No. _____ Date 02.10.14
Subject Geometry
Class 10
Lesson topic:The relative position of the two planes. Parallelism of planes
Lesson objectives: to acquaint with the concept of parallelism of planes, to study the sign of parallelism of a plane and the properties of parallel planes
Lesson type: learning new material
DURING THE CLASSES
1. Organizational moment.
Greeting students, checking the readiness of the class for the lesson, organizing the attention of students, disclosing the general goals of the lesson and its plan.
2. Formation of new concepts and methods of action.
The two planes are calledparallel, if they have no common points, i.e. if α = α (fig. 20).
Theorem 1.
Through a point not lying in a plane, you can draw only one plane parallel to this plane.
Proof.
Let the plane be givenbut
and point A, BUT
but
... In plane but
take two intersecting linesand and
b
: but
,
b
, but
= B
(Fig. 21.) Then, by Theorem 1 (§2, Section 2.1.) through the pointBUT
directbut
1
and b
1
such that but
1
||
but
and b
1
||
b
Hence, according to the axiomCIIIthere is only one plane passing through intersecting straight linesbut
1
and b 1
... It now remains to show that α, i.e. α =
.
Let it not be so, i.e. planes intersect in a straight line with.Then at least one of the straight linesbut orb not parallel to a straight linewith. For definiteness, we assume thatbut with andbut with = S.
Consequently,a 1 c and, as in the proof of Theorem 2 in §2, we havea 1 c = WITH, those.but 1 a = C.
This contradicts the fact that a, ||but ... Therefore, α = α ... The theorem is proved.
Theorem 2.
If you intersect two parallel planes with a third plane, then their straight lines of intersection will be parallel, i.e. α,
a = α,
b =
=>
but||
b(rice.22
).
So, two planes in space can be mutually located in two versions:
planes intersect in a straight line;
the planes are parallel.
Parallelism of planes
Theorem 3.
If two intersecting straight lines of one plane are respectively parallel to two straight lines of another plane, then these planes are parallel.
Theorem 4. Segments of parallel straight lines bounded by parallel planes are equal,between themselves.
3. Application. Formation of skills and abilities.
Objectives: To ensure that students apply the knowledge and methods of action that they need for SR, to create conditions for students to identify individual ways of using what they have learned. Page 24 No. 87,88,89,90 (1)
4. Homework information stage.
Objectives: To provide students with an understanding of the purpose, content and ways of doing homework. Page 22 p3 # 90 (2)
5. Summing up the results of the lesson.
Objective: To provide a qualitative assessment of the work of the class and individual students.
6. Stage of reflection.
Def. Two planes in space are called parallel if they do not intersect, otherwise they intersect.
Theorem 1: If two intersecting straight lines of one plane are respectively parallel to two straight lines of another plane, then these planes are parallel.
Proof:
Let and be given planes, a1 and a2 are straight lines in the plane intersecting at point A, b1 and b2 are, respectively, straight lines parallel to them in
plane. Let us assume that the planes and are not parallel, i.e. intersect along some straight line with. By the theorem, the lines a1 and a2, as parallel to the lines b1 and b2, are parallel to the plane, and therefore they are not
intersect the straight line c lying in this plane. Thus, in the plane through point A there are two straight lines (a1 and a2) parallel to the straight line c. But this is impossible according to the parallel axiom. We have come to the contradiction of the CHTD.
Perpendicular planes: Two intersecting planes are called perpendicular if the third plane, perpendicular to the line of intersection of these planes, intersects them along perpendicular lines.
Theorem 2: If a plane passes through a straight line perpendicular to another plane, then these planes are perpendicular.
Proof:
Let be a plane, b is a straight line perpendicular to it, a plane passing through a straight line b, c is a straight line along which planes and intersect. Let us prove that the planes and are perpendicular. Let us draw in the plane through the point of intersection of the straight line b with the plane a straight line a,
perpendicular to the straight line with. Let us draw through the straight lines a and into the plane. It is perpendicular to the straight line c, because straight line with perpendicular to straight lines a and b. Since the straight lines a and b are perpendicular, the planes and are perpendicular. h.t.d.
