The article talks about the concept of a straight line on a plane. Let's look at the basic terms and their designations. Let's work with the relative position of a line and a point and two lines on a plane. Let's talk about axioms. Finally, we will discuss methods and methods for defining a straight line on a plane.
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Straight line on a plane - concept
First you need to have a clear understanding of what a plane is. Any surface of something can be classified as a plane, only it differs from objects in its boundlessness. If we imagine that the plane is a table, then in our case it will not have boundaries, but will be infinitely huge.
If you touch the table with a pencil, a mark will remain, which can be called a “dot”. Thus, we get an idea of a point on the plane.
Let's consider the concept of a straight line on a plane. If you draw a straight line on a sheet, it will appear on it with a limited length. We did not get the entire straight line, but only part of it, since in fact it does not have an end, just like a plane. Therefore, the depiction of lines and planes in the notebook is formal.
We have an axiom:
Definition 1
Points can be marked on each straight line and in each plane.
Points are designated in both large and small Latin letters. For example, A and D or a and d.
For a point and a line, only two possible locations are known: a point on a line, in other words, that the line passes through it, or a point not on a line, that is, the line does not pass through it.
To indicate whether a point belongs to a plane or a point to a line, use the sign “∈”. If the condition is given that the point A lies on the line a, then it has the following form of writing A ∈ a. In the case when point A does not belong, then another entry A ∉ a.
Fair judgment:
Definition 2
Through any two points located in any plane, there is a single straight line that passes through them.
This statement is considered an akisoma, and therefore does not require proof. If you consider this yourself, you can see that with two existing points there is only one option for connecting them. If we have two given points A and B, then the line passing through them can be called by these letters, for example, line A B. Consider the figure below.
A straight line located on a plane has a large number of points. This is where the axiom comes from:
Definition 3
If two points of a line lie in a plane, then all other points of this line belong to the plane.
The set of points located between two given points is called a straight segment. It has a beginning and an end. A two-letter designation has been introduced.
If it is given that points A and P are the ends of a segment, then its designation will take the form P A or A P. Since the designations of a segment and a line coincide, it is recommended to add or finish the words “segment”, “straight line”.
A shorthand notation for membership involves the use of the signs ∈ and ∉. In order to fix the location of a segment relative to a given line, use ⊂. If the condition states that the segment A P belongs to the line b, then the entry will look like this: A P ⊂ b.
The case where three points simultaneously belong to one line occurs. This is true when one point lies between two others. This statement is considered to be an axiom. If points A, B, C are given, which belong to the same line, and point B lies between A and C, it follows that all given points lie on the same line, since they lie on both sides of point B.
A point divides a line into two parts, called rays. We have an axiom:
Definition 4
Any point O located on a straight line divides it into two rays, with any two points of one ray lying on one side of the ray relative to point O, and others on the other side of the ray.
The arrangement of straight lines on a plane can take the form of two states.
Definition 5
coincide.
This opportunity arises when straight lines have common points. Based on the axiom written above, we have that a straight line passes through two points and only one. This means that when 2 straight lines pass through given 2 points, they coincide.
Definition 6
Two straight lines on a plane can cross.
This case shows that there is one common point, which is called the intersection of lines. The intersection is designated by the sign ∩. If there is a notation form a ∩ b = M, then it follows that the given lines a and b intersect at the point M.
When straight lines intersect, we deal with the resulting angle. The section where straight lines intersect on a plane to form an angle of 90 degrees, that is, a right angle, is subject to separate consideration. Then the lines are called perpendicular. The form of writing two perpendicular lines is as follows: a ⊥ b, which means that line a is perpendicular to line b.
Definition 7
Two straight lines on a plane can be parallel.
Only if two given lines have no common intersections, and therefore no points, are they parallel. A notation is used that can be written for a given parallelism of lines a and b: a ∥ b.
A straight line on a plane is considered together with vectors. Particular importance is attached to zero vectors that lie on a given line or on any of the parallel lines; they are called direction vectors of a line. Consider the figure below.
