How to determine if lines intersect. Mutual arrangement of lines in space

Oh-oh-oh-oh-oh ... well, it's tinny, as if you read the sentence to yourself =) However, then relaxation will help, especially since I bought suitable accessories today. Therefore, let's proceed to the first section, I hope, by the end of the article I will keep a cheerful mood.

Mutual arrangement of two straight lines

The case when the hall sings along in chorus. Two lines can:

1) match;

2) be parallel: ;

3) or intersect at a single point: .

Help for dummies : please remember the mathematical sign of the intersection , it will occur very often. The entry means that the line intersects with the line at the point.

How to determine the relative position of two lines?

Let's start with the first case:

Two lines coincide if and only if their respective coefficients are proportional, that is, there is such a number "lambda" that the equalities

Let's consider straight lines and compose three equations from the corresponding coefficients: . From each equation it follows that, therefore, these lines coincide.

Indeed, if all the coefficients of the equation multiply by -1 (change signs), and all the coefficients of the equation reduce by 2, you get the same equation: .

The second case when the lines are parallel:

Two lines are parallel if and only if their coefficients at the variables are proportional: , but.

As an example, consider two straight lines. We check the proportionality of the corresponding coefficients for the variables :

However, it is clear that .

And the third case, when the lines intersect:

Two lines intersect if and only if their coefficients of the variables are NOT proportional, that is, there is NOT such a value of "lambda" that the equalities are fulfilled

So, for straight lines we will compose a system:

From the first equation it follows that , and from the second equation: , hence, the system is inconsistent(no solutions). Thus, the coefficients at the variables are not proportional.

Conclusion: lines intersect

In practical problems, the solution scheme just considered can be used. By the way, it is very similar to the algorithm for checking vectors for collinearity, which we considered in the lesson. The concept of linear (non) dependence of vectors. Vector basis. But there is a more civilized package:

Example 1

Find out the relative position of the lines:

Decision based on the study of directing vectors of straight lines:

a) From the equations we find the direction vectors of the lines: .

, so the vectors are not collinear and the lines intersect.

Just in case, I will put a stone with pointers at the crossroads:

The rest jump over the stone and follow on, straight to Kashchei the Deathless =)

b) Find the direction vectors of the lines:

The lines have the same direction vector, which means they are either parallel or the same. Here the determinant is not necessary.

Obviously, the coefficients of the unknowns are proportional, while .

Let's find out if the equality is true:

Thus,

c) Find the direction vectors of the lines:

Let's calculate the determinant, composed of the coordinates of these vectors:
, therefore, the direction vectors are collinear. The lines are either parallel or coincide.

The proportionality factor "lambda" is easy to see directly from the ratio of collinear direction vectors. However, it can also be found through the coefficients of the equations themselves: .

Now let's find out if the equality is true. Both free terms are zero, so:

The resulting value satisfies this equation (any number generally satisfies it).

Thus, the lines coincide.

Answer:

Very soon you will learn (or even have already learned) to solve the considered problem verbally literally in a matter of seconds. In this regard, I see no reason to offer something for an independent solution, it is better to lay one more important brick in the geometric foundation:

How to draw a line parallel to a given one?

For ignorance of this simplest task, the Nightingale the Robber severely punishes.

Example 2

The straight line is given by the equation . Write an equation for a parallel line that passes through the point.

Decision: Denote the unknown line by the letter . What does the condition say about it? The line passes through the point. And if the lines are parallel, then it is obvious that the directing vector of the line "ce" is also suitable for constructing the line "de".

We take out the direction vector from the equation:

Answer:

The geometry of the example looks simple:

Analytical verification consists of the following steps:

1) We check that the lines have the same direction vector (if the equation of the line is not properly simplified, then the vectors will be collinear).

2) Check if the point satisfies the resulting equation.

Analytical verification in most cases is easy to perform verbally. Look at the two equations and many of you will quickly figure out how the lines are parallel without any drawing.

Examples for self-solving today will be creative. Because you still have to compete with Baba Yaga, and she, you know, is a lover of all kinds of riddles.

Example 3

Write an equation for a line passing through a point parallel to the line if

There is a rational and not very rational way to solve. The shortest way is at the end of the lesson.

We did a little work with parallel lines and will return to them later. The case of coinciding lines is of little interest, so let's consider a problem that is well known to you from the school curriculum:

How to find the point of intersection of two lines?

