Crossed straight lines theorem. Crossed straight lines. Examples of tasks with and without solutions. How to build a line of space perpendicular to a given

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, which will 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 name those straight lines that never intersect, even at infinity.

In other words, parallel ones 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 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 perpendicular if they intersect at an angle of 90 degrees.

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

Crossed straight lines. Secant

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

Any crossing lines have vertical corners 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.

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 lay 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. Consequently, lines a and b do not lie in the same plane, i.e. interbreed.

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

In less than a minute, I created a new Vord file and continued such an exciting topic. You need to catch the moments of the working mood, so there will be no lyrical introduction. There will be a prosaic whipping =)

Two straight spaces can:

1) interbreed;

2) intersect at a point;

3) be parallel;

4) match.

Case No. 1 is fundamentally different from other cases. Two straight lines intersect if they do not lie in the same plane... Raise one hand up and extend the other hand forward - here's an example of crossing straight lines. In points 2-4, the straight lines must lie in one plane.

How to find out the relative position of straight lines in space?

Consider two straight spaces:

- a straight line given by a point and a direction vector;
- a straight line specified by a point and a direction vector.

For a better understanding, let's perform a schematic drawing:

The drawing shows crossed straight lines as an example.

How to deal with these straight lines?

Since the points are known, it is easy to find the vector.

If straight interbreed, then vectors not coplanar(see lesson Linear (non) dependence of vectors. Basis of vectors), and, therefore, the determinant composed of their coordinates is nonzero. Or, which is actually the same, will be nonzero: .

In cases No. 2-4, our construction "falls" into one plane, while the vectors coplanar, and the mixed product of linearly dependent vectors is equal to zero: .

We spin the algorithm further. Let's pretend that , therefore, the lines either intersect, or parallel, or coincide.

If the direction vectors collinear, then the lines are either parallel or coincide. As a final nail, I propose the following technique: take any point of one straight line and substitute its coordinates into the equation of the second straight line; if the coordinates "fit", then the straight lines coincide; if they "did not fit," then the straight lines are parallel.

The flow of the algorithm is simple, but practical examples still do not hurt:

Example 11

Find out the relative position of two straight lines

Solution: as in many problems of geometry, it is convenient to draw up the solution according to the points:

1) We take out the points and direction vectors from the equations:

2) Find the vector:

Thus, the vectors are coplanar, which means that the lines lie in the same plane and can intersect, be parallel or coincide.

4) Check the direction vectors for collinearity.

Let's compose a system of the corresponding coordinates of these vectors:

From of each the equation implies that, therefore, the system is consistent, the corresponding coordinates of the vectors are proportional, and the vectors are collinear.

Conclusion: straight lines are parallel or coincide.

5) Let us find out if the lines have common points. Take a point belonging to the first straight line and substitute its coordinates into the equations of the straight line:

Thus, the lines have no common points, and they have no choice but to be parallel.

Answer:

An interesting example for an independent solution:

Example 12

Find out the relative position of the straight lines

This is an example for a do-it-yourself solution. Note that the second line has a letter as a parameter. It is logical. In the general case, these are two different lines, therefore each line has its own parameter.

And again I urge you not to skip examples, I will flog the problems I offer are far from random ;-)

Problems with a straight line in space

In the final part of the lesson, I will try to consider the maximum number of different problems with spatial lines. In this case, the starting order of the narrative will be respected: first we will consider problems with intersecting straight lines, then with intersecting straight lines, and at the end we will talk about parallel straight lines in space. However, I must say that some tasks of this lesson can be formulated at once for several cases of the arrangement of straight lines, and in this regard, the division of the section into paragraphs is somewhat arbitrary. There are simpler examples, there are more complex examples, and I hope everyone will find what they need.

Crossed straight lines

Let me remind you that straight lines intersect if there is no plane in which they both lie. When I was thinking over the practice, a monster problem came to my mind, and now I am glad to present to your attention a dragon with four heads:

Example 13

Given are straight lines. Required:

a) prove that straight lines are crossed;

b) find the equations of a straight line passing through a point perpendicular to these straight lines;

c) compose the equations of the straight line that contains common perpendicular intersecting straight lines;

d) find the distance between the straight lines.

Solution: The road will be mastered by the walking:

a) Let us prove that the lines intersect. Let's find the points and direction vectors of these lines:

Find the vector:

Let's calculate mixed product of vectors:

Thus, the vectors not coplanar, which means that the lines intersect, as required.

Probably, everyone has noticed long ago that for crossing lines, the verification algorithm turns out to be the shortest.

b) Let us find the equations of the straight line that passes through the point and is perpendicular to the straight lines. Let's execute a schematic drawing:

For a change, I have placed a straight line PER straight, see how it is slightly erased at the crossing points. Crossbreeds? Yes, in the general case the straight line "de" will intersect with the original straight lines. Although we are not interested in this moment, we just need to build a perpendicular line and that's it.

