If one of two intersecting lines. Definition
CROSSING STRAIGHT Big Encyclopedic Dictionary
crossing straight- straight lines in space that do not lie in the same plane. * * * CROSSING LINE CROSSING LINE, straight lines in space, not lying in the same plane ... encyclopedic Dictionary
Crossed straight lines- straight 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. n ... Great Soviet Encyclopedia
CROSSING STRAIGHT- straight lines in space that do not lie in the same plane. The corner between S. of the item is called. any of the angles between two parallel straight 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 ... Encyclopedia of mathematics
CROSSING STRAIGHT- straight lines in space, not lying in the same plane ... Natural science. encyclopedic Dictionary
Parallel lines- Contents 1 In Euclidean geometry 1.1 Properties 2 In Lobachevsky geometry ... Wikipedia
Ultraparallel straight- Contents 1 In Euclidean geometry 1.1 Properties 2 In Lobachevsky geometry 3 See also ... Wikipedia
RIEMAN'S GEOMETRY- elliptic geometries, one of the non-Euclidean geometries, that is, a geometric theory based on axioms, the requirements for rykh are different from the requirements for the axioms of Euclidean geometry ... Unlike Euclidean geometry in R. g. ... ... Encyclopedia of mathematics
AG.40. Distance between two crossed lines
In coordinates 
FMP. 3. FULL INCREASE
functions of several variables - an increment acquired by a function when all arguments receive (generally speaking, nonzero) increments. More precisely, let the function f be defined in a neighborhood of the point
n-dimensional variable space 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 all possible increments D x 1,. ... ., D x n arguments x 1,. ...., x n, subject only to the condition that the point x (0) + Dx belongs to the domain of the function f. Along with the P. p.functions, we consider the partial increments D x k f function f at the point x (0) in the variable x k, i.e., such increments of Df for which Dx уj = 0, j = 1, 2,. ... ., k- 1, k + 1,. ... ., n, k - fixed (k = 1, 2,..., n).
FMP. 4. О: The partial increment of the function z = (x, y) with respect to x is the difference of the partial increment with respect to
О: The partial derivative with respect to x of the function z = (x, y) is the limit of the ratio of the partial increment to the increment Ax as the latter tends to zero:
Other designations: Similarly for changes
noah u. 
Noticing that it is determined with constant y, and - with constant x, we can formulate a rule: the partial derivative with respect to x of the function z = (x, y) is the usual derivative with respect to x, calculated under the assumption that y = const. Similarly, to calculate the partial derivative with respect to y, one must assume 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
A function is called continuous at a point if one of the equivalent conditions is satisfied:
2) for an arbitrary sequence ( x n) values converging at n→ ∞ to the 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 for 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 the continuity of a function f at the point x 0 implies 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. In mathematical analysis, partial derivative is one of the generalizations of the concept of a 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 a tangent line parallel to the plane xz.
Sections of the graph shown above by a plane y= 1
It should be noted that the designation 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 the 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 abbreviations in expressions. (for more details see Fichtengolts, "Course of differential and integral calculus").
Geometrically, a partial derivative is a derivative in the direction of one of the coordinate axes. Partial derivative of a function f at a point by coordinate x k is equal to the derivative in the direction where the unit stands on k th place.
LA 76) Syst. ur-ny is called Kramer if the number of equations is equal to the number of unknowns.
LA 77-78) Syst. 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 Kramer system was nonzero
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 Kramer system is nonzero, then the system is defined, and its solution can be found by the formulas 
LA 171) 1. Let us find the solution of the Kramer system of equations by the matrix method; 2 .. Let's write the system in matrix form; 3. Let's calculate the determinant of the system, using its properties: 4. Then write down the inverse matrix A-1; 5. Therefore
LA 172) A homogeneous system of linear equations AX = 0. A homogeneous system is always consistent, since 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 for a certain value of t.
LA 174) The set of solutions is homogeneous. systems are called a fundamental system of solutions if: 1) are linearly independent; 2) any solution to the system is a linear combination of solutions.
AG118... The general equation of the plane is ...
The equation of the plane of the view is called general equation of the plane.
AG119. If the plane a is described by the equation Ax + D = 0, then ...
OL 10.What is an infinitesimal quantity and what are its main properties?
OL 11... What quantity is called infinitely large? What's her connection
with infinitesimal?
PR12.K Which limit ratio is called the first remarkable limit? The first remarkable limit is understood as the limiting relation
OL 13 What is the ultimate ratio called the second remarkable limit?