If straight intersect at the point , then its coordinates are the solution systems of linear equations

How to find the point of intersection of lines? Solve the system.

Here's to you geometric meaning of a system of two linear equations with two unknowns are two intersecting (most often) straight lines on a plane.

Example 4

Find the point of intersection of lines

Decision: There are two ways to solve - graphical and analytical.

The graphical way is to simply draw the given lines and find out the point of intersection directly from the drawing:

Here is our point: . To check, you should substitute its coordinates into each equation of a straight line, they should fit both there and there. In other words, the coordinates of a point are the solution of the system . In fact, we considered a graphical way to solve systems of linear equations with two equations, two unknowns.

The graphical method, of course, is not bad, but there are noticeable disadvantages. No, the point is not that seventh graders decide this way, the point is that it will take time to make a correct and EXACT drawing. In addition, some lines are not so easy to construct, and the intersection point itself can be somewhere in the thirtieth kingdom outside the notebook sheet.

Therefore, it is more expedient to search for the intersection point by the analytical method. Let's solve the system:

To solve the system, the method of termwise addition of equations was used. To develop the relevant skills, visit the lesson How to solve a system of equations?

Answer:

The verification is trivial - the coordinates of the intersection point must satisfy each equation of the system.

Example 5

Find the point of intersection of the lines if they intersect.

This is a do-it-yourself example. It is convenient to divide the problem into several stages. Analysis of the condition suggests that it is necessary:
1) Write the equation of a straight line.
2) Write the equation of a straight line.
3) Find out the relative position of the lines.
4) If the lines intersect, then find the point of intersection.

The development of an action algorithm is typical for many geometric problems, and I will repeatedly focus on this.

Full solution and answer at the end of the tutorial:

A pair of shoes has not yet been worn out, as we got to the second section of the lesson:

Perpendicular lines. The distance from a point to a line.
Angle between lines

Let's start with a typical and very important task. In the first part, we learned how to build a straight line parallel to the given one, and now the hut on chicken legs will turn 90 degrees:

How to draw a line perpendicular to a given one?

Example 6

The straight line is given by the equation . Write an equation for a perpendicular line passing through a point.

Decision: It is known by assumption that . It would be nice to find the direction vector of the straight line. Since the lines are perpendicular, the trick is simple:

From the equation we “remove” the normal vector: , which will be the directing vector of the straight line.

We compose the equation of a straight line by a point and a directing vector:

Answer:

Let's unfold the geometric sketch:

Hmmm... Orange sky, orange sea, orange camel.

Analytical verification of the solution:

1) Extract the direction vectors from the equations and with the help dot product of vectors we conclude that the lines are indeed perpendicular: .

By the way, you can use normal vectors, it's even easier.

2) Check if the point satisfies the resulting equation .

Verification, again, is easy to perform verbally.

Example 7

Find the point of intersection of perpendicular lines, if the equation is known and dot.

This is a do-it-yourself example. There are several actions in the task, so it is convenient to arrange the solution point by point.

Our exciting journey continues:

Distance from point to line

Before us is a straight strip of the river and our task is to reach it in the shortest way. There are no obstacles, and the most optimal route will be movement along the perpendicular. That is, the distance from a point to a line is the length of the perpendicular segment.

The distance in geometry is traditionally denoted by the Greek letter "ro", for example: - the distance from the point "em" to the straight line "de".

Distance from point to line is expressed by the formula

Example 8

Find the distance from a point to a line

Decision: all you need is to carefully substitute the numbers into the formula and do the calculations:

Answer:

Let's execute the drawing:

The distance found from the point to the line is exactly the length of the red segment. If you make a drawing on checkered paper on a scale of 1 unit. \u003d 1 cm (2 cells), then the distance can be measured with an ordinary ruler.

Consider another task according to the same drawing:

The task is to find the coordinates of the point , which is symmetrical to the point with respect to the line . I propose to perform the actions on your own, however, I will outline the solution algorithm with intermediate results:

1) Find a line that is perpendicular to a line.

2) Find the point of intersection of the lines: .

Both actions are discussed in detail in this lesson.

3) The point is the midpoint of the segment. We know the coordinates of the middle and one of the ends. By formulas for the coordinates of the middle of the segment find .

It will not be superfluous to check that the distance is also equal to 2.2 units.

