k AND -1-t = -8-3k

using the first two equations,
1 + 3(-21-9k) = -6-3k
-62 - 27k = -6-3k
-24k = 56
k = 56/-24 = -7/3
then t = -21 - 9(-7/3) = 0

does this satisfy the third equation?
LS = -1-0 = -1
RS = -8-3(-7/3) = -1
yes it does!

so they do intersect and the point of intersection is

x = 1+3(0) = 1
y = 0
z = -1-0 = -1

they intersect at (1,0,-1)