How to determine if straight lines are crossed. The angle between crossing lines - definition, examples of finding. How to find out the relative position of straight lines in space

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.

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Angle between crossed lines - definition.

We will approach the definition of the angle between crossed lines gradually.

First, recall the definition of intersecting lines: two lines in three-dimensional space are called interbreedingif they do not lie in the same plane. From this definition it follows that crossing lines do not intersect, are not parallel, and, moreover, do not coincide, otherwise they would both lie in some plane.

Here are some more auxiliary arguments.

Let two intersecting straight lines a and b be given in three-dimensional space. Let's 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 will construct lines a 2 and b 2, parallel to intersecting lines a and b, respectively, passing through point М 2, different from point М 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 coincide with the straight lines a 2 and b 2, respectively, if you perform a parallel translation, 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 crossing lines.

Definition.

The angle between crossing lines Is the angle between two intersecting straight lines, which are respectively parallel to a given intersecting straight line.

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 crossed lines.

Finding the angle between crossed lines.

Since the angle between intersecting straight lines is determined through the angle between intersecting straight lines, then finding the angle between intersecting straight lines is reduced to finding the angle between the corresponding intersecting straight lines in three-dimensional space.

Undoubtedly, the 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 us 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 the intersecting lines a and b, respectively. Then the required 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 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 crossed 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 lines is calculated:

Answer:

The angle between the given crossing lines is.

Example.

Find the sine and cosine of the angle between crossed straight 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 \u003d 3, AD \u003d 2 and AA 1 \u003d 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 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 of coordinates coincide with the vertex A, the Ox axis coincides with the AD line, the Oy axis with the AB line, and the Oz axis with the AA 1 line.

Then point B has coordinates, point E - (if necessary, see the article), point A1 -, 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. 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.

Theorem. If one straight line lies in a given plane, and the other straight line intersects this plane at a point that does not belong to the first straight line, then these two straight lines intersect. The criterion for intersecting lines Proof. Let the line a lie in the plane, and the line b intersects the plane at a point B that does not belong to the line a. If straight lines a and b lie in the same plane, then point B would also lie in this plane. Since a single plane passes through a straight line and a point outside this straight line, then this plane must be a plane. But then the line b would lie in the plane, which contradicts the condition. Therefore, the lines a and b do not lie in the same plane, i.e. interbreed.

How many pairs of intersecting straight lines are there that contain the edges of a regular triangular prism? Solution: For each edge of the base, there are three edges that intersect with it. For each side rib, there are two ribs that intersect with it. Therefore, the required number of pairs of intersecting lines is equal to Exercise 5

How many pairs of intersecting straight lines containing the edges of a regular hexagonal prism are there? Solution: Each edge of the bases participates in 8 pairs of intersecting straight lines. Each side edge participates in 8 pairs of crossing straight lines. Therefore, the required number of pairs of intersecting lines is equal to Exercise 6

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

Intersecting straight lines

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

Intersecting can be called those lines that have at least one intersection point.

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

Parallelism

Parallel you can call those lines that never intersect, even at infinity.

In other words, parallel ones are those that have no 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 straight 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 a letter that one straight line is parallel to the second, then they use the following notation a || b. This entry 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 this straight line, you can draw a single parallel straight line. But notice, again the amendment is on the plane. If we consider three-dimensional space, then you can draw an infinite number of straight lines that will not intersect, but will intersect.

The statement that was described above is called parallel axiom.

Perpendicularity

Straight lines can only be called if perpendicularif they intersect at an angle of 90 degrees.

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

Crossed straight lines. Secant

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

Any crossing lines have vertical angles and adjacent ones.

If the corners, which are formed by two crossing straight lines, have one side in common, then they are called adjacent:

Adjacent angles add up to 180 degrees.

Crossed straight lines are easy to recognize by these features. Sign 1. If there are four points on two lines that do not lie in the same plane, then these lines intersect (Fig. 1.21).

Indeed, if these lines would intersect or be parallel, then they would lie in the same plane, and then these points would lie in the same plane, which contradicts the condition.

Sign 2. If the line O lies in the plane, and the line b intersects the plane a at some point

M, which does not lie on the straight line a, then the straight lines a and b intersect (Fig. 1.22).

Indeed, taking any two points on the line a and any two points on the line b, we arrive at criterion 1, i.e. a and b interbreed.

Real examples of intersecting straight lines are given by transport interchanges (Fig. 1.23).

In space, there are more pairs of intersecting straight lines than there are pairs of parallel or intersecting straight lines. This can be explained as follows.

Take in space some point A and some straight line a that does not pass through point A. To draw a straight line through point A parallel to line a, it is necessary to draw plane a through point A and straight line a (Proposition 2 in clause 1.1), and then in the plane and draw a straight line b parallel to a straight line a (Fig. 1.24).

There is only one such straight line b. All lines passing through point A and intersecting line O also lie in the plane a and fill it all with the exception of line b. All the other straight lines going through A and filling all the space except the plane a will intersect with the straight line a. We can say that intersecting lines in space are a general case, and intersecting and parallel lines are special cases. "Small perturbations" of crossing lines leave them crossing. But the properties of being parallel or intersecting with "small perturbations" in space are not preserved.

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 the 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.

It is obvious that 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 intersecting 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 have only one point in common.)

2. Lines are parallel. (That is, they have no common points and lie in the same plane.)

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