What is known about the direct "de"? The point belonging to her is known. A direction vector is missing.

By condition, the straight line must be perpendicular to the straight lines, which means that its direction vector will be orthogonal to the direction vectors. Already familiar from Example No. 9 motive, we find the cross product:

Let's compose the equations of the straight line "de" by the point and the direction vector:

Ready. In principle, you can change the signs in the denominators and write the answer in the form , but there is no need for this.

To check, it is necessary to substitute the coordinates of the point into the obtained equations of the straight line, then using dot product of vectors make sure that the vector is really orthogonal to the direction vectors "pe one" and "pe two".

How to find the equations of a straight line containing a common perpendicular?

c) This task will be more difficult. I recommend dummies to skip this point, I don’t want to cool your sincere sympathy for analytic geometry =) By the way, it is probably better for more prepared readers to postpone too, the fact is that, in terms of complexity, the example should be put last in the article, but according to the logic of presentation it should be located here.

So, it is required to find the equations of the straight line, which contains the common perpendicular of the intersecting straight lines.

Is a line segment connecting the given lines and perpendicular to the given lines:

Here is our handsome man: - the common perpendicular of intersecting straight lines. He is the only one. There is no other such. We also need to compose the equations of the straight line that contains the given segment.

What is known about the straight "uh"? Its direction vector, found in the previous paragraph, is known. But, unfortunately, we do not know a single point belonging to the straight line "em", we do not know the ends of the perpendicular - points. Where does this perpendicular line intersect the two original lines? In Africa, in Antarctica? From the initial review and analysis of the condition it is not at all clear how to solve the problem…. But there is a tricky move associated with the use of parametric equations of a straight line.

We will issue the decision according to the points:

1) Let's rewrite the equations of the first straight line in parametric form:

Consider a point. We do not know the coordinates. BUT... If a point belongs to a given straight line, then it corresponds to its coordinates, we denote it by. Then the coordinates of the point will be written in the form:

Life is getting better, one unknown - after all, not three unknowns.

2) The same outrage must be carried out on the second point. Let's rewrite the equations of the second straight line in parametric form:

If a point belongs to a given straight line, then with a very specific value its coordinates must satisfy the parametric equations:

Or:

3) The vector, like the previously found vector, will be the direction vector of the straight line. How to compose a vector by two points was considered in the lesson in time immemorial Vectors for dummies... Now the difference is that the coordinates of the vectors are written with unknown parameter values. So what? Nobody forbids subtracting from the coordinates of the end of the vector the corresponding coordinates of the beginning of the vector.

There are two points: .

Find the vector:

4) Since the direction vectors are collinear, then one vector is linearly expressed through the other with a certain proportionality coefficient "lambda":

Or coordinatewise:

It turned out the most, that neither is the usual system of linear equations with three unknowns, which is solvable in the standard, for example, Cramer's method... But here there is an opportunity to get rid of a little blood, from the third equation we express "lambda" and substitute it into the first and second equations:

In this way: , and we do not need a "lambda". The fact that the values ​​of the parameters turned out to be the same is pure coincidence.

5) The sky is completely clear, substitute the found values to our points:

The direction vector is not particularly needed, since its colleague has already been found.

After a long journey, it's always fun to check.

:

The correct equalities are obtained.

Substitute the coordinates of the point into the equations :

The correct equalities are obtained.

6) Final chord: compose the equations of a straight line along a point (you can take it) and a direction vector:

In principle, you can pick up a "good" point with integer coordinates, but this is already a cosmetic.

How to find the distance between crossed lines?

d) We cut off the fourth dragon's head.

Method one... Not even a way, but a small special case. The distance between crossed lines is equal to the length of their common perpendicular: .

The extreme points of the common perpendicular found in the previous paragraph, and the task is elementary:

Method two... In practice, most often the ends of the common perpendicular are unknown, so a different approach is used. Parallel planes can be drawn through two crossing lines, and the distance between these planes is equal to the distance between these lines. In particular, a common perpendicular sticks out between these planes.

In the course of analytical geometry, from the above considerations, a formula was derived for finding the distance between crossing straight lines:
(instead of our points "uh, one, two" you can take arbitrary points of straight lines).

Mixed product of vectors already found in paragraph "a": .

Vector product of vectors found in item "bae": , let's calculate its length:

In this way:

We proudly lay out the trophies in one row:

Answer:
a) , which means that the lines intersect, which was required to prove;
b) ;
v) ;
G)

What else can you tell about crossing straight lines? An angle is defined between them. But we will consider the universal angle formula in the next paragraph:

Intersecting straight lines of space necessarily lie in the same plane:

The first thought is to lean on the intersection point with all your might. And immediately I thought, why deny yourself the right desires ?! Let's pounce on her now!

