Question

Explain fully please and thank you! =)

Determine the point of intersection of the lines L1: r= (1,3,7) + s(1,2,-2),seR and L2: v=(0,2,-3) +t(1,1,4), teR.

a) P(1,3,1)
b P(1,3,7)
c) The lines coincide
d) no point of intersection

Answer 1

if they intersect:
s+1 = t+0 ----> t = s+1
and
2s+3 = t+2 ----> t = 2s+1
and
-2s+7 = 4t-3

using the first two equations
2s+1 = s+1
s = 0 , then t=1

test in 3rd equation ...
LS = -2s+7 = 0 + 7 = 7
RS = 4t-3 = 4-3 = 1 ≠ LS

so they cannot intersect
Do they coincide?
If they did, they should have the same direction vectors ...
the first has direction (1,2,-2) and the second has direction ( 1,1,4) , so they are not parallel

conclusion: We are looking at two lines which are not parallel and which do not intersect