If two lines in space have common point, then we say that these two lines intersect. In the following figure, lines a and b intersect at point A. Lines a and c do not intersect.
Any two straight lines either have only one common point or have no common points.
Parallel lines
Two 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 figure presented above, lines a and c are parallel.
Parallel Lines Theorem
Through any point in space that does not lie on a given line, there passes a line parallel to the given one and, moreover, only one.
Crossing lines
Two lines that lie in the same plane can either intersect or be parallel. But in space, two straight lines do not necessarily belong to this plane. They can be located in two different planes.
It is obvious that 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 straight lines.
The following figure shows two intersecting straight lines a and b, which lie in different planes.
Test and theorem on skew lines
If one of 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 intersect.
Theorem on skew lines: through each of two intersecting lines there passes a plane parallel to the other line, and, moreover, only one.
Thus, we have considered all possible cases of relative positions of lines in space. There are only three of them.
1. 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 cross. (That is, they are located in different planes.)
In this article, we will first define the angle between crossing lines and provide a graphic illustration. Next, we will answer the question: “How to find the angle between crossing lines if the coordinates of the direction vectors of these lines in rectangular system coordinates"? In conclusion, we will practice finding the angle between intersecting lines when solving examples and problems.
Page navigation.
Angle between intersecting straight lines - definition.
We will approach determining the angle between intersecting straight lines gradually.
First, let us recall the definition of skew lines: two lines in three-dimensional space are called interbreeding, if they do not lie in the same plane. From this definition it follows that intersecting lines do not intersect, are not parallel, and, moreover, do not coincide, otherwise they would both lie in a certain plane.
Let us give further auxiliary reasoning.
Let two intersecting lines a and b be given in three-dimensional space. Let's construct straight 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 space M 1 . Thus, we get two intersecting lines a 1 and b 1. Let the angle between intersecting lines a 1 and b 1 be equal to angle . Now let's construct lines a 2 and b 2, parallel to the skew lines a and b, respectively, passing through a point M 2, 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 straight lines a 1 and b 1 will coincide with straight lines a 2 and b 2, respectively, if a parallel transfer is performed, in which point M 1 moves to point M 2. Thus, the measure of the angle between two straight lines intersecting at a point M, respectively parallel to the given intersecting lines, does not depend on the choice of point M.
Now we are ready to define the angle between intersecting lines.
Definition.
Angle between intersecting lines is the angle between two intersecting lines that are respectively parallel to the given intersecting lines.
From the definition it follows that the angle between crossing lines will also not depend on the choice of point M. Therefore, as a point M we can take any point belonging to one of the intersecting lines.
Let us give an illustration of determining the angle between intersecting lines.
Finding the angle between intersecting 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, you can 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 crossing lines using the coordinate method. That's what we'll consider.
Let Oxyz be introduced in three-dimensional space (although in many problems you have to enter it yourself).
Let us set ourselves a task: find the angle between the crossing lines a and b, which correspond to some equations of a line in space in the rectangular coordinate system Oxyz.
Let's solve it.
Let's take an arbitrary point in three-dimensional space M and assume that straight lines a 1 and b 1 pass through it, parallel to the crossing straight lines a and b, respectively. Then the required angle between the intersecting lines a and b is equal to the angle between the intersecting lines a 1 and b 1 by definition.
Thus, we just have to find the angle between 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 direction vector of a straight line allows us to assert that the sets of direction vectors of parallel lines coincide. Therefore, the direction vectors of straight lines a 1 and b 1 can be taken as 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 straight lines a and b, respectively.
Formula for finding the cosine of the angle between crossing lines a and b have 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 to the examples.
Example.
Find the angle between the crossing 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 the numbers in the denominators of the 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, 
- direct vector . Thus, we have all the necessary data to apply the formula by which the angle between intersecting lines is calculated: 
Answer:
The angle between the given intersecting 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: .
Solution.
The direction vectors of the crossing lines AD and BC are the vectors and . Let's calculate their coordinates as the difference corresponding coordinates end and start points 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 the crossing lines: 
Answer:
In conclusion, we will consider the solution to a problem in which it is necessary to find the angle between crossing lines, and the rectangular coordinate system must be entered independently.
Example.
Given a rectangular parallelepiped ABCDA 1 B 1 C 1 D 1, which has 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 crossing lines BE and A 1 C.
Solution.
Since the ribs rectangular parallelepiped if one vertex is mutually perpendicular, then it is convenient to introduce a rectangular coordinate system and determine the angle between the indicated crossing lines using the coordinate method through the angle between the direction vectors of these lines.
Let us introduce a rectangular coordinate system Oxyz as follows: let the origin coincide with the vertex A, the Ox axis coincide with the straight line AD, the Oy axis with the straight line AB, and the Oz axis with the straight 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 to find the angle between intersecting lines using 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 secondary school.
- Pogorelov A.V., Geometry. Textbook for grades 7-11 in general education institutions.
- 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.
CROSSING STRAIGHTS Big Encyclopedic Dictionary
crossing lines- straight lines in space that do not lie in the same plane. * * * CROSSING STRAIGHTS CROSSING STRAIGHTS, straight lines in space, not lying in the same plane... encyclopedic Dictionary
Crossing lines- straight lines in space that do not lie in the same plane. Parallel planes can be drawn through a linear point, the distance between which is called the distance between the linear points. It is equal to the shortest distance between the points of the straight line... Great Soviet Encyclopedia
CROSSING STRAIGHTS- straight lines in space that do not lie in the same plane. The angle between the S. p. is called. any of the angles between two parallel lines passing through an arbitrary point in space. If a and b are the direction vectors of the S. p., then the cosine of the angle between the S. p. ... Mathematical Encyclopedia
CROSSING STRAIGHTS- straight 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 straight 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 . Unlike Euclidean geometry in R. g.... ... Mathematical Encyclopedia
The relative position of two lines in space.
