A line L1 passes through the point P = (1, 1, 1) and is parallel to the vector a = . Another line L2 passes through Q = (2, 1, 0) and is parallel to the vector b = . Show that these two lines intersect and find the point of intersection.

Question

A line L1 passes through the point P = (1, 1, 1) and is parallel to the vector a = <1, 2, 3>. Another line L2 passes through Q = (2, 1, 0) and is parallel to the vector b = <3, 8, 13>. Show that these two lines intersect and find the point of intersection.

anonymous · Answer

Well, first get the equations of the two lines. The general vector equation of a line is:$$\vec r=\vec r_{0}+t \vec v$$where t is a scalar. We get:$$=<1,1,1>+t<1,2,3>$$and,$$=<2,1,0>+t<3,8,13>$$We know that these lines are not parallel (and therefore will intersect) because the vector <1,2,3> is not a multiple of <3,8,13>.

anonymous · Answer

Don't I have to set them equal to each other to prove that they intersect and are not skew?

anonymous · Answer

To find the intersection point, set the two equations equal and solve for t:$$=<3t+2,8t+1,0>$$

anonymous · Answer

yeah, you're correct.  Forgot we are in 3D here.

anonymous · Answer

I got:
1+t=2+3s
1+2t=1+8s
1+3t=13s

From the first equation: t=1+3s, but when I plug in:
1+2(1+3s)=1+8s...
1+2+6s= 3+6s and 3+6s does not equal 1+8s

But, it says they intersect, so I'm not sure what I'm doing wrong.

anonymous · Answer

yeah, I'm getting problem here too...my reasoning must be off.  Are you sure there's no typo anywhere?

anonymous · Answer

Yep, that's the exact question.

anonymous · Answer

ok

anonymous · Answer

Ok, so say the problem wasn't incorrect, then how would I find the point of intersection once I showed that they intersected?

anonymous · Answer

im checking.  setting equal to each other should work...

anonymous · Answer

not sure what I'm doing wrong here.  Not finding an intersection point.

anonymous · Answer

Ok, so you just plug t or s into the parametric equation for the intersection point?

anonymous · Answer

I think the problem is incorrect.

anonymous · Answer

yeah I think it is incorrect.

anonymous · Answer

Thanks

anonymous · Answer

They did the same thing here so these lines must be skew. http://www.math.ucla.edu/~ronmiech/Calculus_Problems/32A/chap11/section5/716d15/716_15.html

anonymous · Answer

Thank you!

anonymous · Answer

no problem.  Keep in mind that we also needed to show the lines are no parallel before concluding they are skew (we did this).