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. So

─ the condition of parallelism of the plane.

Condition of coincidence of planes:

,

since in this case, multiplying the second equation by l = , we obtain 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 scalar product is equal to 0, i.e. = 0, or

A 1 A 2 + B 1 B 2 + C 1 C 2 \u003d 0.

This is a necessary and sufficient condition for the planes to be perpendicular.

Angle between two planes.

Angle between two planes

A 1 x + B 1 y + C 1 z + D 1 \u003d 0,

A 2 x + B 2 y + C 2 z + D 2 \u003d 0

is the angle between their normal vectors and , so

cosj = =
.

straight line in space.

Vector-parametric equation of a straight line.

Definition. Direction vector straight Any vector lying on a line or parallel to it is called.

Compose the equation of a 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).

Set aside from the point M 0 the vector . Let M(x; y; z) be an arbitrary point of the given line, and ─ its radius-vector of the point М 0 . Then , , That's why . 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) \u003d (x 0; y 0; z 0) + (a 1; a 2; a 3) t. From here we get parametric equations of the straight line

x \u003d x 0 + a 1 t,

y = y 0 + a 2 t, (4)

Canonical equations of a straight line.

From equations (4) we express t:

t = , t = , t = ,

where we get canonical equations of the 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 the directing vector of the straight line, you can take the vector = (x 2 - x 1; y 2 ​​- y 1; z 2 - z 1). Since the 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)

Angle between two lines.

Consider two straight lines with direction vectors = (a 1; a 2; a 3) and .

The angle between lines is equal to the angle between their direction vectors, so

cosj = =
(7)

The condition of perpendicularity of lines:

a 1 in 1 + a 2 in 2 + a 3 in 3 = 0.

Condition of parallel lines:

l,

. (8)

Mutual arrangement of lines in space.

Let two lines be given
and
.

Obviously, the lines lie in the same plane if and only if the vectors , and coplanar, i.e.

= 0 (9)

If in (9) the first two rows are proportional, then the lines are parallel. If all three lines are proportional, then the lines coincide. If condition (9) is satisfied and the first two rows are not proportional, then the lines intersect.

If
¹ 0, then the lines are skew.

Problems on a straight line and a plane in space.

A straight line is the intersection of two planes.

Let two planes be given

A 1 x + B 1 y + C 1 z + D 1 \u003d 0,

A 2 x + B 2 y + C 2 z + D 2 \u003d 0

If the planes are not parallel, then the condition is violated

.

Let, for example, 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 line, we fix some value

z = z 0 and solving the system

,

we get the values ​​\u200b\u200bx \u003d x 0, y \u003d y 0. So, the desired point is M (x 0; y 0; z 0).

Required equation

.

Mutual arrangement of a straight line and a plane.

Let the straight line x = x 0 + a 1 t, y = y 0 + a 2 t, z = z 0 + a 3 t be given

and plane

A 1 x + B 1 y + C 1 z + D 1 \u003d 0.

To find common points of a 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 A 1 a 1 + B 1 a 2 + C 1 a 3 ¹ 0, then the system has a unique solution

t = t 0 = -
.

In this case, the line and the plane intersect at a single point M 1 (x 1; y 1; z 1), where

x 1 \u003d x 0 + a 1 t 0, y 1 \u003d y 0 + a 2 t 0, z 1 \u003d z 0 + a 3 t 0.

If A 1 a 1 + B 1 a 2 + C 1 a 3 \u003d 0, A 1 x 0 + B 1 y 0 + C 1 z 0 + D 1 ¹ 0, then the line and the plane do not have common points, i.e. . are parallel.

If A 1 a 1 + B 1 a 2 + C 1 a 3 \u003d 0, A 1 x 0 + B 1 y 0 + C 1 z 0 + D 1 \u003d 0, then the line belongs to the plane.

The angle between a line and a plane.

Mutual arrangement of planes in space

With the mutual arrangement of two planes in space, one of two mutually exclusive cases is possible.

1. Two planes have a common point. Then, by the axiom of the intersection of two planes, they have a common line. Axiom R5 says: if two planes have a common point, then the intersection of these planes is their common line. From this axiom it follows that for planes Such planes are called intersecting.

The two planes do not have a common point.

