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Points, Lines, Planes, and Space Help (page 3)

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By — McGraw-Hill Professional
Updated on Oct 3, 2011

Intersecting Planes

Suppose that two different planes X and Y intersect; that is, they have points in common. Then the two planes intersect in a unique line L . The following two statements are always true, as shown in Fig. 7-5:

  • Planes X and Y share a common line L
  • L is the only line that planes X and Y have in common

 

An Expanded Set of Rules Points, Lines, Planes, and Space Intersecting Planes

Fig. 7-5 . The intersection of two planes X and Y determines a unique line L . The planes extend infinitely in 2D.

Parallel and Skew Lines

Parallel Lines In 3d Space

By definition, two different lines L and M in three-space are parallel lines if and only if both of the following are true:

  • Lines L and M do not intersect
  • Lines L and M lie in the same plane X

If two lines are parallel and they lie in a given plane X , then X is the only plane in which the two lines lie. Thus, two parallel lines define a unique plane in Euclidean three-space.

An Expanded Set of Rules Points, Lines, Planes, and Space Half Planes

Fig. 7-4 . A half plane X , defined by two parallel lines, L and M . The half plane extends infinitely in 2D on the “M” side of L .

Skew Lines

By definition, two lines L and M in three-space are skew lines if and only if both of the following are true:

  • Lines L and M do not intersect
  • Lines L and M do not lie in the same plane

Imagine an infinitely long, straight, two-lane highway and an infinitely long, straight power line propped up on utility poles. Further imagine that the power line and the highway center line are both infinitely thin, and that the power line doesn’t sag between the poles. Suppose the power line passes over the highway somewhere. Then the center line of the highway and the power line define skew lines.

Points, Lines, Planes, and Space Practice Problems

PROBLEM 1

Find an example of a theoretical plane region with a finite, nonzero area but an infinite perimeter.

SOLUTION 1

Examine Fig. 7-6. Suppose the three lines PQ, RS , and TU (none of which are part of the plane region X , but are shown only for reference) are mutually parallel, and that the distances d 1 , d 2 , d 3 , . . . are such that d 2 = d 1 /2, d 3 = d 2 /2, and in general, for any positive integer n , d ( n +1) = d n /2. Also suppose that the length of line segment PV is greater than the length of line segment PT . Then plane region X has an infinite number of sides, each of which has a length greater than the length of line segment PT , so its perimeter is infinite. But the interior area of X is finite and nonzero, because it is obviously less than the interior area of quadrilateral PQSR but greater than the area of quadrilateral TUSR .

 

An Expanded Set of Rules Points, Lines, Planes, and Space Skew Lines

Fig. 7-6 . Illustration for Problem 1.

PROBLEM 2

How many planes can mutually intersect in a given line L?

SOLUTION 2

In theory, an infinite number of planes can all intersect in a common line. Think of the line as an “Euclidean hinge,” and then imagine a plane that can swing around this hinge. Each position of the “swinging plane” represents a unique plane in space.

Practice problems for these concepts can be found at:  Points, Lines, and Planes Practice Test.

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