The relative position of the lines. Relative position of lines Intersection of a line and a plane

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. In the formulation of the necessary and sufficient condition for the parallelism of lines, 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 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 geometry methods. 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 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 the 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.

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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)