Answer:
Yes lines are intersecting, point of intersection is <4,2,0>.
Step-by-step explanation:
Given parametric equations of line are:
[tex]r_{1}(t) = <3, 0, 2> + t <1, 2,-2> \\ r_{1}(t) = <3+t, 0+2t, 2-2t>\\ r_{1}(t) = <3+t, 2t, 2-2t>---(1)[/tex]
[tex]r_{2}(s) = <0, 1,-1> + s <4, 1, 1>\\ r_{2}(s) = <0+4s, 1+s,-1+s>\\ r_{2}(s) = <4s, 1+s,-1+s>---(2)\\[/tex]
If lines are intersecting then parametric coordinates of (1) are equal to (2)
[tex]3+t=4s---(A)\\2t=1+s---(B)\\2-2t=s-1---(C)\\[/tex]
Considering A and B to find values of t and s
From A
t=4s-3---(D)
Putting in (B)
2(4s-3)=1+s
8s-6=1+s
7s=7
s=1
Then
t=4-3
t=1
If lines are intersecting then these values of s and t must satisfy (C)
2-2(1)=1-1
0=0
This shows lines are intersecting.
At this value of t, (1) becomes
[tex]r_{1}(1) = <3+1, 2, 2-2>\\=<4,2,0>[/tex]
Putting s=1 in (2)
[tex]r_{2}(1)=4, 1+1,-1+1>\\=<4,2,0>[/tex]
Point of intersection is <4,2,0>.
