斜直线

Skew Lines

Supppose we had a space. In 3-D geometry, the definition of a pair of parallel lines is a pair of lines that don't intersect and lie on the same plane. But that leads us to wonder. What if they don't lie on the same plane?

A pair of skew lines is a pair of lines that don't intersect, and also don't lie on the same plane. You can think of skew lines as lines that go in completely different directions, like "random" lines. Here's an example photo that should help you understand them.

Please try problems 2, 3, 9, 10, 15, and 18 on this worksheet for extra practice. Have a wonderful Black Friday!

Example of Skew Lines: when are they skew? $\\$

Skew lines are lines that do not intersect, and are not parallel.

Here is an example of two skew lines.

When lines are on the same plane, they are either parallel or they intersect.

Therefore, skew lines must be on separate planes.

However, if two lines look like they are on separate planes, are they automatically skew?

$A$ and $B$ are two planes. Line $l$ is on plane $A$; Line $k$ is on plane $B$. Are $l$ and $k$ skew?

The answer is No.

I find two ways that are helpful to use when checking.

1. If you merge the two planes, are the lines parallel? $\\$ It's not hard to see if you merge $A$ and $B$, $l$ and $k$ are parallel.

2. Can you create a plane where both lines are on? $\\$ The answer is yes!

Try these two methods and see how they work!