1. General equation of the plane

Definition. A plane is a surface, all points of which satisfy the general equation: Ax + By + Cz + D = 0, where A, B, C are the coordinates of the vector

N = Ai + Bj + Ck is the normal vector to the plane. The following special cases are possible:

A = 0 - the plane is parallel to the Ox axis

B = 0 - the plane is parallel to the Oy axis C = 0 - the plane is parallel to the Oz axis

D = 0 - the plane passes through the origin

A = B = 0 - the plane is parallel to the xOy plane A = C = 0 - the plane is parallel to the xOz plane B = C = 0 - the plane is parallel to the yOz plane A = D = 0 - the plane passes through the Ox axis

B = D = 0 - the plane passes through the Oy axis C = D = 0 - the plane passes through the Oz axis

A = B = D = 0 - the plane coincides with the xOy plane A = C = D = 0 - the plane coincides with the xOz plane B = C = D = 0 - the plane coincides with the yOz plane

2. Equation of a surface in space

Definition. Any equation connecting the x, y, z coordinates of any point on a surface is an equation of this surface.

3. Equation of a plane passing through three points

In order for a single plane to be drawn through any three points in space, it is necessary that these points do not lie on one straight line.

Consider points M1 (x1, y1, z1), M2 (x2, y2, z2), M3 (x3, y3, z3) in the general Cartesian system

coordinates.
For an arbitrary point M (x, y, z)	lay in the same plane with dots
M 1, M 2, M 3, it is necessary that the vectors M 1 M 2, M 1 M 3, M 1 M be coplanar, i.e.
M1 M = (x - x1; y - y1; z - z1)
(M 1 M 2, M 1 M 3, M 1 M) = 0. Thus, M 1 M 2	= (x 2 - x 1; y 2		- y 1; z 2 - z 1)
M1 M 3	= (x 3 - x 1; y 3 - y 1; z 3 - z 1)
		x - x1	y - y1	z - z1
Equation of a plane passing through three points:		x 2 - x 1	y 2 - y 1	z 2 - z 1
		x 3 - x 1	y 3 - y 1	z 3 - z 1

4. Equation of a plane in two points and a vector collinear to the plane

Let the points M1 (x1, y1, z1), M2 (x2, y2, z2) and the vector a = (a 1, a 2, a 3) be given.

Let's compose the equation of the plane passing through the given points M1 and M2 and an arbitrary

point M (x, y, z) parallel to the vector a.
Vectors M1 M = (x - x1; y - y1; z - z1)	and the vector a = (a, a					must be
M 1M 2 = (x 2 - x 1; y 2 ​​- y 1; z 2 - z 1)
			x - x1				y - y1	z - z1
coplanar, i.e. (M 1 M, M 1 M 2, a) = 0 Plane equation:		x 2 - x 1					y 2 - y 1	z 2 - z 1

5. Equation of the plane by one point and two vectors collinear to the plane

Let there be given two vectors a = (a 1, a 2, a 3) and b = (b 1, b 2, b 3), collinear to the plane. Then for an arbitrary point M (x, y, z) belonging to the plane, the vectors a, b, MM 1 must be coplanar.

6. Equation of a plane by a point and a normal vector

Theorem. If a point M 0 (x 0, y 0, z 0) is given in the space, then the equation of the plane passing through the point M 0 perpendicular to the normal vector N (A, B, C) has the form: A (x - x 0) + B (y - y 0) + C (z - z 0) = 0.

7. Equation of the plane in segments

If in the general equation Ax + By + Cz + D = 0 divide both sides by (-D)

x -

y -

z - 1 = 0, replacing -

C, we get the equation of the plane

in segments:

one . The numbers a, b, c are the intersection points of the plane, respectively

with axes x, y, z.

8. Equation of the plane in vector form

r n = p, where r = xi + yj + zk is the radius vector of the current point M (x, y, z),

n = i cosα + j cos β + k cosγ is a unit vector with a perpendicular direction,

lowered to the plane from the origin. α, β and γ are the angles formed by this vector with the x, y, z axes. p is the length of this perpendicular. In coordinates, this equation has the form:

x cosα + y cos β + z cosγ - p = 0

9. Distance from point to plane

The distance from an arbitrary point M 0 (x 0, y 0, z 0) to the plane Ax + By + Cz + D = 0 is equal to:

d = Ax0 + By0 + Cz0 + D

A2 + B2 + C 2

Example. Find the equation of the plane passing through the points A (2, -1,4) and B (3,2, -1) perpendicular to the plane x + y + 2z - 3 = 0.