How to find the point of intersection of spatial lines?

Example 14

Find the point of intersection of lines

Solution: Let's rewrite the equations of straight lines in parametric form:

This task was discussed in detail in Example No. 7 of this lesson (see. Equations of a straight line in space). And the straight lines themselves, by the way, I took from Example No. 12. I won’t lie, I’m too lazy to invent new ones.

The solution is standard and has already been encountered when we grind out the equations of the common perpendicular of intersecting straight lines.

The intersection point of the straight lines belongs to the straight line, therefore its coordinates satisfy the parametric equations of the given straight line, and they correspond to quite specific parameter value:

But the same point belongs to the second straight line, therefore:

We equate the corresponding equations and make simplifications:

A system of three linear equations with two unknowns is obtained. If the lines intersect (which is proved in Example 12), then the system is necessarily compatible and has a unique solution. It can be solved Gaussian method, but we will not sin with such kindergarten fetishism, we will do it easier: from the first equation we will express “te zero” and substitute it into the second and third equations:

The last two equations turned out to be, in fact, the same, and it follows from them that. Then:

Substitute the found value of the parameter into the equations:

Answer:

To check, we substitute the found value of the parameter into the equations:
The same coordinates were obtained as required to be verified. Meticulous readers can substitute the coordinates of a point in the original canonical equations of straight lines.

By the way, it was possible to do the opposite: find the point through the "es zero", and check - through the "te zero".

A well-known mathematical sign says: where they discuss the intersection of straight lines, it always smells like perpendiculars.

How to construct a line of space perpendicular to a given one?

(lines intersect)

Example 15

a) Make up the equations of a straight line passing through a point perpendicular to a straight line (lines intersect).

b) Find the distance from a point to a straight line.

Note : the clause "lines intersect" - essential... Through point
you can draw infinitely many perpendicular straight lines that will intersect with the straight "ale". The only solution takes place in the case when a straight line is drawn through this point, perpendicular to two given by a straight line (see Example No. 13, point "b").

a) Solution: The unknown straight line is denoted by. Let's execute a schematic drawing:

What is known about the straight line? By condition, a point is given. In order to compose the equations of a straight line, it is necessary to find the direction vector. As such a vector, a vector is quite suitable, and we will deal with it. More precisely, let's take the unknown end of the vector by the scruff.

1) Let us take out the directing vector from the equations of the straight line "el", and rewrite the equations themselves in parametric form:

Many have guessed that now for the third time in a lesson the magician will get a white swan out of his hat. Consider a point with unknown coordinates. Since it is a point, its coordinates satisfy the parametric equations of the straight line "el" and a specific value of the parameter corresponds to them:

Or in one line:

2) By condition, the straight lines must be perpendicular, therefore, their direction vectors are orthogonal. And if the vectors are orthogonal, then their scalar product is equal to zero:

What happened? The simplest linear equation with one unknown:

3) The value of the parameter is known, we find the point:

And the direction vector:
.

4) Let us compose the equations of a straight line by a point and a direction vector:

The denominators of the proportion turned out to be fractional, and this is exactly the case when it is appropriate to get rid of fractions. I'll just multiply them by -2:

Answer:

Note : a more rigorous ending of the solution is formed as follows: we compose the equations of a straight line along a point and a direction vector. Indeed, if a vector is a directing vector of a straight line, then a vector collinear to it, naturally, will also be a directing vector of a given straight line.

The check consists of two stages:

1) check the direction vectors of straight lines for orthogonality;

2) we substitute the coordinates of the point into the equations of each straight line, they must "fit" both there and there.

A lot was said about typical actions, so I checked on a draft.

By the way, I still forgot the point - to build a point "siu" symmetrical to the point "en" relative to the straight line "el". However, there is a good "flat analog", which can be found in the article The simplest problems with a straight line on a plane... Here all the difference will be in the additional "zeta" coordinate.

How to find the distance from a point to a line in space?

b) Solution: Find the distance from a point to a straight line.

Method one... This distance is exactly equal to the length of the perpendicular:. The solution is obvious: if the points are known , then:

Method two... In practical tasks, the base of the perpendicular is often a secret behind seven seals, so it is more rational to use a ready-made formula.

The distance from a point to a straight line is expressed by the formula:
, where is the directing vector of the straight line "el", and - arbitrary point belonging to the given line.

1) From the equations of the straight line we get the direction vector and the most accessible point.

2) The point is known from the condition, sharpen the vector:

3) Find cross product and calculate its length:

4) Calculate the length of the direction vector:

5) Thus, the distance from a point to a straight line: