Parallelism of lines in space. (2019)

In this lesson we will give basic definitions and theorems on the topic of parallel lines in space.
At the beginning of the lesson, we will consider the definition of parallel lines in space and prove the theorem that through any point in space it is possible to draw only one line parallel to a given one. Next we prove the lemma about two parallel lines intersecting a plane. And with its help we will prove the theorem about two lines parallel to a third line.

Topic: Parallelism of lines and planes

Lesson: Parallel lines in space. Parallelism of three lines

We have already studied parallel lines in planimetry. Now we need to define parallel lines in space and prove the corresponding theorems.

Definition: Two lines in space are called parallel if they lie in the same plane and do not intersect (Fig. 1.).

Designation of parallel lines: a || b.

1. Which lines are called parallel?

2. Prove that all lines intersecting two given parallel lines lie in the same plane.

3. A line intersects lines AB And B.C. at right angles. Are the lines parallel? AB And B.C.?

4. Geometry. Grades 10-11: textbook for students educational institutions(basic and profile levels) / I. M. Smirnova, V. A. Smirnov. - 5th edition, corrected and expanded - M.: Mnemosyne, 2008. - 288 p. : ill.

In this article we will understand the concept of a straight line in three-dimensional space, consider options relative position straight lines and let us dwell on the main methods of defining a straight line in space. For a better understanding, we provide graphic illustrations.

Page navigation.

A straight line in space is a concept.

After we have given the definition of parallel lines in space, we should talk about the direction vectors of a straight line due to their importance. Any non-zero vector lying on this line or on a line that is parallel to this one will be called the direction vector of the line. The direction vector of a straight line is very often used when solving problems involving a straight line in space.

Finally, two lines in three-dimensional space can be intersecting. Two lines in space are called skew if they do not lie in the same plane. This relative position of two lines in space leads us to the concept of an angle between intersecting lines.

Methods for defining a straight line in space.

There are several ways to uniquely determine a straight line in space. Let's list the main ones.

We know from the axiom that a straight line passes through two points, and only one. Thus, if we mark two points in space, this will allow us to unambiguously determine the straight line passing through them.

If a rectangular coordinate system is introduced in three-dimensional space and a straight line is specified by indicating the coordinates of its two points, then we have the opportunity to create an equation for a straight line passing through two given points.

The second method of defining a line in space is based on the theorem: through any point in space that does not lie on a given line, there passes a line parallel to the given one, and only one.

Thus, if we specify a line (or a segment of this line) and a point not lying on it, then we will uniquely define a line parallel to the given one and passing through the given point.

You can specify the point through which the line passes and its direction vector. This will also allow you to unambiguously determine the straight line.

If a straight line is specified in this way relative to a fixed rectangular coordinate system, then we can immediately write down its canonical equations of a straight line in space and the parametric equations of a straight line in space.

The following method of defining a line in space is based on the axiom of stereometry: if two planes have a common point, then they have a common straight line on which all the common points of these planes lie.

Thus, by defining two intersecting planes, we uniquely define a straight line in space.

Another way to define a line in space follows from the theorem (you can find its proof in the books listed at the end of this article): if a plane and a point not lying in it are given, then there is a single line passing through this point and perpendicular to given plane.

Thus, to determine a straight line, you can specify the plane to which the desired straight line is perpendicular and the point through which this straight line passes.

If a line is specified in this way relative to the introduced rectangular coordinate system, then it will be useful to know the material of the article on the equation of a line passing through a given point perpendicular to a given plane.

Bibliography.

• Atanasyan L.S., Butuzov V.F., Kadomtsev S.B., Poznyak E.G., Yudina I.I. Geometry. Grades 7 – 9: textbook for general education institutions.
• Atanasyan L.S., Butuzov V.F., Kadomtsev S.B., Kiseleva L.S., Poznyak E.G. Geometry. Textbook for 10-11 grades of secondary school.
• Bugrov Ya.S., Nikolsky S.M. Higher mathematics. Volume One: Elements linear algebra and analytical geometry.
• Ilyin V.A., Poznyak E.G. Analytic geometry.

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Two lines in space can be located in different ways. First of all, it may happen that two straight lines have a common point. Then they obviously lie in the same plane. Indeed, to construct such a plane, it is enough to draw it through three points: point A of the intersection of the indicated lines (Fig. 323) and points C and B, taken respectively on the lines. Having two common points with each of the lines, the plane will contain both lines.

Now let these lines have no common points. This does not mean that they are parallel, since the definition of parallelism stipulates that the lines belong to the same plane. To solve the question of the location of our straight lines, let’s draw a plane K through one of them, for example, and an arbitrarily taken point A onto another straight line. Two cases are possible:

1) The constructed plane contains the entire second straight line (Fig. 324). In this case, the straight lines belong to the same plane and do not intersect and are therefore parallel.

2) Plane X intersects the line at point A. Then both lines do not lie in the same plane. Such lines are called crossing lines (Fig. 325).

So, there are three main possible cases of the relative position of two lines.

1. The lines lie in the same plane and intersect.

2. The lines lie in the same plane and are parallel.

3. Straight lines intersect, that is, they do not lie in the same plane.

Example. From the 12 edges of a cube, pairs of straight lines can be formed. Of these, 24 pairs are crossing, 24 are intersecting and 18 are pairs of parallel lines. The reader will verify the correctness of this from the model or drawing.

Note that in space the postulate of parallel lines remains valid:

Through a point outside a line there is only one line parallel to it.

In fact, a straight line and a point given outside it determine the plane in which the desired straight line, parallel to the given one, must lie; its uniqueness follows from the postulate of parallels.

Note that two well-known proposals of planimetry related to the properties of parallel lines will require special justification for the case of space (see paragraph 232).

If two lines are parallel to a third, then they are parallel to each other; two angles with respectively parallel and identically directed sides are equal.

Regarding the second of these proposals, we note that the definition of the angle between crossing lines is based on it: the angle between two crossing lines is the angle between two lines parallel to them and drawn through an arbitrary point M. Obviously, such a definition is based on the assumption that the angle is independent of the choice of point M (see paragraph 232).

A perpendicular dropped from a given point onto a line is understood as a straight line drawn from a given point at right angles to a given line and intersecting it. Through a point that does not lie on a line, one can draw a single perpendicular to it.

Indeed, the required perpendicular must lie in the plane defined by the given line and point, and therefore the provisions of planimetry are applicable to it. However, from a point lying on a line, an infinite number of perpendiculars can be drawn to it: one in each plane drawn through this line.