The desired equation of the plane is: Ax + By + Cz + D = 0, the normal vector to this plane is n 1 (A, B, C). Vector AB (1,3, -5) belongs to the plane. The plane given to us,

perpendicular to the sought-for has a normal vector n 2 (1,1,2). Because points A and B belong to both planes, and the planes are mutually perpendicular, then

n = AB × n

− 5

- j

− 5

11 i - 7 j - 2 k.

− 5

Thus, the normal vector n is 1 (11, -7, -2). Because point A belongs to the desired plane, then its coordinates must satisfy the equation of this plane, i.e.

11.2 + 7.1 - 2.4 + D = 0; D = - 21. In total, we get the equation of the plane: 11x - 7 y - 2z - 21 = 0

10. Equation of a line in space

Both on a plane and in space, any line can be defined as a set of points whose coordinates in a certain coordinate system selected in space satisfy the equation:

F (x, y, z) = 0. This equation is called the equation of a line in space.

In addition, a line in space can be defined differently. It can be viewed as the line of intersection of two surfaces, each of which is given by some equation.

Let F (x, y, z) = 0 and Ф (x, y, z) = 0 be the equations of surfaces intersecting along the line L.

F (x, y, z) = 0

Then a pair of equations Ф (x, y, z) = 0 will be called the equation of a line in space.

11. Equation of a straight line in space along a point and a direction vector r 0 = M 0 M.

Because the vectors М 0 М and S are collinear, then the relation М 0 М = ​​St is true, where t is some parameter. Total, you can write: r = r 0 + St.

Because this equation is satisfied by the coordinates of any point of the straight line, then the resulting equation is a parametric equation of the straight line.

x = x0 + mt

This vector equation can be represented in coordinate form: y = y 0 + nt

z = z0 + pt

Transforming this system and equating the values ​​of the parameter t, we obtain the canonical

equations of a straight line in space:	x - x0		y - y0		z - z0

Definition. The direction cosines of a straight line are the direction cosines of the vector S, which can be calculated by the formulas:

	cosα =			; cos β =		; cosγ =
					N 2 + p 2		m 2 + n 2 + p 2

From here we get: m: n: p = cosα: cos β: cosγ.

The numbers m, n, p are called the slopes of the line. Because S is a nonzero vector, then m, n and p cannot be zero at the same time, but one or two of these numbers can be zero. In this case, in the equation of the straight line, the corresponding numerators should be equated to zero.

12. Equation of a straight line in space passing through two points

If on a straight line in space we mark two arbitrary points M 1 (x 1, y 1, z 1) and

M 2 (x 2, y 2, z 2), then the coordinates of these points must satisfy the straight line equation obtained above:

x 2 - x 1		y 2 - y 1		z 2 - z 1

It can be set in different ways (one point and a vector, two points and a vector, three points, etc.). It is with this in mind that the equation of the plane can have various forms. Also, if certain conditions are met, the planes can be parallel, perpendicular, intersecting, etc. We will talk about this in this article. We will learn how to draw up the general equation of the plane and more.

Normal form of the equation

Let's say there is a space R 3 that has a rectangular coordinate system XYZ. Let us set a vector α, which will be released from the initial point O. Through the end of the vector α we draw a plane P, which will be perpendicular to it.

We denote on an arbitrary point Q = (x, y, z). Let us write the radius vector of the point Q with the letter p. In this case, the length of the vector α is equal to p = IαI and Ʋ = (cosα, cosβ, cosγ).

This is a unit vector, which is directed to the side, like the vector α. α, β and γ are the angles that are formed between the vector Ʋ and the positive directions of the space axes x, y, z, respectively. The projection of any point QϵП onto the vector Ʋ is a constant value, which is equal to p: (p, Ʋ) = p (p≥0).

