Check that the lines lie in the same plane. The condition for two straight lines to belong to the same plane. Distance from point to line

This article is about parallel lines and parallel lines. First, the definition of parallel lines on a plane and in space is given, notations are introduced, examples and graphic illustrations of parallel lines are given. Next, the signs and conditions for parallelism of lines are discussed. In conclusion, solutions to typical problems of proving the parallelism of lines are shown, which are given by certain equations of a line in a rectangular coordinate system on a 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-dimensional space are called parallel, if they lie in the same plane and do not have common points.

Please note that the clause “if they lie in the same plane” in the definition of parallel lines in space is very important. Let us clarify this point: two lines in three-dimensional space that do not have common points and do not lie in the same plane are not parallel, but intersecting.

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 rails on level ground can also be considered as parallel lines.

To denote parallel lines, use the symbol “”. That is, if lines a and b are parallel, then we can briefly write a b.

Please note: 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 voice a statement that plays an important role in the study of parallel lines on a plane: through a point not lying on a given line, there passes the only straight line parallel to the given one. This statement is accepted as a fact (it cannot be proven 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 valid: through any point in space that does not lie on a given line, there passes a single straight line parallel to the given one. This theorem is easily proven using the above axiom of parallel lines (you can find its proof in the geometry textbook for grades 10-11, which is listed at the end of the article in the list of references).

For the case in space, the theorem is valid: through any point in space that does not lie on a given line, there passes a single straight line parallel to the given one. This theorem can be easily proven using the above parallel line axiom.

Parallelism of lines - signs and conditions of parallelism.

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

There are also necessary and sufficient conditions for the parallelism of lines on a 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. What is a “necessary condition for parallel lines”? From the name “necessary” it is clear that the fulfillment of this condition is necessary for parallel lines. In other words, if the necessary condition for the lines to be parallel is not met, then the lines are not parallel. Thus, necessary and sufficient condition for parallel lines 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 parallelism of lines, and on the other hand, this is a property that parallel lines have.

Before formulating a necessary and sufficient condition for the parallelism of lines, it is advisable to recall several auxiliary definitions.

Secant line is a line that intersects each of two given non-coinciding lines.

When two straight lines intersect with a transversal, eight undeveloped ones are formed. The so-called lying crosswise, corresponding And one-sided angles. Let's show them in the drawing.

Theorem.

If two straight lines in a plane are intersected by a transversal, then for them to be parallel it is necessary and sufficient that the intersecting angles be equal, or the corresponding angles are equal, or the sum of one-sided angles is equal to 180 degrees.

Let us show a graphic illustration of this necessary and sufficient condition for the parallelism of lines on a plane.

You can find proofs of these conditions for the parallelism of 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 straight lines and the secant lie in the same plane.

Here are a few more theorems that are often used to prove 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 criterion 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 criterion is discussed in geometry lessons in the 10th grade.

Let us illustrate the stated theorems.

Let us present another theorem that allows us to prove the parallelism of lines on a 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, criteria and necessary and sufficient conditions formulated above are excellent for proving the parallelism of lines using the methods of geometry. That is, to prove the parallelism of two given lines, you need to show that they are parallel to a third line, or show the equality of crosswise lying angles, etc. Many similar problems are solved in geometry lessons in high school. However, it should be noted that in many cases it is convenient to use the coordinate method to prove the parallelism of lines on a plane or in three-dimensional space. Let us formulate the necessary and sufficient conditions for the parallelism of lines that are specified in a rectangular coordinate system.

Parallelism of lines in a rectangular coordinate system.

In this paragraph of the article we will formulate necessary and sufficient conditions for parallel lines in a rectangular coordinate system, depending on the type of equations defining these lines, and we will also provide detailed solutions to characteristic problems.

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

Theorem.

For two non-coinciding 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 on a plane is reduced 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 direction vectors of lines a and b, and And are normal vectors of lines a and b, respectively, then the necessary and sufficient condition for the parallelism of lines a and b will be written as , or , or , where t is some real number. In turn, the coordinates of the guides and (or) normal vectors of lines a and b are found using the known equations of lines.

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

If line a corresponds to the equation of a line with an angular coefficient of the form , and line b - , then the normal vectors of these lines have coordinates and , and the condition for parallelism of these lines takes the form . Consequently, if lines on a plane in a rectangular coordinate system are parallel and can be specified by equations of lines with angular coefficients, then the angular coefficients of the lines will be equal. And vice versa: if non-coinciding lines on a plane in a rectangular coordinate system can be specified by equations of a line with equal angular coefficients, then such lines are parallel.

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

Let's look at solutions to several examples.

Example.

Are the lines parallel? And ?

Solution.

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

Answer:

No, the lines are not parallel.

Example.

Are straight lines and parallel?

Solution.