3. Two planes coincide

3. Vectors on the plane and in space

A vector is a directed line segment. Its length is considered the length of the segment. If two points M1 (x1, y1, z1) and M2 (x2, y2, z2) are given, then the vector

If two vectors are given and then

1. Lengths of vectors

2. Sum of vectors:

3. The sum of two vectors a and b is the diagonal of the parallelogram built on these vectors, coming from a common point of their application (parallelogram rule); or a vector connecting the beginning of the first vector with the end of the last - according to the triangle rule. The sum of three vectors a, b, c is the diagonal of the parallelepiped built on these vectors (the rule of the parallelepiped).

Consider:

• 1. The origin of coordinates is at point A;
• 2. The side of the cube is a single segment.
• 3. We direct the OX axis along the AB edge, OY along the AD edge, and the OZ axis along the AA1 edge.

For the bottom plane of the cube

Def. Two planes in space are said to be parallel if they do not intersect, otherwise they intersect.

Theorem1: If two intersecting lines of one plane are respectively parallel to two lines of another plane, then these planes are parallel.

Proof:

Let and be given planes, a1 and a2 - lines in the plane intersecting at the point A, b1 and b2 - lines parallel to them, respectively, in

planes. Let us assume that the planes and are not parallel, i.e. intersect along some line. By the theorem, the lines a1 and a2, being parallel to the lines b1 and b2, are parallel to the plane, and therefore they are not

intersect the line c lying in this plane. Thus, two straight lines (a1 and a2) pass through the point A in the plane, parallel to the line c. But this is impossible according to the parallel axiom. We have arrived at a contradiction of the CTD.

Perpendicular planes: Two intersecting planes are called perpendicular if a third plane, perpendicular to the line of intersection of these planes, intersects them along perpendicular lines.

Theorem2: If a plane passes through a line perpendicular to another plane, then these planes are perpendicular.

Proof:

Let be a plane, β be a line perpendicular to it, be a plane passing through the line β, c be a line along which the 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 line in with the plane the line a,

perpendicular to the straight line. Let's draw through the lines a and into the plane. It is perpendicular to the line c, because line c is perpendicular to lines a and b. Since the 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) plane equation

in vector form:

where is a unit vector, is the distance of P. from the origin. Equation (2) can be obtained from equation (1) by multiplying by the normalizing factor

(signs and opposite).

43. Equations of a straight line in space: General equations, canonical and parametric equations.

Canonical equations:

We 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 and are collinear. This means that the coordinates of these vectors are proportional:

These equations are called canonical. Note that one or two of the direction vector coordinates may be zero. But we perceive 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 the planes

Parametric:

Postponing from the point vectors for different values, collinear to the directing vector, we will get different points of our straight line at the end of the postponed vectors. From equality it follows:

The variable is called a parameter. Since for any point of the line there is a corresponding parameter value and since different points of the line correspond to different values ​​of the parameter, there is a one-to-one correspondence between the parameter values ​​and points of the line. When the parameter runs through all real numbers from to, the corresponding point runs through the entire 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- this is a non-empty set L, on which operations are introduced

addition, that is, each pair of elements of the set is associated with an element of the same set, denoted by

multiplication by a scalar (that is, an element of the field P), that is, any element and any element will be matched with the element from, denoted.

In this case, the following conditions are imposed on the operation:

For any ( commutativity of addition);

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 (the existence of an opposite element).

(associativity of multiplication by a scalar);

(multiplication by a neutral (by multiplication) field elementPsaves the vector).

(distributivity of multiplication by a vector with respect to addition of scalars);

(distributivity of multiplication by a scalar with respect to vector addition).

Set elements L called vectors, and field elements P-scalars. Properties 1-4 coincide with the axioms of the abelian group.

The simplest properties

The vector space is an abelian group by addition.

The neutral element is the only one that results from group properties.

for anyone .

For any opposite element is the only one that follows from the group properties.

for anyone .

for any and

for anyone .

The elements of a 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 dimension of a linear space, connection between them.

End sum of the view

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 non-trivial 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 some finite subset of it is linearly dependent, and linearly independent if any of its finite subsets is linearly independent.

The number of elements (cardinality) of the maximum linearly independent subset of the 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 the basis (the Hamel basis or the linear basis). The elements of a basis 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. ​​Vector coordinates in a given basis. Linear operations with vectors in coordinate form

item 4. Linear operations with vectors incoordinateformrecords.

Let be a basis space and be its two arbitrary vectors. Let and be the representation of these vectors in coordinate form. Let, further, be an arbitrary real number. In these notations, the following theorem holds.