The above equation makes sense when p = 0. The only thing is that the plane P in this case will intersect the point O (α = 0), which is the origin, and the unit vector Ʋ emitted from the point O will be perpendicular to P, despite its direction, which means that the vector Ʋ is determined with accurate to the sign. The previous equation is the equation of our plane P, expressed in vector form. But in coordinates, its appearance will be like this:

P here is greater than or equal to 0. We have found the equation of the plane in space in the normal form.

General equation

If the equation in coordinates is multiplied by any number that is not zero, we get an equation equivalent to the given one, which defines the same plane. It will look like this:

Here A, B, C are numbers that are nonzero at the same time. This equation is referred to as the general plane equation.

Equations of planes. Special cases

The general equation can be modified in the presence of additional conditions. Let's take a look at some of them.

Let us assume that the coefficient A is equal to 0. This means that this plane is parallel to the given axis Ox. In this case, the form of the equation will change: Vu + Cz + D = 0.

Similarly, the form of the equation will change under the following conditions:

• First, if B = 0, then the equation will change to Ax + Cz + D = 0, which will indicate that it is parallel to the Oy axis.
• Secondly, if C = 0, then the equation is transformed into Ax + Vy + D = 0, which will speak of parallelism to the given axis Oz.
• Third, if D = 0, the equation will look like Ax + Vy + Cz = 0, which means that the plane intersects O (origin).
• Fourth, if A = B = 0, then the equation will change to Cz + D = 0, which will prove parallel to Oxy.
• Fifth, if B = C = 0, then the equation becomes Ax + D = 0, which means that the plane to Oyz is parallel.
• Sixth, if A = C = 0, then the equation will take the form Vy + D = 0, that is, it will report parallelism to Oxz.

Equation view in line segments

In the case when the numbers A, B, C, D are different from zero, the form of equation (0) can be as follows:

x / a + y / b + z / c = 1,

in which a = -D / A, b = -D / B, c = -D / C.

We get in the end It is worth noting that this plane will intersect the Ox axis at a point with coordinates (a, 0,0), Oy - (0, b, 0), and Oz - (0,0, c).

Taking into account the equation x / a + y / b + z / c = 1, it is easy to visually represent the location of the plane relative to a given coordinate system.

Normal vector coordinates

The normal vector n to the plane P has coordinates that are the coefficients of the general equation of this plane, that is, n (A, B, C).

In order to determine the coordinates of the normal n, it is enough to know the general equation of the given plane.

When using the equation in segments, which has the form x / a + y / b + z / c = 1, as well as when using the general equation, you can write down the coordinates of any normal vector of a given plane: (1 / a + 1 / b + 1 / With).

It should be noted that the normal vector helps to solve various problems. The most common problems include the problem of proving the perpendicularity or parallelism of planes, the problem of finding the angles between planes or angles between planes and lines.

The form of the plane equation according to the coordinates of the point and the normal vector

A nonzero vector n perpendicular to a given plane is called normal (normal) for a given plane.

Suppose that in coordinate space (rectangular coordinate system) Oxyz are given:

• point Мₒ with coordinates (xₒ, yₒ, zₒ);
• zero vector n = A * i + B * j + C * k.

It is necessary to formulate an equation for a plane that will pass through the point Mₒ perpendicular to the normal n.

In space, choose any arbitrary point and denote it by M (x y, z). Let the radius vector of any point M (x, y, z) be r = x * i + y * j + z * k, and the radius vector of the point Mₒ (xₒ, yₒ, zₒ) - rₒ = xₒ * i + yₒ * j + zₒ * k. The point M will belong to the given plane if the vector МₒМ is perpendicular to the vector n. Let us write the orthogonality condition using the dot product:

[МₒМ, n] = 0.

Since МₒМ = r-rₒ, the vector equation of the plane will look like this:

This equation can have another form. For this, the properties of the dot product are used, and the left side of the equation is transformed. = -. If we denote it as c, then we get the following equation: - c = 0 or = c, which expresses the constancy of the projections onto the normal vector of radius vectors of given points that belong to the plane.

Now you can get the coordinate form of writing the vector equation of our plane = 0. Since r-rₒ = (x-xₒ) * i + (y-yₒ) * j + (z-zₒ) * k, and n = A * i + B * j + C * k, we have:

It turns out that we have an equation of a plane passing through a point perpendicular to the normal n:

A * (x- xₒ) + B * (y- yₒ) C * (z-zₒ) = 0.

