# Thread: any two skew lines lie in parallel planes ???

1. ## any two skew lines lie in parallel planes ???

Guys

The following question appears in the book Calculus, Early Transcendentals by Edwards and Penney:

"Prove that any two skew lines lie in parallel planes."

I am confused because:

1) Two skew lines can exist in non-parallel planes. Imagine two perpendicular planes intersecting at line L. Draw a line (L1) on the first plane where L1 is parallel to L. Draw another line (L2) on the second plane where L2 intersects L. L1 and L2 are skew lines because they are not parallel and do not intersect and yet the two planes are not parallel.

2) If what the question means is that of all the planes containing the first skew line (call this set of planes P1) and of all the planes containing the second skew line (call this set of planes P2), there exist a pair of parallel planes where one of them is an element of P1 and another an element of P2, the question becomes meaningless as this is true for any pair of lines in space.

Any clarification, hints, etc., will be highly appreciated.

Many thanks.

2. ## Re: any two skew lines lie in parallel planes ???

Originally Posted by kwtam338
"Prove that any two skew lines lie in parallel planes."
Let $\vec{D}~\&~\vec{E}$ be the direction vectors of the two lines.
If the planes containing the two skew lines each have normal $\vec{D}\times\vec{E}$ they will be parallel.

3. ## Re: any two skew lines lie in parallel planes ???

Originally Posted by Plato
Let $\vec{D}~\&~\vec{E}$ be the direction vectors of the two lines.
If the planes containing the two skew lines each have normal $\vec{D}\times\vec{E}$ they will be parallel.
Dear Plato

Thanks for your reply. I imagine your reply is an approach to a proof of case (2) in my original post. Am I right in saying that your approach works for any two lines in space regardless of whether they are skew or not?

What really bugs me is that the statement "any two skew lines lie in parallel planes" seems to be either:
(a) false if I take it to mean that any two skew lines must lie in paralellel planes
(b) meaningless if I interpret it as in case (2) of my original post.

But thanks again anyway for replying.

4. ## Re: any two skew lines lie in parallel planes ???

Originally Posted by kwtam338
I imagine your reply is an approach to a proof of case (2) in my original post. Am I right in saying that your approach works for any two lines in space regardless of whether they are skew or not?
What really bugs me is that the statement "any two skew lines lie in parallel planes" seems to be either:
(a) false if I take it to mean that any two skew lines must lie in paralellel planes
(b) meaningless if I interpret it as in case (2) of my original post.
My reply had absolutely nothing to do with either of your two cases. I simply proved the statement.
Given two skew lines there exists two parallel planes each containing one of the lines.

5. ## Re: any two skew lines lie in parallel planes ???

Originally Posted by Plato
I simply proved the statement.
Given two skew lines there exists two parallel planes each containing one of the lines.
OK. But "given any two lines there exist two parallel planes each containing one of the lines." So what is the point of asking for a proof specifically for two skew lines?

To further illustrate my point, consider the followiing analogy:
1) It is already proved that the sum of any two sides of any triangle is greater than the third side.
2) Now you are asked to prove that the sum of any two sides of a right-angled triangle is greater than the third side.
What is the point?

6. ## Re: any two skew lines lie in parallel planes ???

Originally Posted by kwtam338
"Prove that any two skew lines lie in parallel planes."
Look carefully at the question.
You were asked to prove something.
This question usually is more or less a lemma to the following.
Prove that there is a unique line segment perpendicular to each of two skew lines.