Can Two Lines Intersect? A Proof

[Petrus]

Hello MHB,
I am working with old exam and got one problem that gives 5 points (total 30 points) and it says
line \(\displaystyle l_1\) and \(\displaystyle l_2\) gives of
\(\displaystyle (x,y,z)=(1,0,1)+t(2,3,0)\) and \(\displaystyle (x,y,z)=(2,0,-2)+t(1,2,1)\)
prove that \(\displaystyle l_1\) and \(\displaystyle l_2\) intersect each other.
this is how I solved:
There is an intersect only if this equation got a solution:
\(\displaystyle 1+2t=2+s\)
\(\displaystyle 3t=2s\)
\(\displaystyle 1=-2+s\)
from equation 3 we get that \(\displaystyle s=3\) and if we put \(\displaystyle s=3\) in equation 2 we get that \(\displaystyle t=2\) and if we put all those in equation we see it's true.
Well do you think this is good explain? It's pretty much 5 points that is a lot and that's why I want to ask for advice if this would be enough for 5 points acording to you

Regards,
\(\displaystyle |\pi\rangle\)

 

[MarkFL]

Re: prove two line intersect

Your method is valid, I think I would show at what point the two lines intersect.

 

[Petrus]

Re: prove two line intersect

Your method is valid, I think I would show at what point the two lines intersect.

Thanks mark for fast responed! At point \(\displaystyle (x,y,z)=(5,6,1)\)

Regards,
\(\displaystyle |\pi\rangle\)

 

[Ackbach]

Re: prove two line intersect

And it might also be helpful to think geometrically and algebraically about what the no-solutions or infinite-solutions possibilities look like. To what do those correspond?

 

[Petrus]

Re: prove two line intersect

And it might also be helpful to think geometrically and algebraically about what the no-solutions or infinite-solutions possibilities look like. To what do those correspond?

Hello Ackbach,
This is a exemple

Regards,
\(\displaystyle |\pi\rangle\)

 

[Ackbach]

Re: prove two line intersect

I would agree, although you're working in three dimensions. What additional possibility does that introduce?

 

[Petrus]

Re: prove two line intersect

I would agree, although you're working in three dimensions. What additional possibility does that introduce?

Hello Ackbach,
z is the same, with other words it's on same plane

Regards,
\(\displaystyle |\pi\rangle\)

 

[Ackbach]

Re: prove two line intersect

Hello Ackbach,
z is the same, with other words it's on same plane

Regards,
\(\displaystyle |\pi\rangle\)

I think you're getting at it. We would say that you can have non-parallel non-intersecting lines (not possible in Euclidean two-dimensional geometry). We call those skew lines.

 

[Petrus]

Re: Prove two line intersect

Hello MHB,
I was thinking about distance between two line with intercept (I never had any exercise that have been asked) but what I think how it would be.
The distance will go from infinity to zero when it reach the point \(\displaystyle (5,6,1)\) and the go to infinity, what I mean is it will first be really big then it will be smaler and smaler until it got the point \(\displaystyle (5,6,1)\) then it will be zero then it will be bigger and bigger. I have no clue if you can actually say like this but I have never been asked for a distance with two line that got a intersect, Is this correct?

Regards,
\(\displaystyle |\pi\rangle\)

 

[MarkFL]

Re: Prove two line intersect

Yes, consider the plane containing the two lines, and make the origin of this plane the point of intersection of the two lines, with one of the lines lying along the $x$-axis. The other line can then be written within this system as:

\(\displaystyle y=kx\) where \(\displaystyle 0<|k|\)

Then, for some point $(x,0)$ on the horizontal line, its shortest (perpendicular) distance $d$ to the other line is given by:

\(\displaystyle d(x)=\frac{|kx|}{\sqrt{k^2+1}}\)

We now see that:

\(\displaystyle \lim_{x\to\pm\infty}d(x)=\infty\)

\(\displaystyle \lim_{x\to0}d(x)=0\)

And this agrees with what you stated. (Sun)

 

[Petrus]

Re: Prove two line intersect

Hello,
Thanks Ackbach and MarkFL for taking your time and helping me!:)

Regards,
\(\displaystyle |\pi\rangle\)