The form of the equation of the plane according to the coordinates of two points and a vector collinear to the plane

Let's set two arbitrary points M ′ (x ′, y ′, z ′) and M ″ (x ″, y ″, z ″), as well as a vector a (a ′, a ″, a).

Now we will be able to form an equation for a given plane, which will pass through the existing points M ′ and M ″, as well as any point M with coordinates (x, y, z) parallel to a given vector a.

Moreover, the vectors M′M = (x-x ′; y-y ′; zz ′) and M ″ M = (x ″ -x ′; y ″ -y ′; z ″ -z ′) must be coplanar with the vector a = (a ′, a ″, a ‴), which means that (M′M, M ″ M, a) = 0.

So, our equation of a plane in space will look like this:

View of the equation of a plane intersecting three points

Suppose we have three points: (x ′, y ′, z ′), (x ″, y ″, z ″), (x ‴, y ‴, z ‴), which do not belong to the same straight line. It is necessary to write the equation of the plane passing through the given three points. The theory of geometry asserts that this kind of plane really exists, but it is the only one and inimitable. Since this plane intersects the point (x ′, y ′, z ′), the form of its equation will be as follows:

Here A, B, C are nonzero at the same time. Also, the given plane intersects two more points: (x ″, y ″, z ″) and (x ‴, y ‴, z ‴). In this regard, the following conditions must be met:

Now we can compose a homogeneous system with unknowns u, v, w:

In our case x, y or z is an arbitrary point that satisfies equation (1). Considering equation (1) and the system of equations (2) and (3), the system of equations indicated in the figure above is satisfied by the vector N (A, B, C), which is nontrivial. That is why the determinant of this system is equal to zero.

Equation (1), which we got, this is the equation of the plane. It passes through 3 points exactly, and it is easy to check. To do this, we need to expand our determinant by the elements located in the first line. From the existing properties of the determinant it follows that our plane simultaneously intersects three initially specified points (x ′, y ′, z ′), (x ″, y ″, z ″), (x ‴, y ‴, z ‴). That is, we have solved the task set before us.

Dihedral angle between planes

The dihedral angle is a spatial geometric shape formed by two half-planes that originate from one straight line. In other words, this is a part of the space that is limited by these half-planes.

Let's say we have two planes with the following equations:

We know that the vectors N = (A, B, C) and N¹ = (A¹, B¹, C¹) are perpendicular according to the given planes. In this regard, the angle φ between the vectors N and N¹ is equal to the angle (dihedral), which is between these planes. Scalar product looks like:

NN¹ = | N || N¹ | cos φ,

precisely because

cosφ = NN¹ / | N || N¹ | = (AA¹ + BB¹ + CC¹) / ((√ (A² + B² + C²)) * (√ (A¹) ² + (B¹) ² + (C¹) ²)).

It is enough to take into account that 0≤φ≤π.

In fact, two planes that intersect form two angles (dihedral): φ 1 and φ 2. Their sum is equal to π (φ 1 + φ 2 = π). As for their cosines, their absolute values ​​are equal, but they differ in signs, that is, cos φ 1 = -cos φ 2. If in equation (0) we replace A, B and C with numbers -A, -B and -C, respectively, then the equation that we obtain will determine the same plane, the only angle φ in the equation cos φ = NN 1 / | N || N 1 | will be replaced by π-φ.

Perpendicular plane equation

Perpendicular planes are planes between which the angle is 90 degrees. Using the material outlined above, we can find the equation of a plane perpendicular to another. Suppose we have two planes: Ax + Vy + Cz + D = 0 and A¹x + B¹y + C¹z + D = 0. We can argue that they will be perpendicular if cosφ = 0. This means that NN¹ = AA¹ + BB¹ + CC¹ = 0.

Parallel Plane Equation

Parallel are two planes that do not contain common points.

The condition (their equations are the same as in the previous paragraph) is that the vectors N and N¹, which are perpendicular to them, are collinear. This means that the following proportionality conditions are met:

A / A¹ = B / B¹ = C / C¹.

If the proportionality conditions are extended - A / A¹ = B / B¹ = C / C¹ = DD¹,

this indicates that these planes coincide. This means that the equations Ax + By + Cz + D = 0 and A¹x + B¹y + C¹z + D¹ = 0 describe one plane.