Non-zero vectors located on lines perpendicular to a given one are otherwise called normal line vectors. There is a detailed description in the article of the normal vector of a line on a plane. Consider the picture below.
If there are 3 lines on a plane, their location can be very different. There are several options for their location: the intersection of all, parallelism, or the presence of different intersection points. The figure shows the perpendicular intersection of two lines relative to one.
To do this, we present the necessary factors that prove their relative position:
- if two lines are parallel to a third, then they are all parallel;
- if two lines are perpendicular to a third, then these two lines are parallel;
- If on a plane a straight line intersects one parallel line, then it will also intersect another.
Let's look at this in the pictures.
A straight line on a plane can be specified in several ways. It all depends on the conditions of the problem and on what its solution will be based. This knowledge can help for the practical arrangement of straight lines.
Definition 8
The straight line is defined using the specified two points located in the plane.
From the considered axiom it follows that through two points it is possible to draw a straight line and, moreover, only one single one. When a rectangular coordinate system specifies the coordinates of two divergent points, then it is possible to fix the equation of a straight line passing through the two given points. Consider a drawing where we have a line passing through two points. 
Definition 9
A straight line can be defined through a point and a line to which it is parallel.
This method exists because through a point it is possible to draw a straight line parallel to a given one, and only one. The proof is already known from a school course in geometry.
If a line is given relative to a Cartesian coordinate system, then it is possible to construct an equation for a line passing through a given point parallel to a given line. Let's consider the principle of defining a straight line on a plane.
Definition 10
The straight line is specified through the specified point and the direction vector.
When a straight line is specified in a rectangular coordinate system, it is possible to compose canonical and parametric equations on the plane. Let us consider in the figure the location of the straight line in the presence of a direction vector.
The fourth point in specifying a straight line makes sense when the point through which it should be drawn and the straight line perpendicular to it are indicated. From the axiom we have:
Definition 11
Through a given point located on a plane, only one straight line will pass, perpendicular to the given one.
And the last point related to specifying a line on a plane is given the specified point through which the line passes, and in the presence of a normal vector of the line. Given the known coordinates of a point located on a given line and the coordinates of the normal vector, it is possible to write down the general equation of the line.
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In this article we will dwell in detail on one of the primary concepts of geometry - the concept of a straight line on a plane. First, let's define the basic terms and designations. Next, we will discuss the relative position of a line and a point, as well as two lines on a plane, and present the necessary axioms. In conclusion, we will consider ways to define a straight line on a plane and provide graphic illustrations.
Page navigation.
A straight line on a plane is a concept.
Before giving the concept of a straight line on a plane, you should clearly understand what a plane is. Concept of a plane allows you to get, for example, a flat surface on a table or a wall at home. It should, however, be borne in mind that the dimensions of the table are limited, and the plane extends beyond these boundaries to infinity (as if we had an arbitrarily large table).
If we take a well-sharpened pencil and touch its tip to the surface of the “table”, we will get an image of a point. This is how we get representation of a point on a plane.
Now you can move on to the concept of a straight line on a plane.
Place a sheet of clean paper on the table surface (on a plane). In order to draw a straight line, we need to take a ruler and draw a line with a pencil as far as the size of the ruler and sheet of paper we are using allows us to do. It should be noted that in this way we will only get part of the line. We can only imagine an entire straight line extending into infinity.
The relative position of a straight line and a point.
We should start with the axiom: on every straight line and in every plane there are points.
Points are usually denoted in capital Latin letters, for example, points A and F. In turn, straight lines are denoted in small Latin letters, for example, straight lines a and d.
Possible two options for the relative position of a line and a point on a plane: either the point lies on the line (in this case it is also said that the line passes through the point), or the point does not lie on the line (it is also said that the point does not belong to the line or the line does not pass through the point).
To indicate that a point belongs to a certain line, use the symbol “”. For example, if point A lies on line a, then we can write . If point A does not belong to line a, then write .
The following statement is true: there is only one straight line passing through any two points.
This statement is an axiom and should be accepted as a fact. In addition, this is quite obvious: we mark two points on paper, apply a ruler to them and draw a straight line. A straight line passing through two given points (for example, through points A and B) can be denoted by these two letters (in our case, straight line AB or BA).
