Question

Let L1 be the line passing through the points Q1=Q1(1, −2, −4) and Q2=Q2(3, 4, 2)and let L2 be the line passing through the point P1=P1(15, 7, 14) with direction vector [−3, 2, −1]. Determine whether L1 and L2 intersect. If so, find the point of intersection Q.

Answer 1

Ian: How do we find a representation of the vector connecting two points?
Find it, then write L1 in the same format we used in the previous problem: \[vector.r _{0}+t(vector.v)\]

Answer 2

Rewrite L2 in the same format.
Can you imagine what needs to be done next to answer this question?

Answer 3

So L1 = <2,6,6> and L2 = <15,7,14> + t<-3,2,-1>?

Answer 4

L2 = <15,7,14> + t<-3,2,-1> looks great.
L1 needs work. Compare your L1 to your L2: your L2 has 2 vector component parts, whereas your L1 has only 1.
What is the vector connecting the 2 given points?

Answer 5

I don't get it

Answer 6

Let L1 be the line passing through the points Q1=Q1(1, −2, −4) and Q2=Q2(3, 4, 2) results in a direction vector w = < 3-1, 4+2, 2+4>. Agree with that? Same result as your direction vector, or different?

Answer 7

I should have check that myself. You're right on the direction vector. But, Ian, isn't there supposed to be another vector component of L1, namely, the vector from the origin to either of the points through which L1 passes?

Answer 8

so are you saying L1 should be either <1, -2, -4> + t<2, 6, 6> or <3, 4, 2> + t<2, 6, 6>?

Answer 9

Yes. You are given 2 points and need both points to determine the direction vector of the line connecting those two points. But after that you can pick either one of the points and ignore the other.

so now please choose one of your representations for L1 and write it here, along with your L2.

as before, obtain the x component from each and set these comps equal to each other; solve for t. Can you think of what you'll need to do next? What's the point of this problem?

Answer 10

L1=<1, -2, -4> + t<2, 6, 6>
L2 =<15,7,14> + t<-3,2,-1>
and then set 1+2t = 15-3t and solve for t
t = 14/5
so then using that you can solve for the x value right

Answer 11

Yes, and also solve (if possible!) for y and z. If you can do that, your lines intersect; otherwise, they do not.

Answer 12

It didn't work...
so if t = 14/5 then you put it back in to 1+2t and get 33/5 for the x value right?
The correct answer has an x value of 6.
Where did I go wrong?

Answer 13

Or, more correctly, "where did you (mathmale) go wrong?" I goofed. If you use parameter t in L1, then use parameter s (in place of t) in L2.

The goal is to determine whether or not you can find t and s such that x of L1 = x of L2, and so on.

Use simultaneous equations to determine s and t. Familiar with this process?

Answer 14

Oh, that makes sense. Thanks!

Answer 15

Let me know how this goes. I may be logging off OpenStudy very soon, but if you want to contact me, you need only post here again, or private message me. Best to you. MM