For any two cross-lived straight lines there is a plane. Sign of cross-lines. Determination of an ellipse, canonical equation. Conclusion of the canonical equation. Properties

Lecture: Intersecting, parallel and crossing straight lines; Perpendicularity of lines

Intersecting straight

If there are several straight lines on the plane, then they either sooner or later intersect arbitrarily or at right angles, or be parallel. Let's figure it out with each case.

Crossing can be called those straight, in which there will be at least one intersection point.

You ask why at least one can not direct two or three times directly. You're right! But direct can completely coincide with each other. In this case, the common points will be an infinite set.

Parallelism

Parallel You can call those straight, which will never cross, even at infinity.

In other words, parallel are those that have not a single common point. Please note that this definition is valid only if the direct are in the same plane, if they do not have common points, being in different planes, they are considered crossing.

Examples of parallel straight lines in life: two opposite edges of the monitor screen, lines in notebooks, as well as many other parts of things having square, rectangular and other forms.

When they want to show on a letter that one straight parallel second, then use the following designation A || B. This entry says that straight and parallel to the direct b.

When studying this topic, it is important to understand another statement: through some point on a plane that does not belong to this direct, you can conduct a single parallel straight. But pay attention to the amendment again - on the plane. If we consider the three-dimensional space, then you can spend an infinite set of direct, which will not intersect, but will be crossingly.

The statement that was described above is called axiom of parallelism direct.

Perpendicularity

Direct can be called only if perpendicularIf they intersect at an angle of 90 degrees.

In space through some point on a straight line, an infinite set perpendicular straight lines can be carried out. However, if we are talking about a plane, then after one point on a straight line can be carried out the only perpendicular straight line.

Crossed straight. Secant

If some direct intersect at some point at an arbitrary angle, they can be called crossing.

Any cross-lines have vertical angles and adjacent.

If the angles, which are formed by two cross-lived straight, one side is common, then they are called adjacent:

Related angles in the amount give 180 degrees.

If two direct in space have a common point, they say that these two straight lines intersect. In the following figure, the straight lines A ib intersect at the point A. Direct A and C do not intersect.

Any, two direct either have only one common point, or do not have common points.

Parallel straight

Two straight lines are called parallel if they lie in the same plane and do not intersect. To indicate parallel direct use a special icon - ||.

Recording A || B means that straight and parallel to the direct b. The figure below, direct A and C parallel.

Theorem on parallel straight lines

Through any point of space that does not lie on this line, there is a straight line, parallel to this and, moreover, only one.

Straight crossing

Two straight lines, which lie in the same plane, can either intersect or be parallel. But in space, two directs do not have to belong to the plane. They can be located in two different planes.

It is obvious that the straight lines located in different planes do not intersect and are not parallel straight. Two straight, which are not lying in the same plane are called crossing straight.

The following figure shows two cross-lived straight lines A and B, which lie in different planes.

Sign and theorem on cross-lived direct

If one of the two lines lies in some plane, and the other direct crosses this plane at a point that is not lying on the first straight, then these straight crossing.

Theorem on crossing straight lines: Through each of the two cross-lived straight lines, a plane, parallel to another straight line, and moreover, only one.

Thus, we considered all possible cases of the relaxation of direct in space. There are only three of them.

1. Straight intersect. (That is, they have only one common point.)

2. Direct parallel. (That is, they do not have common points and lie in the same plane.)

3. Straight crosslinks. (That is, they are located in different planes.)

Text decoding lesson:

You are already known two cases of the mutual location of direct in space:

1. Locking straight;

2. Parallel straight.

Recall their definitions.

Definition. Direct in space are called intersecting if they lie in the same plane and have one common point

Definition. Direct in space are called parallel if they lie in the same plane and do not have common points.

Common for these definitions is that the straight lies lie in the same plane.

There is no always in space. We can deal with several planes, and not all sorts of two straight lines will lie in the same plane.

For example, the edges of the Cuba ABCDA1B1C1D1

AB and A1D1 lie in different planes.

Definition. Two straight lines are called crossing, if there is no such plane, which used through these straight. From the definition it is clear that these directs do not intersect and not parallel.

We prove the theorem that expresses a sign of cross-lines.

Theorem (sign of cross-lone).

If one of the lines lies in a certain plane, and the other direct crosses this plane at the point that does not belong to this straight line, then these straight crossing.