Theorem. (On linear operations with vectors in coordinate form.)

Let Ln be an arbitrary n-dimensional space, B = (e1,….,en) be a fixed basis in it. Then any vector x belonging to Ln has a one-to-one correspondence with a column of its coordinates in this basis.

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For two planes, the following variants of mutual arrangement are possible: they are parallel or intersect in a straight line.

It is known from stereometry that two planes are parallel if two intersecting lines of one plane are respectively parallel to two intersecting lines of the other plane. This condition is called a sign of parallel planes.

If two planes are parallel, then they intersect some third plane along parallel lines. Based on this, parallel planes R and Q their traces are parallel straight lines (Fig. 50).

When two planes R and Q parallel to the axis X, their horizontal and frontal traces with an arbitrary mutual arrangement of the planes will be parallel to the x axis, i.e., mutually parallel. Consequently, under such conditions, the parallelism of traces is a sufficient sign characterizing the parallelism of the planes themselves. For the parallelism of such planes, you need to make sure that their profile traces are also parallel. P w and Q w. planes R and Q in figure 51 are parallel, and in figure 52 they are not parallel, despite the fact that P v || Q v , and P h y || Q h .

In the case when the planes are parallel, the horizontals of one plane are parallel to the horizontals of the other. In this case, the fronts of one plane must be parallel to the fronts of the other, since these planes have parallel traces of the same name.

In order to construct two planes intersecting with each other, it is necessary to find the line along which the two planes intersect. To construct this line, it is enough to find two points belonging to it.

Sometimes, when the plane is given by traces, it is easy to find these points using a diagram and without additional constructions. Here, the direction of the defined straight line is known, and its construction is based on the use of one point on the diagram.

End of work -

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Descriptive geometry. Lecture notes lecture. About projections

Lecture information about projections the concept of projections reading a drawing .. central projection .. an idea of ​​\u200b\u200bthe central projection can be obtained by studying the image that the human eye gives ..

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All topics in this section:

The concept of projections
Descriptive geometry is a science that is the theoretical foundation of drawing. In this science, methods of depicting various bodies and their elements on a plane are studied.

Parallel projection
Parallel projection is a type of projection that uses parallel projecting rays. When constructing parallel projections, you need to set on

Projections of a point onto two projection planes
Consider the projections of points onto two planes, for which we take two perpendicular planes (Fig. 4), which we will call the horizontal frontal and planes. Flat data intersection line

Missing projection axis
To explain how to obtain on the model projections of a point onto perpendicular projection planes (Fig. 4), it is necessary to take a piece of thick paper in the form of an elongated rectangle. It needs to be bent between

Projections of a point onto three projection planes
Consider the profile plane of projections. Projections on two perpendicular planes usually determine the position of the figure and make it possible to find out its real dimensions and shape. But there are times when

Point coordinates
The position of a point in space can be determined using three numbers, called its coordinates. Each coordinate corresponds to the distance of a point from some plane pr

Projection of a straight line
Two points are needed to define a line. A point is defined by two projections on the horizontal and frontal planes, i.e. a straight line is defined using the projections of its two points on the horizontal

Straight traces
The trace of a straight line is the point of its intersection with some plane or surface (Fig. 20). The horizontal trace of a line is a point H

Various positions of the line
A line is called a line in general position if it is neither parallel nor perpendicular to any projection plane. The projections of a line in general position are also neither parallel nor perpendicular.

Mutual arrangement of two straight lines
Three cases of arrangement of lines in space are possible: 1) the lines intersect, that is, they have a common point; 2) the lines are parallel, that is, they do not have a common point, but lie in the same plane

Perpendicular lines
Consider the theorem: if one side of a right angle is parallel to the projection plane (or lies in it), then the right angle is projected onto this plane without distortion. We present a proof for

Determination of the position of the plane
For an arbitrarily located projection plane, its points fill all three projection planes. Therefore, it makes no sense to talk about the projection of the entire plane, you need to consider only projections

Plane traces
The trace of the plane P is the line of its intersection with a given plane or surface (Fig. 36). The line of intersection of the plane P with the horizontal plane is called

Plane contours and fronts
Among the lines that lie in a certain plane, two classes of lines can be distinguished, which play an important role in solving various problems. These are straight lines, which are called horizontals.