The relative position of two lines in space is characterized by the following three possibilities.
Lines lie in the same plane and have no common points - parallel lines.
The lines lie on the same plane and have one common point - the lines intersect.
In space, two straight lines can also be located in such a way that they do not lie in any plane. Such lines are called skew (they do not intersect or are parallel).
EXAMPLE:
PROBLEM 434 Triangle ABC lies in a plane, a
Triangle ABC lies in the plane, but point D is not in this plane. Points M, N and K are respectively the midpoints of segments DA, DB and DCTheorem. If one of two lines lies in a certain plane, and the other intersects this plane at a point that does not lie on the first line, then these lines intersect.
In Fig. 26 straight line a lies in the plane, and straight line c intersects at point N. Lines a and c are intersecting.
Theorem. Through each of two intersecting lines there passes only one plane parallel to the other line.
In Fig. 26 lines a and b intersect. A straight line is drawn and a plane is drawn (alpha) || b (in plane B (beta) the straight line a1 || b is indicated).
Theorem 3.2.
Two lines parallel to a third are parallel.
This property is called transitivity parallelism of lines.
Proof
Let lines a and b be simultaneously parallel to line c. Let us assume that a is not parallel to b, then line a intersects line b at some point A, which does not lie on line c by condition. Consequently, we have two lines a and b, passing through a point A, not lying on a given line c, and at the same time parallel to it. This contradicts axiom 3.1. The theorem has been proven.
Theorem 3.3.
Through a point not lying on a given line, one and only one line can be drawn parallel to the given one.
Proof
Let (AB) be a given line, C a point not lying on it. Line AC divides the plane into two half-planes. Point B lies in one of them. In accordance with axiom 3.2, it is possible to subtract an angle (ACD) from the ray C A equal to angle(CAB), to another half-plane. ACD and CAB are equal internal crosswise lying with the lines AB and CD and the secant (AC) Then, by Theorem 3.1 (AB) || (CD). Taking into account axiom 3.1. The theorem has been proven.
The property of parallel lines is given by the following theorem, converse to Theorem 3.1.
Theorem 3.4.
If two parallel lines are intersected by a third line, then the intersecting interior angles are equal.
Proof
Let (AB) || (CD). Let's assume that ACD ≠ BAC. Through point A we draw a straight line AE so that EAC = ACD. But then, by Theorem 3.1 (AE ) || (CD ), and by condition – (AB ) || (CD). In accordance with Theorem 3.2 (AE ) || (AB). This contradicts Theorem 3.3, according to which through a point A that does not lie on the line CD, one can draw a unique line parallel to it. The theorem has been proven.
Figure 3.3.1.Based on this theorem, the following properties can be easily justified.
If two parallel lines are intersected by a third line, then the corresponding angles are equal.
If two parallel lines are intersected by a third line, then the sum of the interior one-sided angles is 180°.
Corollary 3.2.
If a line is perpendicular to one of the parallel lines, then it is also perpendicular to the other.
The concept of parallelism allows us to introduce the following new concept, which will be needed later in Chapter 11.
The two rays are called equally directed, if there is a line such that, firstly, they are perpendicular to this line, and secondly, the rays lie in the same half-plane relative to this line.
The two rays are called oppositely directed, if each of them is equally directed with a ray complementary to the other.
We will denote identically directed rays AB and CD: and oppositely directed rays AB and CD -
Figure 3.3.2.
Sign of crossing lines.
If one of 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 intersect.
Cases of mutual arrangement of lines in space.
There are four different cases of arrangement of two lines in space:
– straight crossing, i.e. do not lie in the same plane;
– straight lines intersect, i.e. lie in the same plane and have one common point;
– parallel lines, i.e. lie in the same plane and do not intersect;
- the lines coincide.
Let us obtain the characteristics of these cases of the relative position of lines given by the canonical equations
Where — points belonging to lines And accordingly, a— direction vectors (Fig. 4.34). Let us denote bya vector connecting given points.The following characteristics correspond to the cases of relative position of lines listed above:
– straight and crossing vectors are not coplanar;
– straight lines and intersecting vectors are coplanar, but vectors are not collinear;
– direct and parallel vectors are collinear, but vectors are not collinear;
– straight lines and coincident vectors are collinear.
These conditions can be written using the properties of mixed and vector products. Recall that the mixed product of vectors in the right rectangular coordinate system is found by the formula:
and the determinant intersects is zero, and its second and third rows are not proportional, i.e.– straight and parallel second and third lines of the determinant are proportional, i.e. and the first two lines are not proportional, i.e.
– straight lines and all lines of the determinant coincide and are proportional, i.e.
If one of two lines lies in a plane, and the other intersects this plane at a point not belonging to the first line, then these two lines intersect.
Proof
Let a belong to α, b intersects α = A, A does not belong to a (Drawing 2.1.2). Let us assume that lines a and b are non-crossing, that is, they intersect. Then there exists a plane β to which the lines a and b belong. In this plane β lie a line a and a point A. Since the line a and the point A outside it define a single plane, then β = α. But b drives β and b does not belong to α, therefore the equality β = α is impossible.