Let us reduce the canonical equation of a straight line to the equation of a straight line with an angular coefficient: . Obviously, the equations of the lines and are not the same (in this case, the given lines would be the same) and the angular coefficients of the lines are equal, therefore, the original lines are parallel.

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

For the lines L 1: and L 2: to belong to the same plane  so that the vectors M 1 M 2 =(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, according to the condition of coplanarity of three vectors, the mixed product M 1 M 2 ·s 1 ·s 2 =Δ==0 (8)

Because the condition for 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 positions of lines:

Direction vector of 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 therefore 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 , This means that the lines are either parallel or coincident.

To check whether the straight 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,

therefore the lines are parallel.

Distance from a point to a line.

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

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

Let's build a parallelogram using vectors q And M 0 M 1 . Then the distance from point M 1 to straight 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=

The relative position 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:

a straight line and a plane intersect at one point;

the straight line and the plane are parallel;

the straight line lies in the plane.

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

α: Ах+Бу+Сz+D=0

The equations of the straight line 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. The intersection of a line and a plane.

If a line and a plane intersect, then the direction vector of the line q is 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, through their coordinates,

Am+Bn+Cp≠0 (10)

Let's determine the coordinates of point M - points of intersection of straight line L and plane α.

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

Let's 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 point M:

t M = -→ (11)

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

Angle φ between straight line L :

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

: Ах+Ву+Сz+D=0 with normal vector n=(A;B;C) ranges from 0˚ (in the case of a parallel line and plane) to 90˚ (in the case of a perpendicular line and plane). (The angle between the vector q and its projection onto the plane α).

– angle between vectors q And n.

Because the angle  between the straight 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 straight line L)

Chapter IV. Straight lines and planes in space. Polyhedra

§ 46. Mutual arrangement of lines in space

In space, two different lines may or may not lie in the same plane. Let's look at relevant examples.

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

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

Two different 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 .

Thus, 1 1 || 1 2 if, firstly, there is a plane R such that
1 1 R And 1 2 R and secondly, or 1 1 1 2 = or 1 1 = 1 2 .

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

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

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

Let 1 1 || 1 2 and 1 2 || 1 3. It is necessary to prove that 1 1 || 1 3

If straight 1 1 , 1 2 , 1 3 lie in the same plane, then this statement is proven in planimetry. We will assume that straight 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 straight line and point M R 3, which intersects the plane R 2 along some straight line l. Let's prove that l coincides with 1 3. We will prove it “by contradiction”.

Let's assume that the straight line 1 does not coincide with a straight line 1 3. Then 1 intersects a line 1 2 at some point A. It follows that the plane R 3 passes through point A R 1 and straight 1 1 R 1 and therefore coincides with the plane R 1 . This conclusion contradicts the fact that point M R 3 does not belong to the plane R 1 .
Therefore, our assumption is incorrect, and therefore 1 = 1 3 .

Thus, it has been proven that straight lines 1 1 and 1 3 lie in the same plane R 3. Let us prove that the straight lines 1 1 and 1 3 do not intersect.

Indeed, if 1 1 and 1 3 intersected, 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.

Task. Prove that angles with codirectional sides have equal dimensions.

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

On the rays AM and A 1 M 1 we will lay out segments AB and A 1 B 1 equal in length. Then

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

like opposite sides of a parallelogram.

Similarly, on the rays AN and A 1 N 1 we will plot 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 |.
Hence, /\ ABC /\ A 1 B 1 C 1 and .

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For two lines in space, four cases are possible:

The straight lines coincide;

The lines are parallel (but do not coincide);

Lines intersect;

Straight lines cross, i.e. have no common points and are not parallel.

Let's consider two ways to describe straight lines: canonical equations and general equations. Let the lines L 1 and L 2 be given by 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 = (z - z 2)/n 2 (6.9)

For each line from its canonical equations we immediately determine the 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 = m 1 /m 2 = n 1 /n 2. (6.10)

If the lines coincide, then the vector M 1 M 2 is collinear to the direction vectors:

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

This double equality also means that the point M 2 belongs to the line L 1. Consequently, the condition for the lines to coincide is to satisfy 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. 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 out of four cases, since for Δ ≠ 0 the lines do not belong to the same plane and therefore intersect.

Let's put all the conditions together:

The relative position of the 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. In the case of parallel or crossing, there are no direct solutions. The last two cases can be separated by finding the direction vectors of the lines. To do this, it is enough to calculate two vector artwork 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 direction vector s 1 of straight line L 1 is found using the canonical equations of this straight line: s 1 = (1; 3; -2). The direction vector s 2 of the straight line L 2 is calculated using the vector product of the normal vectors of the planes whose intersection it is:

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

Angle between straight lines. The angle between two straight lines can be found using direction vectors straight The acute angle between straight lines is equal to the angle between their direction vectors (Fig. 6.5) or is additional to it if the angle between the direction vectors is obtuse. Thus, if for 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 through the 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 scalar product in coordinates, we get