Difficulties here may arise in calculations, but in the tower a microcalculator helps out a lot, allowing you to count ordinary fractions. Have advised many times and will recommend again.

How to find the distance between two parallel lines?

Example 9

Find the distance between two parallel lines

This is another example for an independent solution. A little hint: there are infinitely many ways to solve. Debriefing at the end of the lesson, but better try to guess for yourself, I think you managed to disperse your ingenuity well.

Angle between two lines

Whatever the corner, then the jamb:


In geometry, the angle between two straight lines is taken as the SMALLER angle, from which it automatically follows that it cannot be obtuse. In the figure, the angle indicated by the red arc is not considered to be the angle between intersecting lines. And its “green” neighbor or oppositely oriented crimson corner.

If the lines are perpendicular, then any of the 4 angles can be taken as the angle between them.

How are the angles different? Orientation. First, the direction of "scrolling" the corner is fundamentally important. Secondly, a negatively oriented angle is written with a minus sign, for example, if .

Why did I say this? It seems that you can get by with the usual concept of an angle. The fact is that in the formulas by which we will find the angles, a negative result can easily be obtained, and this should not take you by surprise. An angle with a minus sign is no worse, and has a very specific geometric meaning. In the drawing for a negative angle, it is imperative to indicate its orientation (clockwise) with an arrow.

How to find the angle between two lines? There are two working formulas:

Example 10

Find the angle between lines

Decision and Method one

Consider two straight lines given by equations in general form:

If straight not perpendicular, then oriented the angle between them can be calculated using the formula:

Let's pay close attention to the denominator - this is exactly scalar product direction vectors of straight lines:

If , then the denominator of the formula vanishes, and the vectors will be orthogonal and the lines will be perpendicular. That is why a reservation was made about the non-perpendicularity of the lines in the formulation.

Based on the foregoing, the solution is conveniently formalized in two steps:

1) Calculate the scalar product of directing vectors of straight lines:
so the lines are not perpendicular.

2) We find the angle between the lines by the formula:

Using the inverse function, it is easy to find the angle itself. In this case, we use the oddness of the arc tangent (see Fig. Graphs and properties of elementary functions):

Answer:

In the answer, we indicate the exact value, as well as the approximate value (preferably both in degrees and in radians), calculated using a calculator.

Well, minus, so minus, it's okay. Here is a geometric illustration:

It is not surprising that the angle turned out to be of a negative orientation, because in the condition of the problem the first number is a straight line and the “twisting” of the angle began precisely from it.

If you really want to get a positive angle, you need to swap the straight lines, that is, take the coefficients from the second equation , and take the coefficients from the first equation . In short, you need to start with a direct .

With the help of this online calculator you can find the point of intersection of lines on the plane. A detailed solution with explanations is given. To find the coordinates of the point of intersection of the lines, specify the type of the equation of the lines ("canonical", "parametric" or "general"), enter the coefficients of the equations of the lines into the cells and click the "Solve" button. See the theoretical part and numerical examples below.

×

A warning

Clear all cells?

Close Clear

Data entry instruction. Numbers are entered as whole numbers (examples: 487, 5, -7623 etc.), decimal numbers (eg 67., 102.54 etc.) or fractions. The fraction must be typed in the form a/b, where a and b (b>0) are integer or decimal numbers. Examples 45/5, 6.6/76.4, -7/6.7, etc.

Point of intersection of lines in the plane - theory, examples and solutions

1. Point of intersection of straight lines given in general form.

Oxy L 1 and L 2:

Let's build an augmented matrix:

If a B" 2=0 and WITH" 2 =0, then the system of linear equations has many solutions. Hence the direct L 1 and L 2 match. If a B" 2=0 and WITH" 2 ≠0, then the system is inconsistent and, therefore, the lines are parallel and do not have a common point. If B" 2 ≠0, then the system of linear equations has a unique solution. From the second equation we find y: y=WITH" 2 /B" 2 and substituting the resulting value into the first equation, we find x: x=−With 1 −B 1 y. Get the point of intersection of the lines L 1 and L 2: M(x, y).

2. Point of intersection of lines given in canonical form.

Let a Cartesian rectangular coordinate system be given Oxy and let lines be given in this coordinate system L 1 and L 2:

Let's open the brackets and make the transformations:

By a similar method, we obtain the general equation of the straight line (7):

From equations (12) it follows:

How to find the intersection point of lines given in the canonical form is described above.