42. Normal equation of the plane and its properties
Normal (normalized) equation of the plane
in vector form:
where is the unit vector, is the distance of the P. from the origin of coordinates. Equation (2) can be obtained from equation (1) by multiplying by the normalizing factor
(signs and are opposite).
43. Equations of a straight line in space: General equations, canonical and parametric equations.
Canonical equations:
Let us derive the equation of a straight line passing through a given point and parallel to a given direction vector. Note that a point lies on this line if and only if the vectors are collinear. This means that the coordinates of these vectors are proportional:
These equations are called canonical. Note that one or two coordinates of the direction vector may be equal to zero. But we take it as a proportion: we understand it as equality.
General equations:
(A1x + B1y + C1z + D1 = 0
(A2x + B2y + C2z + D2 = 0
Where the coefficients A1-C1 are not proportional to A2-C2, which is equivalent to setting it as a line of intersection of planes
Parametric:
Deferring from the point vectors for different values, collinear to the direction vector, we will receive at the end of the delayed vectors different points of our line. From equality it follows:
A variable is called a parameter. Since for any point of the straight line there is a corresponding value of the parameter and since different points of the straight line correspond to different values of the parameter, then there is a one-to-one correspondence between the values of the parameter and the points of the straight line. When the parameter runs through all real numbers from before, the corresponding point runs through the whole line.
44. The concept of linear space. Axioms. Examples of linear spaces
An example of a linear space is the set of all geometric vectors.
Linear, or vectorspace above the field P is a non-empty set L on which the operations are entered
addition, that is, each pair of elements of a set is associated with an element of the same set, denoted by and
multiplication by a scalar (that is, the field element P), that is, any element and any element is assigned an element from, denoted by.
In this case, the following conditions are imposed on the operations:
For any ( addition commutability);
For any ( addition associativity);
there is an element such that for any ( existence of a neutral element with respect to addition), in particular L not empty;
for any there is an element such that 
(existence of the opposite element).
(scalar multiplication associativity);
(multiplication by a neutral (by multiplication) field elementPpreserves vector).
(distributivity of multiplication by a vector with respect to scalar addition);
(scalar multiplication with respect to vector addition).
Elements of the set L are called vectors, and the field elements P-scalars... Properties 1-4 coincide with the axioms of an abelian group.
Simplest properties
Vector space is an abelian addition group.
The neutral element is the only one that follows from the group properties.
for anyone .
For any, the opposite element is the only one that follows from the group properties.
for anyone .
for any and.
for anyone .
The elements of linear space are called vectors. A space is called real if the operation of multiplying vectors by a number in it is defined only for real numbers, and complex if this operation is defined only for complex numbers.
45. Basis and dimensions of linear space, connection between them.
The final sum of the form
is called a linear combination of elements with coefficients.
A linear combination is called nontrivial if at least one of its coefficients is nonzero.
Elements are called linearly dependent if there is a nontrivial linear combination of them equal to θ. Otherwise, these elements are called linearly independent.
An infinite subset of vectors from L is called linearly dependent if its some finite subset is linearly dependent, and linearly independent if any of its finite subset is linearly independent.
The number of elements (cardinality) of the maximal linearly independent subset of a space does not depend on the choice of this subset and is called the rank, or dimension, of the space, and this subset itself is called a basis (a Hamel basis or a linear basis). Basis elements are also called basis vectors. Basis properties:
Any n linearly independent elements of an n-dimensional space form a basis of this space.
Any vector can be represented (uniquely) as a finite linear combination of basic elements:
46. Coordinates of the vector in the given basis. Linear operations on vectors in coordinate form
item 4. Linear operations with vectors incoordinateformrecords.
Let be the basis of the space and be its two arbitrary vectors. Blanks - records these vectors in coordinate form. Let, further, be an arbitrary real number. In this notation, the following theorem holds.
