Finding whether two lines intersect each other in 3dimensional space

[hangainlover]

Homework Statement

do the lines (x,y,z) = (5+2t, 3+2t,1-t) and (x,y,z) = (13-3r, 13-4r, 4-2r) intersect? If so, at what point? If not, how do we know?

Homework Equations

The Attempt at a Solution

I just do not know where to begin...
I mean what do you do with the variables t and r.
Do those values constanly change?
If you want to direct me to some useful information to help me understand this concept.. feel free to do so

 

[union68]

If they do intersect, then the x component of the first line must equal the x component of the second line, and similar for the y and z components.

So, perhaps you should set the equations equal to each other...

 

[hangainlover]

does this mean
5+2t=13-3r
3+2t=13-4r
1-t=4-2r ?
oh i guess when t=1 and r=2, those two lines intersect each other

 

[HallsofIvy]

Yes, it happens that when t= 1 and t= 2, all three equations are satisfied. And you can do more. By putting t= 1 into the equations for that line or by putting r= 2 into the equations for the second line, you get x= 5+2= 7, y= 3+ 2= 5, and z= 1-t= 0 or, equivalently, 13- 6= 7, y= 13- 8= 5, 4- 4= 0 so the two lines intersect at (7, 5, 0).

Of course, in three dimensions, "most" lines do NOT intersect. You could always solve two of the equations, say, the x and y equations, for s and t. But then you would have to check in the z equation to see if that also was satisfied.