It should be understood that on a straight line defined on a plane there are infinitely many different points, and all these points lie in the same plane. This statement is established by the axiom: if two points of a line lie in a certain plane, then all points of this line lie in this plane.
The set of all points located between two points given on a line, together with these points, is called straight line segment or simply segment. The points limiting the segment are called the ends of the segment. A segment is denoted by two letters corresponding to the endpoints of the segment. For example, let points A and B be the ends of a segment, then this segment can be designated AB or BA. Please note that this designation for a segment coincides with the designation for a straight line. To avoid confusion, we recommend adding the word “segment” or “straight” to the designation.
To briefly record whether a certain point belongs or does not belong to a certain segment, the same symbols and are used. To show that a certain segment lies or does not lie on a line, use the symbols and, respectively. For example, if segment AB belongs to line a, you can briefly write .
We should also dwell on the case when three different points belong to the same line. In this case, one, and only one point, lies between the other two. This statement is another axiom. Let points A, B and C lie on the same line, and point B lies between points A and C. Then we can say that points A and C are on opposite sides of point B. We can also say that points B and C lie on the same side of point A, and points A and B lie on the same side of point C.
To complete the picture, we note that any point on a line divides this line into two parts - two beam. For this case, an axiom is given: an arbitrary point O, belonging to a line, divides this line into two rays, and any two points of one ray lie on the same side of the point O, and any two points of different rays lie on opposite sides of the point O.
The relative position of lines on a plane.
Now let’s answer the question: “How can two straight lines be located on a plane relative to each other?”
Firstly, two straight lines on a plane can coincide.
This is possible when the lines have at least two common points. Indeed, by virtue of the axiom stated in the previous paragraph, there is only one straight line passing through two points. In other words, if two straight lines pass through two given points, then they coincide.
Secondly, two straight lines on a plane can cross.
In this case, the lines have one common point, which is called the point of intersection of the lines. The intersection of lines is denoted by the symbol “”, for example, the entry means that lines a and b intersect at point M. Intersecting lines lead us to the concept of angle between intersecting lines. Separately, it is worth considering the location of straight lines on a plane when the angle between them is ninety degrees. In this case, the lines are called perpendicular(we recommend the article perpendicular lines, perpendicularity of lines). If line a is perpendicular to line b, then short notation can be used.
Thirdly, two straight lines on a plane can be parallel.
From a practical point of view, it is convenient to consider a straight line on a plane together with vectors. Of particular importance are non-zero vectors lying on a given line or on any of the parallel lines; they are called directing vectors of a straight line. The article Directing vector of a straight line on a plane gives examples of directing vectors and shows options for their use in solving problems.
You should also pay attention to non-zero vectors lying on any of the lines perpendicular to this one. Such vectors are called normal line vectors. The use of normal line vectors is described in the article normal line vector on a plane.
When three or more straight lines are given on a plane, many different options for their relative positions arise. All lines can be parallel, otherwise some or all of them intersect. In this case, all lines can intersect at a single point (see the article on a bunch of lines), or they can have different points of intersection.
We will not dwell on this in detail, but will present without proof several remarkable and very often used facts:
- if two lines are parallel to a third line, then they are parallel to each other;
- if two lines are perpendicular to a third line, then they are parallel to each other;
- If a certain line on a plane intersects one of two parallel lines, then it also intersects the second line.
Methods for defining a straight line on a plane.
Now we will list the main ways in which you can define a specific straight line on a plane. This knowledge is very useful from a practical point of view, since the solution to many examples and problems is based on it.
Firstly, a straight line can be defined by specifying two points on a plane.
Indeed, from the axiom discussed in the first paragraph of this article, we know that a straight line passes through two points, and only one.
If the coordinates of two divergent points are indicated in a rectangular coordinate system on a plane, then it is possible to write down the equation of a straight line passing through two given points.
Secondly, a line can be specified by specifying the point through which it passes and the line to which it is parallel. This method is fair, since through a given point on the plane there passes a single straight line parallel to a given straight line. The proof of this fact was carried out in geometry lessons in high school.
