What Skew Lines Are and Aren't

Date: 01/14/2004 at 22:53:04
From: Anita
Subject: What are skew lines and how do you find them

What are skew lines and how do you find them?  I can not draw a figure
but I hope you can understand what I mean.

Date: 01/15/2004 at 13:07:51
From: Doctor Peterson
Subject: Re: What are skew lines and how do you find them

Hi, Anita.

Skewness is not a property of a single line by itself, but a way in
which two lines can be related.  I'll illustrate by having you look 
around the room you're in, assuming it is the usual sort of 
rectangular room with a ceiling.

Look down at the floor in the corner; you'll see this:

  +----------
  |
  |
  |
  |
  |
  |

Those two lines INTERSECT.  That is one way in which two lines can be 
related.  Note that they are also coplanar (since they both lie on the 
floor, which is (part of) a plane; any two intersecting lines lie in 
the same plane.

Now look at the bottom edges of two opposite walls; they will look 
like this:

  --------------





  --------------

These lines are PARALLEL.  They are always the same distance apart; if 
they extended forever, they would never meet.  And they, too, are
coplanar.

Now twist your head around and look at one of those floor edges, as 
well as one of the edges of the ceiling--but on an adjacent wall, not
the same wall.  This is harder to draw, but it will look something 
like this:

          /
        /
      /
    /
  /
           \
              \
                 \

They are NOT coplanar!  They do NOT intersect!  They are NOT always
the same distance apart!  That is what SKEW lines are: a pair of 
noncoplanar lines, which therefore are neither parallel nor
intersecting.  If you chose any two lines in the world, most likely 
they would be skew lines.

The following page has a nice picture of a pair of skew lines that you
may be able to move around with your mouse to see how they look from 
different directions:

    http://mathworld.wolfram.com/SkewLines.html 

Have fun!  And write back if you have any more questions.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/