AG.40. Distance between two intersecting lines
In coordinates 
FMP.3. FULL INCREMENT
functions of several variables - the increment acquired by the function when all arguments receive (generally, non-zero) increments. More precisely, let the function f be defined in a neighborhood of the point
n-dimensional space of variables x 1,. . ., x p. Increment
function f at the point x (0) , where
called full increment if it is considered as a function of n possible increments D x 1, . . ., D x n arguments x 1 , . .., x p, subject only to the condition that the point x (0) + Dx belongs to the domain of the function f. Along with the linear increments of functions, we consider partial increments D x k f function f at the point x (0) in the variable x k, i.e., such increments Df, for which Dx yj =0, j=1, 2, . . ., k- 1, k+1, . . ., n, k - fixed (k=1, 2, . . ., n).
FMP.4. A: The partial increment of the function z \u003d (x, y) with respect to x is the difference with the partial increment with respect to
A: The partial derivative with respect to x of the function z \u003d (x, y) is the limit of the ratio of the partial increment to the increment Ax when the latter tends to zero:
Other designations: Similarly for variables
noah u. 
Noticing that is determined at constant y, and - at constant x, we can formulate the rule: the partial derivative with respect to x of the function z \u003d (x, y) is the usual derivative with respect to x, calculated under the assumption that y \u003d const. Similarly, to calculate the partial derivative with respect to y, one must consider x = const. Thus, the rules for calculating partial derivatives are the same as in the case of a function of one variable.
FMP.5. Continuity of functions. Determining the continuity of a function
The function , is called continuous at the point , if one of the equivalent conditions is satisfied:
2) for an arbitrary sequence ( x n) values , converging at n→ ∞ to a point x 0 , the corresponding sequence ( f(x n)) values of the function converges for n→ ∞ to f(x 0);
3) or f(x) - f(x 0) → 0 as x - x 0 → 0;
4) such that or, which is the same,
f: ]x 0 - δ , x 0 + δ [ → ]f(x 0) - ε , f(x 0) + ε [.
From the definition of continuity of a function f at the point x 0 it follows that
If the function f continuous at every point of the interval ] a, b[, then the function f called continuous on this interval.
FMP.6. AT mathematical analysis, partial derivative- one of the generalizations of the concept of derivative to the case of a function of several variables.
Explicitly, the partial derivative of the function f is defined as follows:
Function Graph z = x² + xy + y². Partial derivative at point (1, 1, 3) at constant y corresponds to the angle of inclination of the tangent line parallel to the plane xz.
Sections of the graph shown above by a plane y= 1
Note that the notation should be understood as whole symbol, in contrast to the usual derivative of a function of one variable, which can be represented as the ratio of the differentials of the function and the argument. However, the partial derivative can also be represented as a ratio of differentials, but in this case it is necessary to indicate by which variable the function is incremented: , where d x f is the partial differential of the function f with respect to the variable x. Often a misunderstanding of the fact of the integrity of a symbol is the cause of errors and misunderstandings, such as, for example, an abbreviation in an expression. (for details, see Fikhtengolts, "Course of differential and integral calculus").
Geometrically, the partial derivative is the derivative along the direction of one of the coordinate axes. Partial derivative of a function f at a point along a coordinate x k is equal to the derivative with respect to the direction, where the unit stands on k-th place.
LA 76) Syst. ur-tion is called Cramer's if the number of equations is equal to the number of unknowns.
LA 77-78) Syst. is called joint if it has at least one solution, and incompatible otherwise.
LA 79-80) Joint system. is called definite if it has only one solution, and indefinite otherwise.
LA 81) ... the determinant of the Cramer system was different from zero
LA 169) In order for the system to be consistent, it is necessary and sufficient that the rank of the matrix be equal to the rank of the extended matrix = .
LA 170) If the determinant of the Cramer system is different from zero, then the system is defined, and its solution can be found by the formulas 
LA 171) 1. Find the solution of the Cramer system of equations by the matrix method; 2.. Let's write the system in matrix form ; 3. Calculate the determinant of the system using its properties: 4. Then write down the inverse matrix A-1; 5. Therefore
LA 172) Homogeneous system of linear equations AX = 0. Homogeneous system is always consistent because it has at least one solution
LA 173) If at least one of the determinants , , is not equal to zero, then all solutions of system (1) will be determined by the formulas , , , where t is an arbitrary number. Each individual solution is obtained at some particular value of t.
LA 174) The set of solutions is homogeneous. systems are called a fundamental system of solutions if: 1) they are linearly independent; 2) any solution of the system is a linear combination of solutions .