Theorem. (About linear operations on vectors in coordinate form.)
Let Ln be an arbitrary n-dimensional space, B = (e1,…., En) a fixed basis in it. Then any vector x belonging to Ln is in one-to-one correspondence with a column of its coordinates in this basis.
Let two planes be given
The first plane has a normal vector (A 1; B 1; C 1), the second plane (A 2; B 2; C 2).
If the planes are parallel, then the vectors and are collinear, i.e. = l for some number l. therefore
─ condition of parallelism of the plane.
Plane coincidence condition:
,
since in this case multiplying the second equation by l =, we get the first equation.
If the parallelism condition is not met, then the planes intersect. In particular, if the planes are perpendicular, then the vectors, are also perpendicular. Therefore, their dot product is 0, i.e. = 0, or
A 1 A 2 + B 1 B 2 + C 1 C 2 = 0.
This is a necessary and sufficient condition for the planes to be perpendicular.
The angle between two planes.
Angle between two planes
A 1 x + B 1 y + C 1 z + D 1 = 0,
A 2 x + B 2 y + C 2 z + D 2 = 0
this is the angle between their normal vectors and, therefore
cosj = 
= 
.
Straight in space.
Vector-parametric equation of a straight line.
Definition. The direction vector of the straight line any vector lying on a straight line or parallel to it is called.
Let us compose the equation of the straight line passing through the point M 0 (x 0; y 0; z 0) and having a direction vector = (a 1; a 2; a 3).
Let us set aside from the point М 0 the vector 
... Let M (x; y; z) be an arbitrary point of a given straight line, and 
─ its radius is the vector of the point M 0. Then 
, 
, so 
... This equation is called vector-parametric equation of a straight line.
Parametric equations of a straight line.
In the vector-parametric equation of the straight line will pass to the coordinate relations (x; y; z) = (x 0; y 0; z 0) + (a 1; a 2; a 3) t. From this we get parametric equations of the straight line
x = x 0 + a 1 t,
y = y 0 + a 2 t, (4)
Canonical equations of the straight line.
From equations (4) we express t:
t =, t = 
, t = 
,
whence we get canonical equations of the straight line
= = (5)
Equation of a straight line passing through two given points.
Let two points M 1 (x 1; y 1; z 1) and M 2 (x 2; y 2; z 2) be given. As a directing vector of a straight line, we can take a vector = 
(x 2 - x 1; y 2 - y 1; z 2 - z 1). Since the straight line passes through the point M 1 (x 1; y 1; z 1), then its canonical equations in accordance with (5) will be written in the form
(6)
The angle between two straight lines.
Consider two straight lines with direction vectors = (a 1; a 2; a 3) and 
.
The angle between straight lines is equal to the angle between their direction vectors, so
cosj = 
= 
(7)
Perpendicularity condition for straight lines:
a 1 in 1 + a 2 in 2 + a 3 in 3 = 0.
The condition for parallelism of lines:
l,
. (8)
Mutual arrangement of straight lines in space.
Given two lines 
and 
.
Obviously, the straight lines lie in the same plane if and only if the vectors, and 
coplanar, i.e.
= 0 (9)
If in (9) the first two lines are proportional, then the straight lines are parallel. If all three lines are proportional, then the straight lines match. If condition (9) is satisfied and the first two lines are not proportional, then the lines intersect.
If ¹ 0, then the lines are intersecting.
Problems on a line and a plane in space.
A straight line as the intersection of two planes.
Let two planes be given
A 1 x + B 1 y + C 1 z + D 1 = 0,
A 2 x + B 2 y + C 2 z + D 2 = 0
If the planes are not parallel, then the condition
.
For example, let ¹.
Let's find the equation of the straight line along which the planes intersect.
As the direction vector of the desired straight line, we can take the vector
= × = 
= 
.
To find a point belonging to the desired straight line, we fix some value
z = z 0 and solving the system
,
we get the values x = x 0, y = y 0. So, the required point M (x 0; y 0; z 0).
The required equation
.
