Parallel, intersecting, or skew

[dswatson]

Let L1 be the line parametrized by r1(t) = <t+1,2t-1,-t+2> and L2 be the line parametrized by r2(t)=<2t+6,-t-1,-2t-3>. Determine if L1 and L2 are the same, parallel, intersecting, or skew. I set x1=x2,y1=y2, and z1=z2 and the t does not equal ct so they are not parallel or the same line.

I then changed the parameters of L2 from t to s and set the components equal.

2s+6=t+1
t=2s+5
then plugged back into the equations.
I got 2s+5,4s+9,-2s-3=2s+6,-s-1,-2s-3

the z terms are the same so does that mean they intersect at some point?

I put this on a test and it was counted as incorrect and I was wondering how to correct it. Thank you in advance.

 

[SammyS]

Let L1 be the line parametrized by r1(t) = <t+1,2t-1,-t+2> and L2 be the line parametrized by r2(t)=<2t+6,-t-1,-2t-3>. Determine if L1 and L2 are the same, parallel, intersecting, or skew. I set x1=x2,y1=y2, and z1=z2 and the t does not equal ct so they are not parallel or the same line.

I then changed the parameters of L2 from t to s and set the components equal.
(This is the correct way to solve this problem.) There is no guarantee the the "t" in one parametrization is the same as the "t" in the other.

2s+6=t+1
t=2s+5
then plugged back into the equations.
I got 2s+5,4s+9,-2s-3=2s+6,-s-1,-2s-3

the z terms are the same so does that mean they intersect at some point?
When the z terms are the same, are the x & y terms also the same? If not, then that shows that the lines do NOT intersect

I put this on a test and it was counted as incorrect and I was wondering how to correct it. Thank you in advance.

Hello dswatson.

Some comments are in red above.

When you found t=2s+5 from the x (or x₁ ) components, the the x should also match for the two lines.

If t=2s+5, then r₁(t) = <2s+5+1,2(2s+5)-1,-(2s+5)+2> = <2s+6, 4s+9, -2s-5>.

Compare with: r₂(s)=<2s+6,-s-1,-2s-3>

Both the x & the z match, but NOT the y. What do you suppose that means?

 

[dswatson]

I would assume that the lines match in the x and z if looking at a graph but the lines would be at different points on the y-axis? is this why they are skew?

 

[SammyS]

Is there any value of s that makes y match for the two lines?