Distance to plane from point

Let's say we have a plane P, which is given by equation (0). It is necessary to find the distance to it from the point with coordinates (xₒ, yₒ, zₒ) = Qₒ. To do this, you need to bring the equation of the plane P into a normal form:

(p, v) = p (p≥0).

In this case, ρ (x, y, z) is the radius vector of our point Q, located on P, p is the length of the perpendicular P, which was released from the zero point, v is the unit vector, which is located in the direction a.

The difference ρ-ρº of the radius vector of some point Q = (x, y, z), belonging to P, as well as the radius vector of a given point Q 0 = (xₒ, yₒ, zₒ) is such a vector, absolute value whose projection onto v is equal to the distance d, which must be found from Q 0 = (xₒ, yₒ, zₒ) to П:

D = | (ρ-ρ 0, v) |, but

(ρ-ρ 0, v) = (ρ, v) - (ρ 0, v) = p- (ρ 0, v).

So it turns out

d = | (ρ 0, v) -p |.

Thus, we will find the absolute value of the resulting expression, that is, the desired d.

Using the parameter language, we get the obvious:

d = | Axₒ + Buₒ + Czₒ | / √ (A² + B² + C²).

If set point Q 0 is located on the other side of the plane P, as well as the origin of coordinates, then between the vector ρ-ρ 0 and v is therefore:

d = - (ρ-ρ 0, v) = (ρ 0, v) -p> 0.

In the case when the point Q 0 together with the origin of coordinates is located on the same side of P, then the created angle is acute, that is:

d = (ρ-ρ 0, v) = p - (ρ 0, v)> 0.

As a result, it turns out that in the first case (ρ 0, v)> p, in the second (ρ 0, v)<р.

The tangent plane and its equation

The tangent plane to the surface at the point of tangency Mº is the plane containing all possible tangents to the curves drawn through this point on the surface.

With this form of the equation of the surface F (x, y, z) = 0, the equation of the tangent plane at the tangent point Mº (xº, yº, zº) will look like this:

F x (xº, yº, zº) (x-xº) + F x (xº, yº, zº) (y-yº) + F x (xº, yº, zº) (z-zº) = 0.

If we set the surface in explicit form z = f (x, y), then the tangent plane will be described by the equation:

z-zº = f (xº, yº) (x- xº) + f (xº, yº) (y-yº).

Intersection of two planes

In the coordinate system (rectangular) Oxyz is located, two planes P ′ and P ″ are given, which intersect and do not coincide. Since any plane in a rectangular coordinate system is determined by the general equation, we will assume that P ′ and P ″ are given by the equations Ax + B′y + C′z + D ′ = 0 and A ″ x + B ″ y + C ″ z + D ″ = 0. In this case, we have the normal n '(A', B ', C') of the plane P 'and the normal n ″ (A ″, B ″, C ″) of the plane P ″. Since our planes are not parallel and do not coincide, these vectors are not collinear. Using the language of mathematics, we can write this condition as follows: n ′ ≠ n ″ ↔ (A ′, B ′, C ′) ≠ (λ * A ″, λ * B ″, λ * C ″), λϵR. Let the straight line that lies at the intersection of P ′ and P ″ will be denoted by the letter a, in this case a = P ′ ∩ P ″.

a is a straight line consisting of the set of all points of the (common) planes P ′ and P ″. This means that the coordinates of any point belonging to the straight line a must simultaneously satisfy the equations A'x + B'y + C'z + D '= 0 and A ″ x + B ″ y + C ″ z + D ″ = 0. This means that the coordinates of the point will be a particular solution of the following system of equations:

As a result, it turns out that the (general) solution of this system of equations will determine the coordinates of each of the points of the straight line, which will act as the intersection point of P ′ and P ″, and determine the straight line a in the Oxyz coordinate system (rectangular) in space.

Consider the plane Q in space. Its position is completely determined by specifying a vector N perpendicular to this plane, and some fixed point lying in the plane Q. Vector N perpendicular to the plane Q is called the normal vector of this plane. If we denote by A, B and C the projection of the normal vector N, then

Let us derive the equation of the plane Q passing through a given point and having a given normal vector. To do this, consider a vector connecting a point with an arbitrary point in the Q plane (Fig. 81).