Mutual arrangement of a straight line and a plane.
Let a straight line x = x 0 + a 1 t, y = y 0 + a 2 t, z = z 0 + a 3 t
and plane
A 1 x + B 1 y + C 1 z + D 1 = 0.
To find the common points of a straight line and a plane, it is necessary to solve the system of their equations
A 1 (x 0 + a 1 t) + B 1 (y 0 + a 2 t) + C 1 (z 0 + a 3 t) + D 1 = 0,
(A 1 a 1 + B 1 a 2 + C 1 a 3) t + (A 1 x 0 + B 1 y 0 + C 1 z 0 + D 1) = 0.
If А 1 а 1 + В 1 а 2 + С 1 а 3 ¹ 0, then the system has a unique solution
t = t 0 = - 
.
In this case, the straight line and the plane intersect at a single point M 1 (x 1; y 1; z 1), where
x 1 = x 0 + a 1 t 0, y 1 = y 0 + a 2 t 0, z 1 = z 0 + a 3 t 0.
If A 1 a 1 + B 1 a 2 + C 1 a 3 = 0, A 1 x 0 + B 1 y 0 + C 1 z 0 + D 1 ¹ 0, then the line and the plane have no common points, i.e. ... are parallel.
If A 1 a 1 + B 1 a 2 + C 1 a 3 = 0, A 1 x 0 + B 1 y 0 + C 1 z 0 + D 1 = 0, then the line belongs to the plane.
The angle between a straight line and a plane.
The angle between two planes. Conditions parallelism and perpendicularity two planes:
let two planes Q 1 and Q 2 be given:
A 1 x + B 1 y + C 1 z + D 1 = 0
A 2 x + B 2 y + C 2 z + D 2 = 0
The angle between the planes is understood as one of the dihedral angles formed by these planes.
If the planes are perpendicular, then so are their normals, i.e. ... But then, i.e.
A 1 A 2 + B 1 B 2 + C 1 C 2 = 0. The resulting equality is condition of perpendicularity of two planes.
If the planes are parallel, then their normals will also be parallel. But then, as you know, the coordinates of the vectors are proportional: 
... That's what it is condition of parallelism of two planes.
Mutual arrangement of straight lines.
Angle between straight lines. Conditions for parallelism and perpendicularity of straight lines.
The half angle between these lines is understood as the angle between the direction vectors S 1 and S 2.
To find the acute angle between the straight lines L 1 and L 2, the numerator of the right side of the formula should be taken modulo.
If straight lines L 1 and L 2 perpendicular, then in this and only in this case we have cos = 0. therefore, the numerator of the fraction = 0, i.e. 
=0.
If straight lines L 1 and L 2 parallel, then their direction vectors S 1 and S 2 are parallel. therefore, the coordinates of these vectors are proportional to:.
The condition under which two straight lines lie in the same plane:
=0.
When this condition is met, the lines either lie in the same plane, that is, they either intersect.
Mutual arrangement of a straight line and a plane.
The angle between a straight line and a plane. Conditions for parallelism and perpendicularity of a straight line and a plane.
Let the plane be given by the equation Ax + By + Cz + D = 0, and the line L by the equations 
... The angle between a straight line and a plane is any of two adjacent angles formed by a straight line and its projection onto a plane. Let's designate through the angle between the plane and the straight line.
.
If the line L is parallel to the plane Q, then the vectors n and S are perpendicular, and therefore, i.e.
0 is parallelism condition straight and plane.
If the line L is perpendicular to the plane Q, then the vectors n and S are parallel. Therefore, the equalities
Are perpendicularity conditions straight and plane.
The intersection of a straight line with a plane. The condition of belonging to a straight plane:
Consider a straight line and the plane Ax + By + Cz + D = 0.
Simultaneous execution of equalities:
Ax 0 + By 0 + Cz 0 + D = 0 are the condition of belonging to a straight plane.
Ellipse.
The locus of points, the sum of the distances from which to two fixed points of the plane (usually called focal points) is constant, is called ellipse.