OL 14 What pairs of equivalent functions do you know?
CR64 Which series is called harmonic? Under what condition does it converge?
A series of 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 given two positive series
If, at least from some place (say, for), the inequality holds:, then the convergence of the series follows from the convergence of the series, or - what 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, for, both series converge or diverge simultaneously.
CR 45 Formulate the necessary 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 kind…. It converges when
A series of 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 straight line (in a given plane, in space) is called a basis on this straight line (on this plane, in space) if any vector lying on a given straight 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 this plane.
AG 7. An ordered system of linearly independent vectors lying on a given straight line (in a given plane, in space) is called a basis on this straight line (on this plane, in space) if any vector lying on a given straight line (in a given plane, space ) can be represented as a linear combination of vectors of this linearly independent system.
Any triplet of non-coplanar vectors forms a basis in space.
AG 8, The coefficients in the expansion of a vector in 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, you need to subtract the coordinates of its beginning from the coordinates of the end of the vector: if,, then.
AG 9.a) We construct a vector (a vector, with a beginning at a point and an end at a point, is called point radius vector ).
AG 10. No, because the radian measure of the angle between two vectors is always between and
AG 11. A scalar is any real number. Dot product two vectors and is called the number equal to the product of their moduli by the cosine of the angle between them.
AG 12. we can calculate distance between points, basis vectors, angle between vectors.
AG 13. The vector 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 in which the vectors and
In this article, we first give the definition of the angle between crossed lines and give a graphic illustration. Next, we will answer the question: "How to find the angle between crossing straight lines, if the coordinates of the direction vectors of these straight lines in a rectangular coordinate system are known?" In conclusion, we will practice finding the angle between crossing lines when solving examples and problems.
Page navigation.
Angle between crossed lines - definition.
We will approach the definition of the angle between crossing straight lines gradually.
First, recall the definition of intersecting 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 crossing lines do not intersect, are not parallel, and, moreover, do not coincide, otherwise they would both lie in a certain plane.
Let us give some more auxiliary reasoning.
Let two intersecting straight lines a and b be given in three-dimensional space. Let us construct lines a 1 and b 1 so that they are parallel to intersecting lines a and b, respectively, and pass through some point of the space M 1. Thus, we get two intersecting lines a 1 and b 1. Let the angle between the intersecting straight lines a 1 and b 1 be equal to the angle. Now we construct lines a 2 and b 2, parallel to intersecting lines a and b, respectively, passing through point M 2, different from point M 1. The angle between the intersecting straight lines a 2 and b 2 will also be equal to the angle. This statement is true, since the straight lines a 1 and b 1 will coincide with the straight lines a 2 and b 2, respectively, if you perform a parallel transfer, in which point M 1 goes to point M 2. Thus, the measure of the angle between two intersecting straight lines at the point M, respectively parallel to the given intersecting straight lines, does not depend on the choice of the point M.
Now we are ready to define the angle between crossed lines.
Definition.
Angle between crossed lines Is the angle between two intersecting straight lines, which are respectively parallel to the given intersecting straight lines.
It follows from the definition that the angle between crossing lines will also not depend on the choice of point M. Therefore, as a point M, you can take any point belonging to one of the intersecting lines.
Here is an illustration of the definition of the angle between crossing lines.
Finding the angle between crossed lines.
Since the angle between intersecting straight lines is determined through the angle between intersecting straight lines, finding the angle between intersecting straight lines is reduced to finding the angle between the corresponding intersecting straight lines in three-dimensional space.
Undoubtedly, methods taught in geometry lessons in high school are suitable for finding the angle between crossed lines. That is, having completed the necessary constructions, you can associate 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 the result is 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 crossing straight lines by the coordinate method. This is what we will consider.
Let Oxyz be introduced in three-dimensional space (however, in many problems it has to be entered independently).
Let's set ourselves the task: find the angle between the crossing straight lines a and b, which correspond to some equations of a straight line in space in the rectangular coordinate system Oxyz.
Let's solve it.
Take an arbitrary point of the three-dimensional space M and assume that straight lines a 1 and b 1 pass through it, parallel to intersecting lines a and b, respectively. Then the sought angle between the intersecting straight lines a and b is equal to the angle between the intersecting straight lines a 1 and b 1 by definition.