Direct AB lies in the plane α. The direct CD crosses the plane α at a point with that does not belong to the straight AV.

Prove that direct AB and DC are cross.

Evidence

Proof We will lead the method from nasty.

Suppose, AB and CD lie in the same plane, denote it β.

Then the plane β passes through direct AB and point C.

By a consequence of the axiom, through direct AB and the point C not lying on it can be plane, and with just one.

But we already have such a plane - the plane α.

Consequently, the plane β and α coincide.

But it is impossible, because Direct CD crosses α, and does not lie in it.

We came to contradiction, therefore, our assumption is incorrect. AB and CD lie in

different planes are crossingly.

Theorem is proved.

So, there are three ways of reciprocal location in space:

A) Straight intersects, i.e. have only one common point.

B) straight parallel, i.e. Lying in the same plane and do not have common points.

C) straight crossbreak, i.e. Do not lie in the same plane.

Consider another theorem of cross-lived direct

Theorem. Through each of the two crossing lines, the plane, parallel to another straight line, and moreover, only one.

AB and CD - cross-lived straight

Prove that there is a plane α such that direct AB lies in the plane α, and the direct CD is parallel to the plane α.

Evidence

Let us prove the existence of such a plane.

1) After a point A, we will spend a direct AE parallel to CD.

2) Since straight AE and AV intersect, then they can be plane. Denote it through α.

3) Since the direct CD is parallel to AE, and AE lies in the plane α, the direct CD ∥ plane α (by the theorem on the perpendicularity of the direct and plane).

The plane α is the desired plane.

We prove that the plane α is the only one satisfying the condition.

Any other plane passing through the straight AV will cross AE, which means that the direct CD parallel to it. Those., Any other plane passing through AB intersects with a direct CD, so it is not parallel to it.

Consequently, the plane α is the only one. Theorem is proved.

Straight crossing Big Encyclopedic Dictionary

straight crossing - straight in space that are not lying in the same plane. * * * Straight straight straight straight, straight in space that are not lying in the same plane ... encyclopedic Dictionary

Straight crossing - straight in space that are not lying in the same plane. Through S. p. You can carry out parallel planes, the distance between which is called the distance between S. p. It is equal to the shortest distance between the points S. P ... Great Soviet Encyclopedia

Straight crossing - straight in space that are not lying in the same plane. The angle between S. p. Naz. Any corners between two parallel direct, passing through an arbitrary point of space. If a and b guide vectors S. p., Then the cosine of the angle between S. P ... Mathematical encyclopedia

Straight crossing - straight in space, not lying in the same plane ... Natural science. encyclopedic Dictionary

Parallel straight - Content 1 in Euclidean geometry 1.1 Properties 2 in Lobachevsky geometry ... Wikipedia

Ultraparal lines - content 1 in Euclidean geometry 1.1 Properties 2 in Lobachevsky geometry 3 cm. Also ... Wikipedia

Riemann Geometry - E L L and P T T and C A S K A I, one of the non-child geometries, i.e., geometrich, theory, based on axioms, requirements for ryy are different from the requirements of the axiom of Euclidean geometry . Unlike the Euclidean geometry in R. G. ... ... Mathematical encyclopedia

Theorem. If one straight lies in this plane, and the other direct crosses this plane at a point that does not belong to the first straight line, then these two straight lines are cross. Sign of cross-country proof. Let direct A lie in the plane, and the straight line b crosses the plane at the point B that does not belong to the straight line a. If the straight a and b were lying in the same plane, then in this plane would also be point B. Since the only plane passes through the straight and point outside, the plane should be this plane. But then the straight line B lying in the plane, which contradicts the condition. Therefore, straight a and b are not lying in the same plane, i.e. Crossed.

How many pairs of cross-live lines containing the ribs of the correct triangular prism? Solution: For each rib, there are three ribs, crossing it. For each side edge there are two ribs, with it crossing. Consequently, the desired number of pairs of cross-lived straight lines is the exercise 5

How many pairs of crossing straight lines containing the ribs of the right hexagonal prism? Solution: Each rib grounds participates in 8 pairs of cross-lines. Each lateral edge participates in 8 pairs of cross-lines. Consequently, the desired number of pairs of cross-lived straight lines is EXERCISE 6