Plane in space - necessary information. Plane a passing through these points

Three planes may not have a single common point (if at least two of them are parallel, and also if their intersection lines are parallel), may have an infinite number of common points (if they all pass through one straight line), or may have only

one common point. In the first case, the system of equations

has no solutions, in the second it has countless solutions, in the third it has only one solution. For research, it is most convenient to use determinants (§ 183, 190), but you can also get by using the means of elementary algebra.

Example 1. Planes

have no common points, since planes (1) and (2) are parallel (§ 125). The system of equations is inconsistent (equations (1) and (2) contradict each other).

Example 2. Investigate whether three planes have common points

We are looking for a solution to system (4)-(6). Eliminating 2 from (4) and (5), we get. Eliminating 2 from (4) and (6), we get. These two equations are inconsistent. This means that the three planes do not have common points. Since there are no parallel planes among them, the three lines along which the planes intersect in pairs are parallel.

Example 3. Investigate whether planes have common points

Proceeding as in example 2, we obtain both times, i.e., in fact, not two, but one equation. It has countless solutions. That means three

I5 Whatever three points are that do not lie on the same line, there is at most one plane passing through these points.

I6 If two points A and B of a line lie in plane a, then each point of line a lies in plane a. (In this case we will say that line a lies in plane a or that plane a passes through line a.

I7 If two planes a and b have a common point A, then they have at least one more common point B.

I8 There are at least four points that do not lie in the same plane.

Already from these 8 axioms it is possible to deduce several theorems of elementary geometries, which are clearly obvious and, therefore, are not proven in a school geometry course and are even sometimes, for logical reasons, included in the axioms of one or another school course

For example:

1. Two lines have at most one common point.

2. If two planes have a common point, then they have a common line on which all common points of these two planes lie

Proof: (for show off):

By I 7 $ B, which also belongs to a and b, because A,B "a, then according to I 6 AB "b. This means that the straight line AB is common to the two planes.

3. Through a line and a point not lying on it, as well as through two intersecting lines, there passes one and only one plane.

4. On each plane there are three points that do not lie on the same line.

COMMENT: Using these axioms you can prove a few theorems and most of them are so simple. In particular, it is impossible to prove from these axioms that the set of geometric elements is infinite.

GROUP II Axioms of order.

If three points are given on a straight line, then one of them can be related to the other two in a relation “lie between”, which satisfies the following axioms:

II1 If B lies between A and C, then A, B, C are different points of the same line and B lies between C and A.

II2 Whatever the two points A and B, there is at least one point C on the line AB such that B lies between A and C.

II3 Among any three points on a line, there is at most one point lying between the other two

According to Hilbert, over the segment AB(BA) we mean a pair of points A and B. Points A and B are called the ends of the segment, and any point lying between points A and B is called the interior point of the segment AB(BA).

COMMENT: But from II 1-II 3 it does not yet follow that every segment has internal points, but from II 2, Þ that the segment has external points.

II4 (Pasch's axiom) Let A, B, C be three points that do not lie on the same line, and let be a straight line in the ABC plane that does not pass through any of the points A, B, C. Then if a straight line a passes through a point on a segment AB, then it also passes through a point on a segment AC or BC.

Sl.1: Whatever the points A and C, there is at least one point D on the line AC lying between A and C.

Document: I 3 Þ$ i.e. not lying on the line AC

Now we can prove two statements

DC3 Statement II 4 also holds if points A, B and C lie on the same straight line.

And the most interesting thing.

Level 4 . Between any two points on a line there is an infinite number of other points (self).

However, it cannot be established that the set of points on a line is uncountable .

Axioms of groups I and II allow us to introduce such important concepts as half-plane, ray, half-space and angle. First we prove the theorem.

Th1. A line a lying in the plane a divides the set of points of this plane that do not lie on the line a into two non-empty subsets so that if points A and B belong to the same subset, then the segment AB has no common points with the line a; if these points belong to different subsets, then the segment AB has a common point with the line a.

Idea: a relation is introduced, namely, A and B Ï A are in the relation Δ if the segment AB has no common points with the line A or these points coincide. Then the sets of equivalence classes with respect to the relation Δ were considered. It is proven that there are only two of them using simple reasoning.

Odr1 Each of the subsets of points defined by the previous theorem is called a half-plane with boundary a.

Similarly, we can introduce the concepts of a ray and a half-space.

Ray- h, and the straight line is .

Odr2 An angle is a pair of rays h and k emanating from the same point O and not lying on the same straight line. so O is called the vertex of the angle, and the rays h and k are the sides of the angle. We denote it in the usual way: Ðhk.

Point M is called an interior point of angle hk if point M and ray k lie in the same half-plane with the boundary and point M and ray k lie in the same half-plane with the boundary. The set of all interior points is called the interior region of an angle.

The outer area of ​​the corner is an infinite set, because all points of a segment with ends on different sides of an angle are interior. The following property is often included in axioms for methodological reasons.

Property: If a ray comes from the vertex of an angle and passes through at least one interior point of this angle, then it intersects any segment with ends on different sides of the angle. (Self)

GROUP III. Axioms of congruence (equality)

On a set of segments and angles, a relation of congruence or equality is introduced (denoted by “=”), satisfying the axioms:

III 1 If a segment AB and a ray emanating from point A / are given, then $ t.B / belonging to this ray, so that AB = A / B / .

III 2 If A / B / =AB and A // B // =AB, then A / B / =A // B // .

III 3 Let A-B-C, A / -B / -C / , AB=A / B / and BC=B / C / , then AC=A / C /

Odr3 If O / is a point, h / is a ray emanating from this point, and l / is a half-plane with boundary , then the triple of objects O / ,h / and l / is called a flag (O / ,h / ,l /).

III 4 Let Ðhk and flag (О / ,h / ,l /) be given. Then in the half-plane l / there is a unique ray k / emanating from the point O / such that Ðhk = Ðh / k / .

III 5 Let A, B and C be three points that do not lie on the same line. If in this case AB = A / B / , AC = A / C / , ÐB / A / C / = ÐBAC, then ÐABC = ÐA / B / C / .

1. Point B/B III 1 is the only one on this beam (self)

2. The congruence relation of segments is an equivalence relation on the set of segments.

3. In an isosceles triangle, the angles at the bases are equal. (According to III 5).

4. Signs of equality of triangles.

5. The angle congruence relation is an equivalence relation on the set of angles. (Report)

6. An exterior angle of a triangle is greater than each angle of the triangle that is not adjacent to it.

7. In each triangle, the larger angle lies opposite the larger side.

8. Any segment has one and only one midpoint

9. Any angle has one and only one bisector

The following concepts can be introduced:

Odr4 An angle equal to its adjacent one is called a right angle.

You can define vertical angles, perpendicular and oblique, etc.

It is possible to prove the uniqueness of ^. You can introduce the concepts > and< для отрезков и углов:

Odr5 If segments AB and A / B / and $ t.C are given, i.e. A / -C-B / and A / C = AB, then A / B / >AB.

Odr6 If two angles Ðhk and Ðh / k / are given, and if through the internal region Ðhk and its vertex one can draw a ray l such that Ðh / k / = Ðhl, then Ðhk > Ðh / k / .

And the most interesting thing is that with the help of the axioms of groups I-III one can introduce the concept of motion (superposition).

It's done something like this:

Let two sets of points p and p / be given. Let us assume that a one-to-one correspondence is established between the points of these sets. Each pair of points M and N of the set p defines a segment MN. Let M / and N / be points of the set p / corresponding to the points MN. Let us agree to call the segment M / N / corresponding to the segment MN.

Odr7 If the correspondence between p and p / is such that the corresponding segments always turn out to be mutually congruent, then sets p and p / are called congruent . Moreover, they also say that each of the sets p and p / is obtained movement from another or that one of these sets can be superimposed on another. The corresponding points of the set p and p / are called overlapping.

Approval1: When moving, points lying on a line transform into points also lying on a certain line.

Utv2 The angle between two segments connecting a point of a set with its two other points is congruent to the angle between the corresponding segments of a congruent set.

You can introduce the concept of rotation, shift, composition of movements, etc.

GROUP IV. Axioms continuity And.

IV 1 (Axiom of Archimedes). Let AB and CD be some segments. Then on the line AB there is a finite set of points A 1, A 2, ..., A n such that the following conditions are satisfied:

1. A-A 1 -A 2, A 1 -A 2 -A 3, ..., A n -2 -A n -1 -A n

2. AA 1 = A 1 A 2 = … = A n-1 A n = CD

3. A-B-A n

IV2 (Cantor’s Axiom) Let an infinite sequence of segments A1B1, A2B2,... be given on an arbitrary line a, of which each subsequent one lies inside the previous one and, in addition, for any segment CD there is a natural number n such that AnBn< СD. Тогда на прямой а существует т.М, принадлежащая каждому из отрезков данной последовательности.

From the conditions of Cantor’s axiom it immediately follows that such a m.M is unique, because if this is not so, and noun. one more t.N, then segment MN

It can be proven that axioms I-III and IV 1 , IV 2 are equivalent to the following proposition of Dedekind.

points that do not lie on the same line? a) Intersect; b) nothing can be said; c) do not intersect; d) coincide; e) have three common points.

2. Which of the following statements is true? a) If two points of a circle lie in a plane, then the entire circle lies in this plane; b) a straight line lying in the plane of the triangle intersects its two sides; c) any two planes have only one common point; d) a plane passes through two points, and only one; e) a line lies in the plane of a given triangle if it intersects two lines containing the sides of the triangle.

3. Can two different planes have only two common points? a) Never; b) I can, but under additional conditions; c) always have; d) the question cannot be answered; d) another answer.

4. Points K, L, M lie on the same line, point N does not lie on it. One plane is drawn through every three points. How many different planes did this result in? a) 1; b) 2; at 3; d) 4; d) infinitely many.

5. Choose the correct statement. a) A plane passes through any three points, and only one; b) if two points of a line lie in a plane, then all points of the line lie in this plane; c) if two planes have a common point, then they do not intersect; d) a plane, and only one, passes through a line and a point lying on it; e) it is impossible to draw a plane through two intersecting straight lines.

6. Name the common straight line of the planes PBM and MAB. a) PM; b) AB; c) PB; d) BM; e) cannot be determined.

7. Lines a and b intersect at point M. Line c, not passing through point M, intersects lines a and b. What can be said about the relative positions of lines a, b and c? a) All straight lines lie in different planes; b) straight lines a and b lie in the same plane; c) all straight lines lie in the same plane; d) nothing can be said; e) line c coincides with one of the lines: either a or b.

8. Lines a and b intersect at point O. A € a, B € b, Y € AB. Choose the correct statement. a) Points O and Y do not lie in the same plane; b) straight lines OY and a are parallel; c) straight lines a, b and point Y lie in the same plane; d) points O and Y coincide; e) points Y and A coincide.

1. What can be said about the relative position of two planes that have three common points that do not lie on the same straight line?
a) Intersect; b) nothing can be said; c) do not intersect; d) coincide; e) have three common points.

2. Which of the following statements is true?
a) If two points of a circle lie in a plane, then the entire circle lies in this plane; b) a straight line lying in the plane of the triangle intersects its two sides; c) any two planes have only one common point; d) a plane passes through two points, and only one; e) a line lies in the plane of a given triangle if it intersects two lines containing the sides of the triangle.

3. Can two different planes have only two common points?
a) Never; b) I can, but under additional conditions; c) always have; d) the question cannot be answered; d) another answer.

4. Points K, L, M lie on the same line, point N does not lie on it. One plane is drawn through every three points. How many different planes did this result in?
a) 1; b) 2; at 3; d) 4; d) infinitely many.

5. Choose the correct statement.
a) A plane passes through any three points, and only one; b) if two points of a line lie in a plane, then all points of the line lie in this plane; c) if two planes have a common point, then they do not intersect; d) a plane, and only one, passes through a line and a point lying on it; e) it is impossible to draw a plane through two intersecting straight lines.