AG118. The general equation of the plane is…
The view plane equation is called the general equation of the plane.
AG119.If the plane a is described by the equation Ax+D=0, then...
PR 10.What is an infinitesimal quantity and what are its main properties?
OL 11. What is called infinitely large? What is her connection
with an infinitesimal?
PR12.K What limiting relation is called the first wonderful limit? The first remarkable limit is the limit relation
OL 13 What limiting relation is called the second remarkable limit?
OL 14 What pairs of equivalent functions do you know?
CR64 What is the harmonic series? Under what condition does it converge?
The series of a species is called harmonic.
CR 65.What is the sum of an infinite decreasing progression?
CR66. What statement is meant by the first comparison theorem?
Let there be two positive rows
If, at least from a certain point (say, for ), the following inequality holds: , then the convergence of the series implies the convergence of the series or, which is the same, the divergence of the series follows from the divergence of the series.
CR67. What statement is meant by the second comparison theorem?
Let's pretend that . If there is a limit
then both series converge or diverge at the same time.
CR 45 Formulate the required criterion for the convergence of the series.
If the series has a finite sum, then it is called convergent.
CR 29 A harmonic series is a series of the form…. It converges when
The series of a species is called harmonic. Thus, the harmonic series converges at and diverges at .
AG 6. An ordered system of linearly independent vectors lying on a given line (in a given plane, in space) is called a basis on this line (on this plane, in space), if any vector lying on a given line (in a given plane, space) ) can be represented as a linear combination of vectors of this linearly independent system.
Any pair of non-collinear vectors lying in a given plane forms a basis on that plane.
AG 7. An ordered system of linearly independent vectors lying on a given line (in a given plane, in space) is called a basis on this line (on this plane, in space), if any vector lying on a given line (in a given plane, space) ) can be represented as a linear combination of vectors of this linearly independent system.
Any triple of non-coplanar vectors forms a basis in space.
AG 8, The coefficients in the expansion of a vector in terms of a basis are called the coordinates of this vector in a given basis. In order to find the coordinates of a vector with a given start and end, it is necessary to subtract the coordinates of its beginning from the coordinates of the end of the vector: if , , then .
AG 9.a) Let's construct a vector (a vector, with the beginning at the point and the end at the point , is called point radius vector ).
AG 10. No, because the radian measure of the angle between two vectors is always enclosed between and
AG 11. A scalar is any real number. Dot product two vectors and is called a number equal to the product of their modules and the cosine of the angle between them.
AG 12. we can calculate distance between points, basis vectors, angle between vectors.
AG 13. The cross product of a vector by a vector is the third vector that has the following properties:
Its length is 
The vector is perpendicular to the plane containing the vectors and
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
In this article, we will first define the angle between skew lines and give a graphic illustration. Next, we answer the question: "How to find the angle between skew lines if the coordinates of the direction vectors of these lines in a rectangular coordinate system are known"? In conclusion, we will practice finding the angle between skew lines when solving examples and problems.
Page navigation.
Angle between skew lines - definition.
We will gradually approach the definition of the angle between intersecting lines.
Let us first recall the definition of skew lines: two lines in three-dimensional space are called interbreeding if they do not lie in the same plane. It follows from this definition that skew lines do not intersect, are not parallel, and, moreover, do not coincide, otherwise they would both lie in some plane.
We present some additional auxiliary arguments.
Let two intersecting lines a and b be given in three-dimensional space. Let us construct the lines a 1 and b 1 so that they are parallel to the skew lines a and b, respectively, and pass through some point in the space M 1 . Thus, we will get two intersecting lines a 1 and b 1 . Let the angle between the intersecting lines a 1 and b 1 be equal to the angle . Now let's construct lines a 2 and b 2 , parallel to skew lines a and b, respectively, passing through the point M 2 , which is different from the point M 1 . The angle between the intersecting lines a 2 and b 2 will also be equal to the angle. This statement is true, since the lines a 1 and b 1 will coincide with the lines a 2 and b 2, respectively, if you perform a parallel transfer, in which the point M 1 goes to the point M 2. Thus, the measure of the angle between two lines intersecting at the point M, respectively parallel to the given skew lines, does not depend on the choice of the point M.
We are now ready to define the angle between skew lines.
Definition.
Angle between skew lines is the angle between two intersecting lines that are respectively parallel to the given skew lines.
It follows from the definition that the angle between the skew lines will also not depend on the choice of the point M . Therefore, as a point M, you can take any point belonging to one of the skew lines.
We give an illustration of the definition of the angle between skew lines.
Finding the angle between skew lines.