Construction of plane traces
Consider the construction of traces of the plane P, which is given by a pair of intersecting lines I and II (Fig. 45). If a line is in the plane P, then its traces lie on the traces of the same name

Various positions of the plane
A plane in general position is a plane that is neither parallel nor perpendicular to any of the projection planes. The traces of such a plane are also neither parallel nor perpendicular.

Straight line parallel to the plane
There can be several positions of a straight line relative to a certain plane. 1. The line lies in some plane. 2. A line is parallel to some plane. 3. Direct transfer

A straight line that intersects a plane
To find the point of intersection of a line and a plane, it is necessary to construct lines of intersection of two planes. Consider the line I and the plane P (Fig. 54).

Prism and pyramid
Consider a straight prism that stands on a horizontal plane (Fig. 56). Her side grains

Cylinder and cone
A cylinder is a figure whose surface is obtained by rotating the straight line m around the i-axis, located in the same plane with this straight line. In the case when the line m

Ball, torus and ring
When some axis of rotation I is the diameter of a circle, then a spherical surface is obtained (Fig. 66).

Lines used in drawing
Three main types of lines (solid, dashed and dash-dotted) of various thicknesses are used in drawing (Fig. 76).

Location of views (projections)
In drawing, six types are used, which are shown in Figure 85. The figure shows the projections of the letter "L".

Deviation from the above rules for the arrangement of views
In some cases, deviations from the rules for constructing projections are allowed. Among these cases, the following can be distinguished: partial views and views located without a projection connection with other views.

Number of projections that define this body
The position of bodies in space, shape and size are usually determined by a small number of appropriately selected points. If, when depicting a projection of a body, pay attention

Rotation of a point about an axis perpendicular to the plane of projections
Figure 91 shows the axis of rotation I, which is perpendicular to the horizontal plane, and a point A arbitrarily located in space. When rotating about the axis I, this point describes

Determination of the natural length of a segment by rotation
A segment parallel to any projection plane is projected onto it without distortion. If you rotate the segment so that it becomes parallel to one of the projection planes, then you can define

The construction of projections of the section figure can be done in two ways
1. You can find the meeting points of the edges of the polyhedron with the cutting plane, and then connect the projections of the found points. As a result of this, projections of the desired polygon will be obtained. In this case,

Pyramid
Figure 98 shows the intersection of the pyramid surface with the frontal projection plane P. Figure 98b shows the frontal projection a of the meeting point of the rib KS with the plane

oblique sections
Oblique sections are understood as a range of tasks for constructing natural types of sections of the body under consideration by the projected plane. To perform an oblique section, it is necessary to dismember

Hyperbola as a section of the surface of a cone by the frontal plane
Let it be required to construct a section of the surface of a cone standing on a horizontal plane by the plane P, which is parallel to the plane V. Figure 103 shows the frontal

Section of the surface of the cylinder
There are the following cases of a section of the surface of a right circular cylinder by a plane: 1) a circle, if the secant plane P is perpendicular to the axis of the cylinder, and it is parallel to the bases

Section of the surface of the cone
In the general case, a circular conical surface includes two completely identical cavities that have a common vertex (Fig. 107c). The generators of one cavity are a continuation of the

Section of the surface of the ball
Any section of the surface of the ball by a plane is a circle, which is projected without distortion only if the cutting plane is parallel to the plane of projections. In the general case, we

oblique sections
Let it be required to construct a natural view of the section by the frontally projecting plane of the body. Figure 110a considers a body bounded by three cylindrical surfaces (1, 3 and 6), the surface

Pyramid
To find traces of a straight line on the surface of some geometric body, you need to draw through a straight auxiliary plane, then find the section of the surface of the body by this plane. The desired will be

Cylindrical helix
Formation of a helix. Consider Figure 113a where the point M moves uniformly along a certain circle, which is a section of a circular cylinder by the plane P. Here this plane

Two bodies of revolution
The method of drawing auxiliary planes is used when constructing a line of intersection of the surfaces of two bodies of revolution. The essence of this method is as follows. Carry out an auxiliary plane

Sections
There are some definitions and rules that apply to sections. A section is a flat figure that was obtained as a result of the intersection of a given body with some

cuts
Definitions and rules that apply to cuts. A cut is such a conditional image of an object when its part, located between the eye of the observer and the cutting plane

Partial cut or tear
The cut is called complete if the depicted object is cut in its entirety, the remaining cuts are called partial, or cuts. In Figure 120, in the left view and on the plan, full sections are made. hairstyle