4. Intersection point of lines defined in different views.

Let a Cartesian rectangular coordinate system be given Oxy and let lines be given in this coordinate system L 1 and L 2:

Let's find t:

We solve the system of linear equations with respect to x, y. To do this, we use the Gauss method. We get:

Example 2. Find the point of intersection of lines L 1 and L 2:

(21)

To find the point of intersection of lines L 1 and L 2 it is necessary to solve the system of linear equations (20) and (21). We represent the equations in matrix form.

Let two lines be given and it is required to find their point of intersection. Since this point belongs to each of the two given lines, its coordinates must satisfy both the equation of the first line and the equation of the second line.

Thus, in order to find the coordinates of the point of intersection of two lines, one should solve the system of equations

Example 1. Find the point of intersection of lines and

Decision. We will find the coordinates of the desired intersection point by solving the system of equations

The intersection point M has coordinates

Let us show how to construct a straight line from its equation. To draw a line, it is enough to know two of its points. To plot each of these points, we give an arbitrary value to one of its coordinates, and then from the equation we find the corresponding value of the other coordinate.

If in the general equation of a straight line, both coefficients at the current coordinates are not equal to zero, then to construct this straight line, it is best to find the points of its intersection with the coordinate axes.

Example 2. Construct a straight line.

Decision. Find the point of intersection of this line with the x-axis. To do this, we solve together their equations:

and we get . Thus, the point M (3; 0) of the intersection of this straight line with the abscissa axis was found (Fig. 40).

Solving then jointly the equation of the given line and the equation of the y-axis

we find the point of intersection of the line with the y-axis. Finally, we construct a line from its two points M and

When solving some geometric problems using the coordinate method, it is necessary to find the coordinates of the point of intersection of lines. Most often, one has to look for the coordinates of the point of intersection of two lines on the plane, but sometimes it becomes necessary to determine the coordinates of the point of intersection of two lines in space. In this article, we will deal with finding the coordinates of the point at which two lines intersect.

Page navigation.

The point of intersection of two lines is a definition.

Let's first define the point of intersection of two lines.

In the section on the relative position of lines on the plane, it is shown that two lines on the plane can either coincide (and they have infinitely many common points), or be parallel (and two lines have no common points), or intersect, having one common point. There are more options for the mutual arrangement of two lines in space - they can coincide (have infinitely many common points), can be parallel (that is, lie in the same plane and do not intersect), can be intersecting (not lying in the same plane), and can also have one common point, that is, intersect. So, two lines both in the plane and in space are called intersecting if they have one common point.

From the definition of intersecting lines it follows determination of the point of intersection of lines: The point where two lines intersect is called the point of intersection of these lines. In other words, the only common point of two intersecting lines is the point of intersection of these lines.

For clarity, we present a graphical illustration of the point of intersection of two lines in the plane and in space.

Top of page

Finding the coordinates of the point of intersection of two lines on the plane.

Before finding the coordinates of the point of intersection of two lines in the plane according to their known equations, we consider an auxiliary problem.

Oxy a and b. We will assume that the direct a corresponds to the general equation of the straight line, and the straight line b- type. Let be some point of the plane, and it is required to find out whether the point is M 0 the point of intersection of the given lines.

Let's solve the problem.

If a M0 a and b, then by definition it also belongs to the line a and direct b, that is, its coordinates must simultaneously satisfy both the equation and the equation . Therefore, we need to substitute the coordinates of the point M 0 into the equations of given lines and see if two true equalities are obtained. If the point coordinates M 0 satisfy both equations and , then is the point of intersection of the lines a and b, otherwise M 0 .

Is the point M 0 with coordinates (2, -3) point of intersection of lines 5x-2y-16=0 and 2x-5y-19=0?

If a M 0 is the point of intersection of the given lines, then its coordinates satisfy the equations of the lines. Let's check this by substituting the coordinates of the point M 0 into the given equations:

We got two true equalities, therefore, M 0 (2, -3)- point of intersection of lines 5x-2y-16=0 and 2x-5y-19=0.

For clarity, we present a drawing that shows straight lines and shows the coordinates of the point of their intersection.

yes, dot M 0 (2, -3) is the point of intersection of the lines 5x-2y-16=0 and 2x-5y-19=0.

Do lines intersect? 5x+3y-1=0 and 7x-2y+11=0 at the point M 0 (2, -3)?