6. Name the common straight line of the planes PBM and MAB.
a) PM; b) AB; c) PB; d) BM; e) cannot be determined.

7. Which of the listed planes does straight line RM intersect (Fig. 1)?
a) DD1C; b) D1PM; c) B1PM; d) ABC; e) CDA.
B1 C1

8.Two planes intersect in a straight line c. Point M lies in only one of the planes. What can be said about the relative position of point M and line c?
a) No conclusion can be drawn; b) straight line c passes through point M; c) point M lies on line c; d) straight line c does not pass through point M; d) another answer.

9. Lines a and b intersect at point M. Line c, not passing through point M, intersects lines a and b. What can be said about the relative positions of lines a, b and c?
a) All straight lines lie in different planes; b) straight lines a and b lie in the same plane; c) all straight lines lie in the same plane; d) nothing can be said; e) line c coincides with one of the lines: either a or b.

10. Lines a and b intersect at point O. A € a, B € b, Y € AB. Choose the correct statement.
a) Points O and Y do not lie in the same plane; b) straight lines OY and a are parallel; c) straight lines a, b and point Y lie in the same plane; d) points O and Y coincide; e) points Y and A coincide.

Are rays AB and AC perpendicular to its edge? 2. Is it true that the linear angle BAC is a dihedral angle if the rays AB and AC lie on the faces of the dihedral angle? 3. Is it true that angle BAC is a linear angle of a dihedral angle if rays AB and AC are perpendicular to its edge, and points E and C lie on the faces of the angle? 4. The linear angle of a dihedral angle is 80 degrees. Is there a straight line in one of the faces of the angle that is perpendicular to the other face? 5. Angle ABC is a linear angle of a dihedral angle with an alpha edge. Is the straight line alpha perpendicular to the ABC plane? Is it true that all lines perpendicular to a given plane and intersecting a given line lie in the same plane?

In planimetry, the plane is one of the main figures, therefore, it is very important to have a clear understanding of it. This article was created to cover this topic. First, the concept of a plane, its graphical representation is given and the designations of planes are shown. Next, the plane is considered together with a point, a straight line or another plane, and options arise from their relative positions in space. In the second and third and fourth paragraphs of the article, all the options for the relative position of two planes, a straight line and a plane, as well as points and planes are analyzed, the basic axioms and graphic illustrations are given. In conclusion, the main methods of defining a plane in space are given.

Page navigation.

Plane - basic concepts, symbols and image.

The simplest and most basic geometric figures in three-dimensional space are a point, a straight line and a plane. We already have an idea of ​​a point and a line on a plane. If we place a plane on which points and lines are depicted in three-dimensional space, then we get points and lines in space. The idea of ​​a plane in space allows us to obtain, for example, the surface of a table or wall. However, a table or wall has finite dimensions, and the plane extends beyond its boundaries to infinity.

Points and lines in space are designated in the same way as on a plane - in large and small Latin letters, respectively. For example, points A and Q, lines a and d. If two points lying on a line are given, then the line can be denoted by two letters corresponding to these points. For example, straight line AB or BA passes through points A and B. Planes are usually denoted by small Greek letters, for example, planes, or.

When solving problems, it becomes necessary to depict planes in a drawing. A plane is usually depicted as a parallelogram or an arbitrary simple closed region.

A plane is usually considered together with points, straight lines or other planes, and various options for their relative positions arise. Let's move on to their description.

The relative position of the plane and the point.

Let's start with the axiom: there are points in every plane. From it follows the first option for the relative position of the plane and the point - the point can belong to the plane. In other words, a plane can pass through a point. To indicate that a point belongs to a plane, the symbol “” is used. For example, if the plane passes through point A, then you can briefly write .

It should be understood that on a given plane in space there are infinitely many points.

The following axiom shows how many points in space must be marked so that they define a specific plane: through three points that do not lie on the same line, a plane passes, and only one. If three points lying in a plane are known, then the plane can be denoted by three letters corresponding to these points. For example, if a plane passes through points A, B and C, then it can be designated ABC.

Let us formulate another axiom, which gives the second version of the relative position of the plane and the point: there are at least four points that do not lie in the same plane. So, a point in space may not belong to the plane. Indeed, by virtue of the previous axiom, a plane passes through three points in space, and the fourth point may or may not lie on this plane. When writing briefly, use the symbol “”, which is equivalent to the phrase “does not belong”.

For example, if point A does not lie in the plane, then use the short notation.

Straight line and plane in space.

Firstly, a straight line can lie in a plane. In this case, at least two points of this line lie in the plane. This is established by the axiom: if two points of a line lie in a plane, then all points of this line lie in the plane. To briefly record the belonging of a certain line to a given plane, use the symbol “”. For example, the notation means that straight line a lies in the plane.

Secondly, a straight line can intersect a plane. In this case, the straight line and the plane have one single common point, which is called the point of intersection of the straight line and the plane. When writing briefly, I denote the intersection with the symbol “”. For example, the notation means that straight line a intersects the plane at point M. When a plane intersects a certain straight line, the concept of an angle between the straight line and the plane arises.

Separately, it is worth focusing on a straight line that intersects the plane and is perpendicular to any straight line lying in this plane. Such a line is called perpendicular to the plane. To briefly record perpendicularity, use the symbol “”. For a more in-depth study of the material, you can refer to the article perpendicularity of a straight line and a plane.

Of particular importance when solving problems related to the plane is the so-called normal vector of the plane. A normal vector of a plane is any non-zero vector lying on a line perpendicular to this plane.

Thirdly, a straight line may be parallel to the plane, that is, it may not have common points in it. When writing concurrency briefly, use the symbol “”. For example, if line a is parallel to the plane, then we can write . We recommend that you study this case in more detail by referring to the article parallelism of a line and a plane.

It should be said that a straight line lying in a plane divides this plane into two half-planes. The straight line in this case is called the boundary of the half-planes. Any two points of the same half-plane lie on the same side of a line, and two points of different half-planes lie on opposite sides of the boundary line.

Mutual arrangement of planes.

Two planes in space can coincide. In this case they have at least three points in common.

Two planes in space can intersect. The intersection of two planes is a straight line, which is established by the axiom: if two planes have a common point, then they have a common straight line on which all the common points of these planes lie.

In this case, the concept of an angle between intersecting planes arises. Of particular interest is the case when the angle between the planes is ninety degrees. Such planes are called perpendicular. We talked about them in the article perpendicularity of planes.

Finally, two planes in space can be parallel, that is, have no common points. We recommend that you read the article parallelism of planes to get a complete understanding of this option for the relative arrangement of planes.

Methods for defining a plane.

Now we will list the main ways to define a specific plane in space.

Firstly, a plane can be defined by fixing three points in space that do not lie on the same straight line. This method is based on the axiom: through any three points that do not lie on the same line, there is a single plane.

If a plane is fixed and specified in three-dimensional space by indicating the coordinates of its three different points that do not lie on the same straight line, then we can write the equation of the plane passing through the three given points.

The next two methods of defining a plane are a consequence of the previous one. They are based on corollaries of the axiom about a plane passing through three points:

• a plane passes through a line and a point not lying on it, and only one (see also the article equation of a plane passing through a line and a point);
• There is only one plane passing through two intersecting lines (we recommend that you read the article: equation of a plane passing through two intersecting lines).

The fourth way to define a plane in space is based on defining parallel lines. Recall that two lines in space are called parallel if they lie in the same plane and do not intersect. Thus, by indicating two parallel lines in space, we will determine the only plane in which these lines lie.

If in three-dimensional space relative to a rectangular coordinate system a plane is specified in the indicated way, then we can create an equation for a plane passing through two parallel lines.

In high school geometry lessons, the following theorem is proven: through a fixed point in space there passes a single plane perpendicular to a given line. Thus, we can define a plane if we specify the point through which it passes and a line perpendicular to it.

If a rectangular coordinate system is fixed in three-dimensional space and a plane is specified in the indicated way, then it is possible to construct an equation for a plane passing through a given point perpendicular to a given straight line.

Instead of a line perpendicular to the plane, you can specify one of the normal vectors of this plane. In this case, it is possible to write