If the coordinate axes are located so that Ox passes through the foci F 1 (C, 0) and F 2 (-C, 0), and O (0,0) coincides with the middle of the segment F 1 F 2, then by F 1 M + F 2 M we get:
canonical ur-e of ellipse 
,
b 2 = - (c 2 -a 2).
a and b are the semi-axes of the ellipse., a-large, b-smaller.
Eccentricity... , (if a> b)
(if a The eccentricity characterizes the bulge of the ellipse. The ellipse has an eccentricity: 0. The case = 0 arises only when c = 0, and this is the case of a circle - this is an ellipse with zero eccentricity. Headmistresses (D) The locus of points, the ratio of the distances from which to the point of the ellipse to the distance from this point of the ellipse to the focus is constant and equal to the value, is called directors. . Note: the circle has no directrix. Hyperbola. The locus of points, the modulus of the difference between the distances from which to two fixed points of the plane is constant, is called hyperbole. Canonical hyperbola equation: The hyperbola is a line of the second order. The hyperbola has 2 asymptotes: and The hyperbola is called equilateral if its semiaxes are equal. (a = b). Canonical equation: Eccentricity- the ratio of the distance between the foci to the value of the real axis of the hyperbola: Since c> a for a hyperbola, the eccentricity of the hyperbola is> 1. The eccentricity characterizes the shape of the hyperbola:. The eccentricity of an equilateral hyperbola is equal. Headmistresses- straight lines. Focal Radii: There are hyperboles that have common asymptotes. Such hyperboles are called conjugate. Parabola. Parabola- the set of all points of the plane, each of which is equally distant from a given point, called the focus, and a given straight line, called the directrix. Distance from focus to directrix - parabola parameter(p> 0) .- semi-focal diameter. The parabola is a line of the second order. M (x, y) is an arbitrary point of the parabola. Let's connect the point M with F, draw the segment MN perpendicular to the directrix. According to the definition of the parabola, MF = MN. Using the formula for the distance between 2 points, we find: Canonical parabola equation: Ellipsoid. We investigate the surface given by the equation: Consider sections of a surface with planes parallel to the xOy plane. The equations of such planes are: z = h, where h is any number. The line obtained in the section is determined by two ur-nes: Examining the surface: A) if then the Line of intersection of the surface with the planes z = h does not exist. B) if, the line of intersection degenerates into two points (0,0, s), and (0,0, -s). The plane z = c, z = - c touches the given surface. C) if, then the equations can be rewritten as: The considered sections make it possible to depict the surface as a closed oval surface. The surface is called ellipsoids. If any semi-axes are equal, the triaxial ellipsoid turns into an ellipsoid of revolution, and if a = b = c, then into a sphere. Hyperboloid and cone. By virtue of the axiom: two planes having a common point have a common straight line - only two cases of planes are possible: 1) the planes have a common straight line, that is, they intersect; 2) the planes have no common point, such planes are called parallel. The existence of parallel planes follows from the following construction. Take in the plane (Fig. 331) any two intersecting straight lines a and b. Through the point M, which does not belong to the plane X, draw lines a and b, respectively, parallel to the data. Let us show that the plane containing these lines is parallel to the plane. Indeed, if these planes intersected along some straight line c, then this straight line, belonging to the plane, would intersect at least one of the straight lines a, and such an intersection point would be the intersection point of one of these straight lines with the plane. Meanwhile, both straight lines are parallel to the plane by construction. Thus, the assumption of the intersection of the planes leads to a contradiction. Therefore, the planes are parallel. this implies Parallelism of planes. If two intersecting straight lines of one plane are respectively parallel to two intersecting straight lines of another plane, then the planes are parallel.
, where .
and 
.
=> 
= =>
=>
y 2 = 2px.
, as you can see, the line of intersection is an ellipse with semiaxes a1 =, b1 =. Moreover, the smaller h, the larger the semiaxis. When n = 0, they reach their maximum values. a1 = a, b1 = b. The equations will take the form: 