Substitute the coordinates of the point M 0 into the equations of lines, by this action we will check whether the point belongs to M 0 both lines at the same time:

Since the second equation, when substituting the coordinates of the point into it M 0 did not turn into a true equality, then the point M 0 does not belong to the line 7x-2y+11=0. From this fact, we can conclude that the point M 0 is not a point of intersection of the given lines.

It is also clearly seen in the drawing that the point M 0 is not a point of intersection of lines 5x+3y-1=0 and 7x-2y+11=0. Obviously, the given lines intersect at a point with coordinates (-1, 2) .

M 0 (2, -3) is not a point of intersection of lines 5x+3y-1=0 and 7x-2y+11=0.

Now we can proceed to the problem of finding the coordinates of the point of intersection of two lines according to the given equations of lines on the plane.

Let a rectangular Cartesian coordinate system be fixed on the plane Oxy and given two intersecting lines a and b equations and respectively. Let us denote the point of intersection of the given lines as M 0 and solve the following problem: find the coordinates of the point of intersection of two lines a and b according to the known equations of these lines and .

Dot M0 belongs to each of the intersecting lines a and b a-priory. Then the coordinates of the point of intersection of the lines a and b satisfy both the equation and the equation . Therefore, the coordinates of the point of intersection of two lines a and b are a solution to a system of equations (see the article solving systems of linear algebraic equations).

Thus, in order to find the coordinates of the point of intersection of two lines defined on the plane by general equations, it is necessary to solve a system composed of equations of given lines.

Let's consider an example solution.

Find the point of intersection of two lines defined in a rectangular coordinate system in the plane by the equations x-9y+14=0 and 5x-2y-16=0.

We are given two general equations of lines, we will compose a system from them: . The solutions of the resulting system of equations are easily found if its first equation is solved with respect to the variable x and substitute this expression into the second equation:

The found solution of the system of equations gives us the desired coordinates of the point of intersection of two lines.

M 0 (4, 2)- point of intersection of lines x-9y+14=0 and 5x-2y-16=0.

So, finding the coordinates of the point of intersection of two lines, defined by general equations on the plane, is reduced to solving a system of two linear equations with two unknown variables. But what if the straight lines on the plane are given not by general equations, but by equations of a different type (see the types of the equation of a straight line on the plane)? In these cases, you can first bring the equations of lines to a general form, and only after that find the coordinates of the intersection point.

Before finding the coordinates of the point of intersection of the given lines, we bring their equations to a general form. The transition from the parametric equations of a straight line to the general equation of this straight line is as follows:

Now we will carry out the necessary actions with the canonical equation of the line:

Thus, the desired coordinates of the point of intersection of the lines are the solution to the system of equations of the form . We use Cramer's method to solve it:

M 0 (-5, 1)

There is another way to find the coordinates of the point of intersection of two lines in the plane. It is convenient to use it when one of the straight lines is given by parametric equations of the form , and the other is given by a straight line equation of a different type. In this case, into another equation instead of variables x and y you can substitute the expressions and , from where you can get the value that corresponds to the point of intersection of the given lines. In this case, the point of intersection of the lines has coordinates .

Let's find the coordinates of the point of intersection of the lines from the previous example in this way.

Determine the coordinates of the point of intersection of the lines and .

Substitute in the equation of the direct expression:

Solving the resulting equation, we get . This value corresponds to the common point of the lines and . We calculate the coordinates of the intersection point by substituting the straight line into the parametric equations:
.

M 0 (-5, 1).

To complete the picture, one more point should be discussed.

Before finding the coordinates of the point of intersection of two lines in the plane, it is useful to make sure that the given lines really intersect. If it turns out that the original lines coincide or are parallel, then there can be no question of finding the coordinates of the intersection point of such lines.

You can, of course, do without such a check, and immediately draw up a system of equations of the form and solve it. If the system of equations has a unique solution, then it gives the coordinates of the point at which the original lines intersect. If the system of equations has no solutions, then we can conclude that the original lines are parallel (since there is no such pair of real numbers x and y, which would simultaneously satisfy both equations of given lines). From the presence of an infinite set of solutions to the system of equations, it follows that the original lines have infinitely many points in common, that is, they coincide.

Let's look at examples that fit these situations.

Find out if the lines and intersect, and if they intersect, then find the coordinates of the intersection point.