If a straight line on a plane is defined in this way relative to the introduced rectangular Cartesian coordinate system, then it is possible to compose its equation. This is written about in the article equation of a line passing through a given point parallel to a given line.
Thirdly, a straight line can be specified by specifying the point through which it passes and its direction vector.
If a straight line is given in a rectangular coordinate system in this way, then it is easy to construct its canonical equation of a straight line on a plane and parametric equations of a straight line on a plane.
The fourth way to specify a line is to indicate the point through which it passes and the line to which it is perpendicular. Indeed, through a given point of the plane there passes a single straight line perpendicular to the given straight line. Let's leave this fact without proof.
Finally, a line in a plane can be specified by specifying the point through which it passes and the normal vector of the line.
If the coordinates of a point lying on a given line and the coordinates of the normal vector of the line are known, then it is possible to write down the general equation of the line.
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 of linear algebra and analytical geometry.
- Ilyin V.A., Poznyak E.G. Analytic geometry.
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There are three options for the relative position of two lines in space: lines can be intersecting, parallel and crossing.
3.1 Intersecting lines
Two different lines are called intersecting if they have a common point. The point of intersection is unique: if two lines have two points in common, then they coincide.
The intersecting lines are shown in Fig. 19 . Lines a and b, as we see, intersect at point A.
Rice. 19. Intersecting lines
Notice that there is a single plane passing through the two intersecting lines. This is also shown in Fig. 19: a single plane passes through lines a and b.
Question. Line a intersects line b, line b intersects line c. Is it true that lines a and c intersect?
3.2 Parallel lines
Since the seventh grade, you remember that ¾parallel lines are those that do not intersect¿. In space, however, for lines to be parallel, one additional condition is needed.
Definition. Two lines in space are called parallel if they lie in the same plane and do not intersect.
Thus, in addition to ¾non-intersection¿, it is required that the lines lie in the same plane. In Fig. 20 shows parallel lines a and b; a (single) plane passes through them.
Rice. 20. Parallel lines
Parallelism has the important property of transitivity. Namely, for three different lines a, b and c the following holds:
a k b and b k c) a k c
(two different lines parallel to a third line are parallel to each other).
3.3 Crossing lines
If two lines intersect or are parallel, then, as we have seen, a plane can be drawn through them (and, moreover, the only one). In space, however, it is generally impossible to draw a plane through two straight lines.
Definition. Two lines are called skew if they are neither parallel nor intersecting.
An equivalent definition is this: two lines are called skew if they do not lie in the same plane.
In Fig. 21 shows crossing lines a and b.
b
Rice. 21. Crossing lines
An important fact is that two parallel planes can be drawn through two intersecting lines. Namely, if lines a and b intersect, then there is a unique pair of planes and such that a, b and k. This is shown in Fig. 21.
All three considered options for the relative arrangement of straight lines can be seen in the triangular prism ABCA1 B1 C1 (Fig. 22).
Rice. 22. The relative position of two straight lines
Namely, lines AB and BC intersect (left figure); lines BC and B1 C1 are parallel (picture in the center); straight lines AB and B1 C1 intersect (right picture).
4 Planes are called parallel if they do not have common points.
Well, according to the axeom of parallel lines... after all, these lines are located in parallel planesTrue, because two planes are called parallel if they do not intersect. This means that these planes do not have a single common point, but the lines lie in these planes, which means they cannot have common points.
Similar tasks:
A point lying in one of the intersecting planes is 6 cm away from the second plane, and 12 cm from the line of their intersection. Calculate the angle between the planes.
Given points M(3;0;-1), K(1;3;0), P(4;-1;2). Find on the axis Oh such a point A to vectors MK And RA were perpendicular.
The two vertices of an equilateral triangle are located in the plane alpha. Angle between plane alpha and the plane of this triangle is equal to fi. The side of the triangle is equal to m. Calculate:
1) the distance from the third vertex of the triangle to the plane alpha;
2) area of projection of the triangle onto the plane alpha.