Thus, it remains for us to find the angle between the intersecting straight lines a 1 and b 1. To apply the formula for finding the angle between two intersecting straight lines in space, we need to know the coordinates of the direction vectors of the straight lines a 1 and b 1.
How can we get them? It's very simple. The definition of the direction vector of a straight line allows us to assert that the sets of direction vectors of parallel straight lines coincide. Therefore, as the direction vectors of the straight lines a 1 and b 1, we can take the direction vectors 
and 
lines a and b, respectively.
So, the angle between two crossed straight lines a and b is calculated by the formula 
where and - direction vectors of straight lines a and b, respectively.
Formula for finding the cosine of the angle between crossed straight lines a and b has the form 
.
Allows you to find the sine of the angle between crossing lines, if the cosine is known: 
.
It remains to analyze the solutions of examples.
Example.
Find the angle between crossing straight lines a and b, which are defined in the rectangular coordinate system Oxyz by the equations 
and 
.
Decision.
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 the 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, 
- directing vector of a straight line ... Thus, we have all the necessary data to apply the formula by which the angle between crossing straight lines is calculated: 
Answer:
The angle between the given crossing lines is equal to.
Example.
Find the sine and cosine of the angle between the crossing lines on which the edges AD and BC of the pyramid ABCD lie, if the coordinates of its vertices are known:.
Decision.
Directing vectors of crossing lines AD and BC are vectors and. Let's calculate their coordinates as the difference of the corresponding coordinates of the points of the end and the beginning of the vector: 
According to the formula 
we can calculate the cosine of the angle between the specified crossing lines:
Now let's calculate the sine of the angle between crossing lines: 
Answer:
In conclusion, we will consider the solution to the problem in which it is required to find the angle between crossing straight lines, and the rectangular coordinate system has to be entered independently.
Example.
Given a rectangular parallelepiped ABCDA 1 B 1 C 1 D 1, in which AB = 3, AD = 2 and AA 1 = 7 units. Point E lies on edge AA 1 and divides it in a ratio of 5 to 2 counting from point A. Find the angle between the crossed lines BE and A 1 C.
Decision.
Since the edges of a rectangular parallelepiped at one vertex are mutually perpendicular, it is convenient to enter a rectangular coordinate system, and determine the angle between the indicated crossing straight lines using the coordinate method through the angle between the direction vectors of these straight lines.
Let us introduce a rectangular coordinate system Oxyz as follows: let the origin coincide with the vertex A, the Ox axis coincides with the AD straight line, the Oy axis with the AB straight line, and the Oz axis — with the AA 1 straight line.
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 to find the angle between the crossing lines along the coordinates of the direction vectors: 
Answer:
Bibliography.
- Atanasyan L.S., Butuzov V.F., Kadomtsev S.B., Kiseleva L.S., Poznyak E.G. Geometry. A textbook for grades 10-11 of secondary school.
- Pogorelov A.V., Geometry. Textbook for grades 7-11 of educational institutions.
- Bugrov Y.S., Nikolsky S.M. Higher mathematics. Volume One: Elements of Linear Algebra and Analytic Geometry.
- Ilyin V.A., Poznyak E.G. Analytic geometry.
If two lines in space have a common point, then they say that these two lines intersect. In the following figure, lines a and b meet 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 straight lines in space are called parallel if they lie in the same plane and do not intersect. To denote parallel lines, use a special icon - ||.
The notation a || b means that line a is parallel to line b. In the picture above, lines a and c are parallel.
Parallel line theorem
Through any point in space that does not lie on a given straight line, there is a straight line parallel to the given one and, moreover, only one.
Crossed straight lines
Two straight 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 this plane. They can be located in two different planes.
Obviously, straight lines located in different planes do not intersect and are not parallel straight lines. Two straight lines that do not lie in the same plane are called crossing lines.
The following figure shows two intersecting straight lines a and b, which lie in different planes.
Criterion and Theorem on Crossed Lines
If one of the two straight lines lies in a certain plane, and the other straight line intersects this plane at a point not lying on the first straight line, then these lines are intersecting.
Crossed Lines Theorem: through each of the two crossing lines there is a plane parallel to the other line, and moreover, only one.
Thus, we have considered all possible cases of mutual arrangement of straight lines in space. There are only three of them.
1. Lines intersect. (That is, they only have one point in common.)
2. Lines are parallel. (That is, they have no points in common and lie in the same plane.)
3. Straight lines are crossed. (That is, they are located in different planes.)
