This article is about parallel lines and about parallel lines. First, the definition of parallel lines in the plane and in space is given, notation is introduced, examples and graphic illustrations of parallel lines are given. Further, the signs and conditions of parallelism of straight lines are analyzed. In conclusion, the solutions of characteristic problems of proving the parallelism of lines are shown, which are given by some equations of the line in rectangular system coordinates on the plane and in three-dimensional space.

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Parallel lines - basic information.

Definition.

Two lines in a plane are called parallel if they do not have common points.

Definition.

Two lines in three dimensions are called parallel if they lie in the same plane and have no common points.

Note that the "if they lie in the same plane" clause in the definition of parallel lines in space is very important. Let's clarify this point: two straight lines in three-dimensional space that do not have common points and do not lie in the same plane are not parallel, but are skew.

Here are some examples of parallel lines. The opposite edges of the notebook sheet lie on parallel lines. The straight lines along which the plane of the wall of the house intersects the planes of the ceiling and floor are parallel. Railroad tracks on level ground can also be thought of as parallel lines.

The symbol "" is used to denote parallel lines. That is, if the lines a and b are parallel, then you can briefly write a b.

Note that if lines a and b are parallel, then we can say that line a is parallel to line b, and also that line b is parallel to line a.

Let us sound the statement that plays important role in the study of parallel lines in the plane: through a point not lying on a given line, there passes the only line parallel to the given one. This statement is accepted as a fact (it cannot be proved on the basis of the known axioms of planimetry), and it is called the axiom of parallel lines.

For the case in space, the theorem is true: through any point in space that does not lie on a given line, there passes a single line parallel to the given one. This theorem can be easily proved using the above axiom of parallel lines (you can find its proof in the geometry textbook 10-11 class, which is listed at the end of the article in the bibliography).

For the case in space, the theorem is true: through any point in space that does not lie on a given line, there passes a single line parallel to the given one. This theorem is easily proved using the axiom of parallel lines given above.

Parallelism of lines - signs and conditions of parallelism.

A sign of parallel lines is an sufficient condition parallelism of lines, that is, such a condition, the fulfillment of which guarantees the parallelism of lines. In other words, the fulfillment of this condition is sufficient to state the fact that the lines are parallel.

There are also necessary and sufficient conditions for parallel lines in the plane and in three-dimensional space.

Let us explain the meaning of the phrase "necessary and sufficient condition for parallel lines".

We have already dealt with the sufficient condition for parallel lines. And what is the "necessary condition for parallel lines"? By the name "necessary" it is clear that the fulfillment of this condition is necessary for the lines to be parallel. In other words, if the necessary condition for parallel lines is not satisfied, then the lines are not parallel. In this way, necessary and sufficient condition for lines to be parallel is a condition, the fulfillment of which is both necessary and sufficient for parallel lines. That is, on the one hand, this is a sign of parallel lines, and on the other hand, this is a property that parallel lines have.

Before stating the necessary and sufficient condition for lines to be parallel, it is useful to recall a few auxiliary definitions.

secant line is a line that intersects each of the two given non-coincident lines.

At the intersection of two lines of a secant, eight non-deployed ones are formed. The so-called lying crosswise, corresponding And one-sided corners. Let's show them on the drawing.

Theorem.

If two straight lines on a plane are crossed by a secant, then for their parallelism it is necessary and sufficient that the crosswise lying angles are equal, or the corresponding angles are equal, or the sum of one-sided angles is equal to 180 degrees.

Let us show a graphical illustration of this necessary and sufficient condition for parallel lines in the plane.

You can find proofs of these conditions for parallel lines in geometry textbooks for grades 7-9.

Note that these conditions can also be used in three-dimensional space - the main thing is that the two lines and the secant lie in the same plane.

Here are a few more theorems that are often used in proving the parallelism of lines.

Theorem.

If two lines in a plane are parallel to a third line, then they are parallel. The proof of this feature follows from the axiom of parallel lines.

There is a similar condition for parallel lines in three-dimensional space.

Theorem.

If two lines in space are parallel to a third line, then they are parallel. The proof of this feature is considered in the geometry lessons in grade 10.

Let us illustrate the voiced theorems.

Let us give one more theorem that allows us to prove the parallelism of lines in the plane.

Theorem.

If two lines in a plane are perpendicular to a third line, then they are parallel.

There is a similar theorem for lines in space.

Theorem.

If two lines in three-dimensional space are perpendicular to the same plane, then they are parallel.

Let us draw pictures corresponding to these theorems.

All the theorems formulated above, signs and necessary and sufficient conditions are perfectly suitable for proving the parallelism of straight lines by methods of geometry. That is, to prove the parallelism of two given lines, it is necessary to show that they are parallel to the third line, or to show the equality of cross-lying angles, etc. Many of these problems are solved in geometry lessons in high school. However, it should be noted that in many cases it is convenient to use the method of coordinates to prove the parallelism of lines in a plane or in three-dimensional space. Let us formulate the necessary and sufficient conditions for the parallelism of lines that are given in a rectangular coordinate system.

Parallelism of lines in a rectangular coordinate system.

In this section of the article, we will formulate necessary and sufficient conditions for parallel lines in a rectangular coordinate system, depending on the type of equations that determine these lines, and we will also give detailed solutions to typical problems.

Let's start with the condition of parallelism of two lines on the plane in the rectangular coordinate system Oxy . His proof is based on the definition of the direction vector of the line and the definition of the normal vector of the line on the plane.

Theorem.

For two non-coincident lines to be parallel in a plane, it is necessary and sufficient that the direction vectors of these lines are collinear, or the normal vectors of these lines are collinear, or the direction vector of one line is perpendicular to the normal vector of the second line.

Obviously, the condition of parallelism of two lines in the plane reduces to (direction vectors of lines or normal vectors of lines) or to (direction vector of one line and normal vector of the second line). Thus, if and are the direction vectors of the lines a and b, and And are the normal vectors of lines a and b, respectively, then the necessary and sufficient condition for parallel lines a and b can be written as , or , or , where t is some real number. In turn, the coordinates of the directing and (or) normal vectors of the straight lines a and b are found from the known equations of the straight lines.

In particular, if the line a in the rectangular coordinate system Oxy on the plane defines the general equation of the line of the form , and the straight line b - , then the normal vectors of these lines have coordinates and respectively, and the condition of parallelism of lines a and b will be written as .

If the straight line a corresponds to the equation of the straight line with the slope coefficient of the form . Therefore, if straight lines on a plane in a rectangular coordinate system are parallel and can be given by equations of straight lines with slope coefficients, then the slope coefficients of the lines will be equal. And vice versa: if non-coincident straight lines on a plane in a rectangular coordinate system can be given by the equations of a straight line with equal slope coefficients, then such straight lines are parallel.

If the line a and the line b in a rectangular coordinate system define the canonical equations of the line on the plane of the form And , or parametric equations of a straight line on a plane of the form And respectively, then the direction vectors of these lines have coordinates and , and the parallelism condition for lines a and b is written as .

Let's take a look at a few examples.

Example.

Are the lines parallel? And ?

Solution.

We rewrite the equation of a straight line in segments in the form of a general equation of a straight line: . Now we can see that is the normal vector of the straight line , and is the normal vector of the straight line. These vectors are not collinear since there is no such real number t , for which the equality ( ). Consequently, the necessary and sufficient condition for the parallelism of lines on the plane is not satisfied, therefore, the given lines are not parallel.

Answer:

No, the lines are not parallel.

Example.

Are lines and parallels?

Solution.

We bring the canonical equation of a straight line to the equation of a straight line with a slope: . Obviously, the equations of the lines and are not the same (in this case, the given lines would be the same) and the slopes of the lines are equal, therefore, the original lines are parallel.

For two straight lines in space, four cases are possible:

Straight lines match;

The lines are parallel (but not the same);

The lines intersect;

The lines intersect, i.e. do not have common points and are not parallel.

Consider two ways of describing lines: canonical equations and general equations. Let the lines L 1 and L 2 be given by the canonical equations:

L 1: (x - x 1) / l 1 = (y - y 1) / m 1 = (z - z 1) / n 1 , L 2: (x - x 2) / l 2 = (y - y 2) / m 2 \u003d (z - z 2) / n 2 (6.9)

For each straight line from its canonical equations, we immediately determine a point on it M 1 (x 1 ; y 1 ; z 1) ∈ L 1 , M 2 (x 2 ; y 2 ​​; z 2) ∈ L 2 and the coordinates of the direction vectors s 1 = (l 1 ; m 1 ; n 1 ) for L 1 , s 2 = (l 2 ; m 2 ; n 2 ) for L 2 .

If the lines coincide or are parallel, then their direction vectors s 1 and s 2 are collinear, which is equivalent to the equality of the ratios of the coordinates of these vectors:

l 1 / l 2 \u003d m 1 / m 2 \u003d n 1 / n 2. (6.10)

If the lines coincide, then the direction vectors are also collinear with the vector M 1 M 2 :

(x 2 - x 1) / l 1 \u003d (y 2 - y 1) / m 1 \u003d (z 2 - z 1) / n 1. (6.11)

This double equality also means that the point M 2 belongs to the line L 1 . Therefore, the condition for the lines to coincide is the fulfillment of equalities (6.10) and (6.11) simultaneously.

If the lines intersect or cross, then their direction vectors are non-collinear, i.e. condition (6.10) is violated. The intersecting lines lie in the same plane and, therefore, vectors s 1 , s 2 and M 1 M 2 are coplanarthird order determinant composed of their coordinates (see 3.2):

Condition (6.12) is satisfied in three cases out of four, since for Δ ≠ 0 the lines do not belong to the same plane and therefore intersect.

Let's bring all the conditions together:

Mutual arrangement lines is characterized by the number of solutions of the system (6.13). If the lines coincide, then the system has infinitely many solutions. If the lines intersect, then this system has a unique solution. There are no direct solutions in the case of parallel or crossing direct solutions. The last two cases can be separated by finding the direction vectors of the lines. To do this, it suffices to calculate two vector works n 1 × n 2 and n 3 × n 4 , where n i = (A i ; B i ; C i ), i = 1, 2, 3.4. If the resulting vectors are collinear, then the given lines are parallel. Otherwise they are interbreeding.

Example 6.4.

The directing vector s 1 of the straight line L 1 is found by the canonical equations of this straight line: s 1 = (1; 3; -2). The directing vector s 2 of the straight line L 2 is calculated using the vector product of the normal vectors of the planes, the intersection of which it is:

Since s 1 \u003d -s 2, then the lines are parallel or coincide. Let us find out which of these situations is realized for given lines. To do this, we substitute the coordinates of the point M 0 (1; 2; -1) ∈ L 1 into general equations direct L 2 . For the first of them, we obtain 1 = 0. Therefore, the point M 0 does not belong to the line L 2 and the lines under consideration are parallel.

Angle between lines. The angle between two lines can be found using direction vectors direct. Acute angle between straight lines equal to the angle between their direction vectors (Fig. 6.5) or is complementary to it if the angle between the direction vectors is obtuse. Thus, if for the lines L 1 and L 2 their direction vectors s x and s 2 are known, then the acute angle φ between these lines is determined by scalar product:

cosφ = |S 1 S 2 |/|S 1 ||S 2 |

For example, let s i = (l i ; m i ; n i ), i = 1, 2. Using formulas (2.9) and (2.14) to calculate vector length and the scalar product in coordinates, we get

Chapter IV. Lines and planes in space. Polyhedra

§ 46. Mutual arrangement of lines in space

In space, two distinct lines may or may not lie in the same plane. Let's consider the corresponding examples.

Let the points A, B, C do not lie on one straight line. Draw a plane through them R and choose some point S that does not belong to the plane R(Fig. 130).

Then the lines AB and BC lie in the same plane, namely, in the plane R, lines AS and CB do not lie in the same plane. Indeed, if they lay in the same plane, then the points A, B, C, S would also lie in this plane, which is impossible, since S does not lie in the plane passing through the points A, B, C.

Two distinct lines that lie in the same plane and do not intersect are called parallel. Coinciding lines are also called parallel. If straight 1 1 and 1 2 parallel, then write 1 1 || 1 2 .

In this way, 1 1 || 1 2 if, first, there is a plane R such that
1 1 R And 1 2 R and second, or 1 1 1 2 = or 1 1 = 1 2 .

Two lines that do not lie in the same plane are called intersecting lines. Obviously, skew lines do not intersect and are not parallel.

Let us prove one important property of parallel lines, which is called the transitivity of parallelism.

Theorem. If two lines are parallel to a third, then they are parallel to each other.

Let be 1 1 || 1 2 and 1 2 || 1 3 . We need to prove that 1 1 || 1 3

If straight 1 1 , 1 2 , 1 3 lie in the same plane, then this statement is proved in planimetry. We will assume that the lines 1 1 , 1 2 , 1 3 do not lie in the same plane.

Through straight lines 1 1 and 1 2 draw a plane R 1 , and through 1 2 and 1 3 - plane R 2 (Fig. 131).

Note that the straight line 1 3 contains at least one point M that does not belong to the plane
R 1 .

Draw a plane through the line and the point M R 3 that intersects with the plane R 2 along some line l. Let's prove that l coincides with 1 3 . We will prove the "method by contradiction".

Let's assume that the line 1 does not match the line 1 3 . Then 1 crosses the line 1 2 at some point A. This implies that the plane R 3 passes through point A R 1 and straight 1 1 R 1 and therefore coincides with the plane R one . This conclusion contradicts the fact that the point M R 3 does not belong to the plane R 1 .
Therefore, our assumption is wrong, and therefore 1 = 1 3 .

Thus, it is proved that the lines 1 1 and 1 3 lie in the same plane R 3 . Let us prove that the lines 1 1 and 1 3 do not intersect.

Indeed, if 1 1 and 1 3 intersect, for example, at point B, then the plane R 2 would pass through a straight line 1 2 and through point B 1 1 and, therefore, would coincide with R 1 , which is impossible.

A task. Prove that angles with codirectional sides are equal.

Let the angles MAN and M 1 A 1 N 1 have co-directed sides: the ray AM is co-directed to the ray A 1 M 1, and the ray AN is co-directed to the ray A 1 N 1 (Fig. 132).

On the rays AM and A 1 M 1 we set aside segments AB and A 1 B 1 equal in length. Then

|| and |BB 1 | = |AA 1 |

as opposite sides of a parallelogram.

Similarly, on the rays AN and A 1 N 1 we set aside segments AC and A 1 C 1 equal in length. Then

|| and |CC 1 | = |AA 1 |

From the transitivity of parallelism it follows that || . And since |BB 1 | = |CC 1 | , then BB 1 C 1 C is a parallelogram, and therefore |BC| = |B 1 C 1 |.
Consequently, /\ ABC /\ A 1 B 1 C 1 and .

The lines lie in the same plane. if they 1) intersect; 2) are parallel.

For the belonging of the lines L 1: and L 2: to the same plane  so that the vectors M 1 M 2 \u003d (x 2 -x 1; y 2 ​​-y 1; z 2 -z 1), q 1 =(l 1 ;m 1 ;n 1 ) and q 2 =(l 2 ;m 2 ;n 2 ) were coplanar. That is, by the condition of the complanarity of three vectors, the mixed product M 1 M 2 s 1 s 2 =Δ==0 (8)

Because the condition of parallelism of two lines has the form: , then for the intersection of lines L 1 and L 2 , so that they satisfy condition (8) and so that at least one of the proportions is violated.

Example. Explore the relative position of the lines:

Direction vector of the straight line L 1 – q 1 =(1;3;-2). Line L 2 is defined as the intersection of 2 planes α 1: x-y-z+1=0; α 2: x+y+2z-2=0. Because line L 2 lies in both planes, then it, and hence its direction vector, is perpendicular to the normals n 1 And n 2 . Therefore, the direction vector s 2 is the cross product of vectors n 1 And n 2 , i.e. q 2 =n 1 X n 2 ==-i-3j+2k.

That. s 1 =-s 2 , means the lines are either parallel or coincide.

To check whether the lines coincide, we substitute the coordinates of the point M 0 (1;2;-1)L 1 into the general equations L 2: 1-2+2+1=0 - incorrect equalities, i.e. point M 0 L 2,

so the lines are parallel.

The distance from a point to a line.

The distance from the point M 1 (x 1; y 1; z 1) to the line L given by the canonical equation L: can be calculated using the cross product.

It follows from the canonical equation of the straight line that the point M 0 (x 0; y 0; z 0) L, and the directing vector of the straight line q=(l;m;n)

Let's build a parallelogram on vectors q And M 0 M 1 . Then the distance from the point M 1 to the line L is equal to the height h of this parallelogram. Because S=| q x M 0 M 1 |=h| q|, then

h= (9)

The distance between two straight lines in space.

L 1: and L 2:

1) L 1 L 2 .

d=

2) L 1 and L 2 - crossing

d=

Mutual arrangement of a straight line and a plane in space.

For the location of a straight line and a plane in space, 3 cases are possible:

line and plane intersect at one point;

line and plane are parallel;

the line lies in a plane.

Let the straight line be given by its canonical equation, and the plane - by the general

α: Ax+By+Cz+D=0

The straight line equations give the point M 0 (x 0; y 0; z 0) L and the direction vector q=(l;m;n), and the plane equation is a normal vector n=(A;B;C).

1. Intersection of a line and a plane.

If a line and a plane intersect, then the direction vector of the line q not parallel to the plane α, and therefore not orthogonal to the normal vector of the plane n. Those. their dot product nq≠0 or, in terms of their coordinates,

Am+Bn+Cp≠0 (10)

Determine the coordinates of the point M - points of intersection of the line L and the plane α.

Let's move from the canonical equation of the straight line to the parametric one: , tR

We substitute these relations into the equation of the plane

A(x 0 +lt)+B(y 0 +mt)+C(z 0 +nt)+D=0

A,B,C,D,l,m,n,x 0 ,y 0 ,z 0 are known, let's find the parameter t:

t(Al+Bm+Cn)= -D-Ax 0 -By 0 -Cz 0

if Am+Bn+Cp≠0, then the equation has a unique solution that determines the coordinates of the point M:

t M = -→ (11)

The angle between a line and a plane. Conditions of parallelism and perpendicularity.

Angle φ between line L :

with guide vector q=(l;m;n) and plane

: Ax+By+Cz+D=0 with normal vector n=(A;B;C) ranges from 0˚ (in case of parallel line and plane) to 90˚ (in case of perpendicularity of line and plane). (Angle between vector q and its projection onto the plane α).

– angle between vectors q And n.

Because the angle  between the line L and the plane  is complementary to the angle , then sin φ=sin(-)=cos =- (the absolute value is considered because the angle φ is acute sin φ=sin(-) or sin φ =sin(+) depending on the direction of the straight line L)

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