The given equations of lines correspond to the equations and . Let's solve the system composed of these equations.

Obviously, the equations of the system are linearly expressed through each other (the second equation of the system is obtained from the first by multiplying both of its parts by 4 ), therefore, the system of equations has an infinite number of solutions. Thus, the equations and define the same line, and we cannot talk about finding the coordinates of the point of intersection of these lines.

equations and are defined in a rectangular coordinate system Oxy the same straight line, so we cannot talk about finding the coordinates of the intersection point.

Find the coordinates of the point of intersection of the lines and, if possible.

The condition of the problem admits that the lines may not intersect. Let's compose a system of these equations. We apply the Gauss method to solve it, since it allows us to establish the compatibility or inconsistency of the system of equations, and in the case of its compatibility, find a solution:

The last equation of the system after the direct course of the Gauss method turned into an incorrect equality, therefore, the system of equations has no solutions. From this we can conclude that the original lines are parallel, and we cannot talk about finding the coordinates of the point of intersection of these lines.

The second solution.

Let's find out if the given lines intersect.

A normal vector is a line, and a vector is a normal vector of a line. Let's check the fulfillment of the condition of collinarity of the vectors and : the equality is true, since, therefore, the normal vectors of the given lines are collinear. Then, these lines are parallel or coincide. Thus, we cannot find the coordinates of the point of intersection of the original lines.

it is impossible to find the coordinates of the point of intersection of the given lines, since these lines are parallel.

Find the coordinates of the point of intersection of the lines 2x-1=0 and if they intersect.

Let's compose a system of equations that are general equations of given lines: . The determinant of the main matrix of this system of equations is different from zero, therefore the system of equations has a unique solution, which indicates the intersection of the given lines.

To find the coordinates of the point of intersection of the lines, we need to solve the system:

The resulting solution gives us the coordinates of the point of intersection of the lines, that is, - the point of intersection of the lines 2x-1=0 and .

Top of page

Finding the coordinates of the point of intersection of two lines in space.

The coordinates of the point of intersection of two lines in three-dimensional space are found similarly.

Let the intersecting lines a and b given in a rectangular coordinate system Oxyz equations of two intersecting planes, that is, a straight line a is determined by the system of the form , and the line b- . Let be M 0- point of intersection of lines a and b. Then the point M 0 by definition belongs to the line a and direct b, therefore, its coordinates satisfy the equations of both lines. Thus, the coordinates of the point of intersection of the lines a and b represent a solution to a system of linear equations of the form . Here we will need information from the section on solving systems of linear equations in which the number of equations does not coincide with the number of unknown variables.

Let's consider examples.

Find the coordinates of the point of intersection of two lines given in space by the equations and .

Let's compose a system of equations from the equations of given lines: . The solution of this system will give us the desired coordinates of the point of intersection of lines in space. Let us find the solution of the written system of equations.

The main matrix of the system has the form , and the extended one - .

Determine the rank of the matrix BUT and matrix rank T. We use the method of bordering minors, while we will not describe in detail the calculation of determinants (if necessary, refer to the article calculating the determinant of a matrix):

Thus, the rank of the main matrix is ​​equal to the rank of the extended matrix and is equal to three.

Therefore, the system of equations has a unique solution.

We take the determinant as the basis minor, so the last equation should be excluded from the system of equations, since it does not participate in the formation of the basis minor. So,

The solution of the resulting system is easily found:

Thus, the point of intersection of lines and has coordinates (1, -3, 0) .

(1, -3, 0) .

It should be noted that the system of equations has a unique solution if and only if the lines a and b intersect. If direct a and b parallel or intersecting, then the last system of equations has no solutions, since in this case the lines have no common points. If straight a and b coincide, then they have an infinite set of common points, therefore, the indicated system of equations has an infinite set of solutions. However, in these cases we cannot talk about finding the coordinates of the point of intersection of the lines, since the lines are not intersecting.

Thus, if we do not know in advance, the given lines intersect a and b or not, it is reasonable to compose a system of equations of the form and solve it using the Gauss method. If we get a unique solution, then it will correspond to the coordinates of the point of intersection of the lines a and b. If the system turns out to be inconsistent, then the direct a and b do not intersect. If the system has an infinite number of solutions, then the direct a and b match.

