lines l1 and l2 are called intersecting if they do not lie in the same plane. Let a and b be the direction vectors of these lines, and the points M1 and M2 belong respectively to the lines and l1 and l2

Then the vectors a, b, M1M2> are not coplanar, and therefore their mixed product is not equal to zero, i.e. (a, b, M1M2>) =/= 0. The converse is also true: if (a, b, M1M2> ) =/= 0, then the vectors a, b, M1M2> are not coplanar, and, consequently, the lines l1 and l2 do not lie in the same plane, i.e., they intersect. Thus, two lines intersect if and only if condition(a, b, M1M2>) =/= 0, where a and b are the direction vectors of the lines, and M1 and M2 are the points belonging respectively to the given lines. The condition (a, b, M1M2>) = 0 is a necessary and sufficient condition for the lines to lie in the same plane. If the lines are given by their canonical equations

then a = (a1; a2; a3), b = (b1; b2; b3), M1 (x1; y1; z1), M2(x2; y2; z2) and condition (2) is written as follows:

Distance between intersecting lines

this is the distance between one of the skew lines and a plane parallel to it passing through the other line. The distance between the skew lines is the distance from some point of one of the skew lines to a plane passing through the other line parallel to the first line.

26. Definition of an ellipse, canonical equation. Derivation of the canonical equation. Properties.

An ellipse is the locus of points in a plane for which the sum of the distances to two focused points F1 and F2 of this plane, called foci, is a constant value. This does not exclude the coincidence of the foci of the ellipse. coordinate system such that the ellipse will be described by the equation (the canonical equation of the ellipse):

It describes an ellipse centered at the origin, whose axes coincide with the coordinate axes.

If on the right side there is a unit with a minus sign, then the resulting equation:

describes an imaginary ellipse. It is impossible to draw such an ellipse in the real plane. Let's denote the foci as F1 and F2, and the distance between them as 2c, and the sum of the distances from an arbitrary point of the ellipse to the foci as 2a

To derive the ellipse equation, we choose the coordinate system Oxy so that the foci F1 and F2 lie on the Ox axis, and the origin of coordinates coincides with the middle of the segment F1F2. Then the foci will have the following coordinates: u Let M(x; y) be an arbitrary point of the ellipse. Then, according to the definition of an ellipse, i.e.

This, in fact, is the equation of an ellipse.

27. Definition of a hyperbola, canonical equation. Derivation of the canonical equation. Properties

A hyperbola is a locus of points in a plane for which the absolute value of the difference between the distances to two fixed points F1 and F2 of this plane, called foci, is a constant. Let M(x;y) be an arbitrary point of the hyperbola. Then according to the definition of a hyperbola |MF 1 – MF 2 |=2a or MF 1 – MF 2 =±2a,

28. Definition of a parabola, canonical equation. Derivation of the canonical equation. Properties. A parabola is a GMT of a plane for which the distance to some fixed point F of this plane is equal to the distance to some fixed straight line, also located in the plane under consideration. F is the focus of the parabola; the fixed straight line is the directrix of the parabola. r=d,

r=; d=x+p/2; (x-p/2) 2 +y 2 =(x+p/2) 2 ; x 2 -xp + p 2 / 4 + y 2 \u003d x 2 + px + p 2 / 4; y 2 =2px;

Properties: 1. The parabola has an axis of symmetry (the axis of the parabola); 2.All

the parabola is located in the right half-plane of the Oxy plane at p>0, and in the left

if p<0. 3.Директриса параболы, определяемая каноническим уравнением, имеет уравнение x= -p/2.

Lecture: Intersecting, parallel and skew lines; perpendicularity of lines

intersecting lines

If there are several straight lines on the plane, then sooner or later they will intersect arbitrarily, or at right angles, or they will be parallel. Let's take a look at each case.

Intersecting lines are those lines that have at least one point of intersection.

You may ask why at least one line cannot intersect another line two or three times. You're right! But the lines can completely coincide with each other. In this case, there will be an infinite number of common points.

Parallelism

Parallel one can name those lines that will never intersect, even at infinity.

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

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

When they want to show in writing that one straight line is parallel to the second, then the following notation a||b is used. This notation says that line a is parallel to line b.

When studying this topic, it is important to understand one more statement: through some point on the plane that does not belong to a given line, one can draw a single parallel line. But pay attention, again the correction is on the plane. If we consider three-dimensional space, then it is possible to draw an infinite number of lines that will not intersect, but will intersect.

The statement described above is called axiom of parallel lines.

Perpendicularity

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

In space, through a certain point on a line, an infinite number of perpendicular lines can be drawn. However, if we are talking about a plane, then through one point on a line, one can draw a single perpendicular line.

Crossed lines. Secant

If some lines intersect at some point at an arbitrary angle, they can be called interbreeding.

Any skew lines have vertical angles and adjacent ones.

If the angles that are formed by two intersecting lines have one side in common, then they are called adjacent:

Adjacent angles add up to 180 degrees.

CROSSING STRAIGHTS Big Encyclopedic Dictionary

intersecting lines are lines in space that do not lie in the same plane. * * * CROSSING DIRECTS CROSSING RIGHTS, straight lines in space, not lying in the same plane ... encyclopedic Dictionary

Crossed lines are lines in space that do not lie in the same plane. Parallel planes can be drawn through the S. p., the distance between which is called the distance between the S. p. It is equal to the shortest distance between the points of the S. p ... Great Soviet Encyclopedia

CROSSING STRAIGHTS are lines in space that do not lie in the same plane. The angle between S. p. any of the angles between two parallel lines passing through an arbitrary point in space. If a and b are direction vectors of S. p., then the cosine of the angle between S. p ... Mathematical Encyclopedia

CROSSING STRAIGHTS- lines in space that do not lie in the same plane ... Natural science. encyclopedic Dictionary

Parallel lines- Contents 1 In Euclidean geometry 1.1 Properties 2 In Lobachevsky geometry ... Wikipedia

Ultraparallel lines- Contents 1 In Euclidean geometry 1.1 Properties 2 In Lobachevsky geometry 3 See also ... Wikipedia

RIEMANN GEOMETRY- elliptical geometry, one of the non-Euclidean geometries, i.e. geometric, a theory based on axioms, the requirements for which are different from the requirements of the axioms of Euclidean geometry . In contrast to Euclidean geometry in R. g. ... ... Mathematical Encyclopedia

If two lines in space have a common point, then these two lines are said to intersect. In the following figure, lines a and b intersect at point A. Lines a and c do not intersect.

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

Parallel lines

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

The notation a||b means that line a is parallel to line b. In the figure above, lines a and c are parallel.

Parallel line theorem

Through any point in space that does not lie on a given line, there passes a line parallel to the given line and, moreover, only one.

Crossed lines

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

Obviously, lines located in different planes do not intersect and are not parallel lines. Two lines that do not lie in the same plane are called crossing lines.

The following figure shows two intersecting lines a and b that lie in different planes.

Sign and the skew lines theorem

If one of the two lines lies in a certain plane, and the other line intersects this plane at a point not lying on the first line, then these lines are skew.

Crossing lines theorem: through each of the two intersecting lines there passes a plane parallel to the other line, and moreover, only one.

Thus, we have considered all possible cases of mutual arrangement of lines in space. There are only three of them.

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

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

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