Since the angle between intersecting lines is determined through the angle between intersecting lines, finding the angle between intersecting lines is reduced to finding the angle between the corresponding intersecting lines in three-dimensional space.
Undoubtedly, the methods studied in geometry lessons in high school. That is, having completed the necessary constructions, it is possible to connect the desired angle with any angle known from the condition, based on the equality or similarity of the figures, in some cases it will help cosine theorem, and sometimes leads to the result definition of sine, cosine and tangent of an angle right triangle.
However, it is very convenient to solve the problem of finding the angle between skew lines using the coordinate method. That is what we will consider.
Let Oxyz be introduced in three-dimensional space (however, in many problems it has to be introduced independently).
Let's set ourselves the task: to find the angle between the intersecting lines a and b, which correspond to some equations of the line in space in the rectangular coordinate system Oxyz.
Let's solve it.
Let's take an arbitrary point of the three-dimensional space M and assume that the lines a 1 and b 1 pass through it, parallel to the intersecting lines a and b, respectively. Then the required angle between intersecting lines a and b is equal to the angle between intersecting lines a 1 and b 1 by definition.
Thus, it remains for us to find the angle between the intersecting lines a 1 and b 1 . To apply the formula for finding the angle between two intersecting lines in space, we need to know the coordinates of the direction vectors of the lines a 1 and b 1 .
How can we get them? And it's very simple. The definition of the directing vector of a straight line allows us to state that the sets of directing vectors of parallel straight lines coincide. Therefore, as the direction vectors of the lines a 1 and b 1, we can take the direction vectors 
and 
straight lines a and b, respectively.
So, the angle between two intersecting lines a and b is calculated by the formula 
, where and are the direction vectors of the lines a and b, respectively.
Formula for finding the cosine of the angle between skew lines a and b has the form 
.
Allows you to find the sine of the angle between skew lines if the cosine is known: 
.
It remains to analyze the solutions of the examples.
Example.
Find the angle between the skew lines a and b , which are defined in the Oxyz rectangular coordinate system by the equations 
and 
.
Solution.
The canonical equations of a straight line in space allow you to immediately determine the coordinates of the directing vector of this straight line - they are given by numbers in the denominators of fractions, that is, 
. Parametric equations of a straight line in space also make it possible to immediately write down the coordinates of the direction vector - they are equal to the coefficients in front of the parameter, that is, 
- direction vector straight . Thus, we have all the necessary data to apply the formula by which the angle between skew lines is calculated: 
Answer:
The angle between the given skew lines is .
Example.
Find the sine and cosine of the angle between the skew lines on which the edges AD and BC of the pyramid ABCD lie, if the coordinates of its vertices are known:.
Solution.
The direction vectors of the crossing lines AD and BC are the vectors and . Let's calculate their coordinates as the difference between the corresponding coordinates of the end and start points of the vector: 
According to the formula 
we can calculate the cosine of the angle between the given skew lines:
Now we calculate the sine of the angle between the skew lines: 
Answer:
In conclusion, we consider the solution of a problem in which it is required to find the angle between skew lines, and the rectangular coordinate system has to be entered independently.
Example.
A rectangular parallelepiped ABCDA 1 B 1 C 1 D 1 is given, in which AB=3 , AD=2 and AA 1 =7 units. Point E lies on the edge AA 1 and divides it in relation to 5 to 2 counting from point A. Find the angle between the skew lines BE and A 1 C.
Solution.
Since the edges of a rectangular parallelepiped are mutually perpendicular at one vertex, it is convenient to introduce a rectangular coordinate system and determine the angle between the indicated skew lines using the coordinate method through the angle between the direction vectors of these lines.
We introduce a rectangular coordinate system Oxyz as follows: let the origin coincide with the vertex A, the Ox axis coincide with the line AD, the Oy axis with the line AB, and the Oz axis with the line AA 1.
Then point B has coordinates, point E - (if necessary, see the article), point A 1 -, and point C -. From the coordinates of these points, we can calculate the coordinates of the vectors and . We have 
, 
.
It remains to apply the formula for finding the angle between the skew lines according to the coordinates of the direction vectors: 
Answer:
Bibliography.
- Atanasyan L.S., Butuzov V.F., Kadomtsev S.B., Kiseleva L.S., Poznyak E.G. Geometry. Textbook for 10-11 grades of high school.
- Pogorelov A.V., Geometry. Textbook for grades 7-11 of educational institutions.
- Bugrov Ya.S., Nikolsky S.M. Higher Mathematics. Volume One: Elements of Linear Algebra and Analytic Geometry.
- Ilyin V.A., Poznyak E.G. Analytic geometry.
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 intersecting 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.