You can do without using the Gauss method. Alternatively, you can calculate the ranks of the main and extended matrices of this system, and based on the data obtained and the Kronecker-Capelli theorem, make a conclusion either about the existence of a single solution, or about the existence of many solutions, or about the absence of solutions. It's a matter of taste.

If the lines and intersect, then determine the coordinates of the point of intersection.

Let's compose a system of given equations: . We solve it by the Gauss method in matrix form:

It became clear that the system of equations has no solutions, therefore, the given lines do not intersect, and there can be no question of finding the coordinates of the point of intersection of these lines.

we cannot find the coordinates of the point of intersection of the given lines, since these lines do not intersect.

When intersecting lines are given by canonical equations of a line in space or parametric equations of a line in space, then you should first obtain their equations in the form of two intersecting planes, and only after that find the coordinates of the intersection point.

Two intersecting lines are given in a rectangular coordinate system Oxyz equations and . Find the coordinates of the point of intersection of these lines.

Let us set the initial straight lines by the equations of two intersecting planes:

To find the coordinates of the point of intersection of the lines, it remains to solve the system of equations. The rank of the main matrix of this system is equal to the rank of the extended matrix and is equal to three (we recommend checking this fact). As the basis minor, we take , therefore, the last equation can be excluded from the system. Having solved the resulting system by any method (for example, the Cramer method), we obtain the solution . Thus, the point of intersection of lines and has coordinates (-2, 3, -5) .

Lesson from the series "Geometric Algorithms"

Hello dear reader!

We continue to get acquainted with geometric algorithms. In the last lesson, we found the equation of a straight line in the coordinates of two points. We have an equation of the form:

Today we will write a function that, using the equations of two straight lines, will find the coordinates of their intersection point (if any). To check the equality of real numbers, we will use the special function RealEq().

Points on the plane are described by a pair of real numbers. When using the real type, it is better to arrange the comparison operations with special functions.

The reason is known: there is no order relation on the Real type in the Pascal programming system, so it is better not to use records of the form a = b, where a and b are real numbers.
Today we will introduce the RealEq() function to implement the “=” (strictly equal) operation:

Function RealEq(Const a, b:Real):Boolean; (strictly equal) begin RealEq:=Abs(a-b)<=_Eps End; {RealEq}

Task. Equations of two straight lines are given: and . Find their point of intersection.

Decision. The obvious solution is to solve the system of equations of lines: Let's rewrite this system a little differently:
(1)

We introduce the notation: , , . Here D is the determinant of the system, and are the determinants obtained by replacing the column of coefficients for the corresponding unknown with a column of free terms. If , then system (1) is definite, that is, it has a unique solution. This solution can be found by the following formulas: , , which are called Cramer's formulas. Let me remind you how the second order determinant is calculated. The determinant distinguishes between two diagonals: the main and secondary. The main diagonal consists of elements taken in the direction from the upper left corner of the determinant to the lower right corner. Side diagonal - from the upper right to the lower left. The second order determinant is equal to the product of the elements of the main diagonal minus the product of the elements of the secondary diagonal.

The code uses the RealEq() function to check for equality. Calculations over real numbers are made with accuracy up to _Eps=1e-7.

Program geom2; Const _Eps: Real=1e-7;(calculation accuracy) var a1,b1,c1,a2,b2,c2,x,y,d,dx,dy:Real; Function RealEq(Const a, b:Real):Boolean; (strictly equal) begin RealEq:=Abs(a-b)<=_Eps End; {RealEq} Function LineToPoint(a1,b1,c1,a2,b2,c2: real; var x,y:real):Boolean; {Определение координат точки пересечения двух линий. Значение функции равно true, если точка пересечения есть, и false, если прямые параллельны. } var d:real; begin d:=a1*b2-b1*a2; if Not(RealEq(d,0)) then begin LineToPoint:=True; dx:=-c1*b2+b1*c2; dy:=-a1*c2+c1*a2; x:=dx/d; y:=dy/d; end else LineToPoint:=False End;{LineToPoint} begin {main} writeln("Введите коэффициенты уравнений: a1,b1,c1,a2,b2,c2 "); readln(a1,b1,c1,a2,b2,c2); if LineToPoint(a1,b1,c1,a2,b2,c2,x,y) then writeln(x:5:1,y:5:1) else writeln("Прямые параллельны."); end.

We have compiled a program with which you can, knowing the equations of the lines, find the coordinates of their intersection point.