For any position of the point M on the Q plane, the MXM vector is perpendicular to the normal vector N of the Q plane. Therefore, the scalar product Let us write the scalar product through the projections. Since, and is a vector, then

and therefore

We have shown that the coordinates of any point of the Q plane satisfy Eq. (4). It is easy to see that the coordinates of points that do not lie on the plane Q do not satisfy this equation (in the latter case). Therefore, we have obtained the desired equation of the plane Q. Equation (4) is called the equation of the plane passing through this point. It is the first degree relative to the current coordinates

So, we have shown that any plane corresponds to an equation of the first degree relative to the current coordinates.

Example 1. Write the equation of the plane passing through a point perpendicular to the vector.

Solution. Here . Based on formula (4), we obtain

or, after simplification,

By giving different values ​​to the coefficients A, B and C of equation (4), we can obtain the equation of any plane passing through a point. The set of planes passing through a given point is called a bundle of planes. Equation (4), in which the coefficients A, B and C can take any values, is called the equation of the bundle of planes.

Example 2. Make an equation for a plane passing through three points (Fig. 82).

Solution. Let us write the equation for the bundle of planes passing through the point

The position of the plane in space will be completely determined if we set its distance from the origin O, that is, the length of the perpendicular OT lowered from point O to the plane, and the unit vector n °, perpendicular to the plane and directed from the origin O to the plane (Fig. 110).

When point M moves along the plane, then its radius vector changes so that it is always connected by some condition. Let's see what this condition is. Obviously, for any point lying on the plane, we have:

This condition holds only for points on the plane; it is violated if the point M lies outside the plane. Thus, equality (1) expresses a property common to all points of the plane and only to them. According to § 7 Ch. 11 we have:

and, therefore, equation (1) can be rewritten as:

Equation (D) expresses a condition under which the point) lies on a given plane, and is called the normal equation of this plane. The radius vector of an arbitrary point M of the plane is called the current radius vector.

Equation (1) of the plane is written in vector form. Passing to the coordinates and placing the origin at the origin of the vectors - point O, we note that the projections of the unit vector on the coordinate axes are the cosines of the angles compiled by the axes with this vector, and the projections of the radius vector of the point M

the coordinates of the point serve, that is, we have:

Equation (D) turns into coordinate:

When translating the vector equation (Γ) of the plane into the coordinate equation (2), we used formula (15), § 9, Ch. 11, expressing the dot product in terms of vector projections. Equation (2) expresses a condition under which the point M (x, y, z) lies on a given plane, and is called the normal equation of this plane in coordinate form. The resulting equation (2) is of the first degree with respect to, that is, any plane can be represented by an equation of the first degree with respect to the current coordinates.

Note that the derived equations (1 ") and (2) remain valid even when, i.e., the given plane passes through the origin. In this case, any of the two unit vectors perpendicular to the plane and differing by one from another direction.

Comment. The normal equation of the plane (2) can be derived without using the vector method.

Take an arbitrary plane and draw straight line I through the origin of coordinates perpendicular to it. Set on this straight line a positive direction from the origin to the plane (if the selected plane passed through the origin, then any direction on the straight line could be taken).

The position of this plane in space is completely determined by its distance from the origin, that is, by the length of the l-axis segment from the origin to the point of its intersection with the plane (in Fig. 111 - the segment) and the angles between the axis and the coordinate axes. When a point moves along a plane with coordinates, its coordinates change so that they are all the time connected by some condition. Let's see what this condition is.

Let us construct in fig. 111 coordinate polyline OPSM of an arbitrary point M of the plane. Take the projection of this broken line onto the l-axis. Noticing that the projection of a broken line is equal to the projection of its closing segment (Chapter I, § 3), we will have.

24. Linear dependence of the columns of the matrix. Properties Linear dependence and independence of rows (columns) of a matrix

Properties of linearly dependent and linearly independent matrix columns

25. Basic minor. Basic minor theorem. Rank theorem.

26. Systems of linear equations. The Kronecker - Capelli theorem on the compatibility of systems.

27. Homogeneous systems of linear equations. Properties of their solutions. General solution to the donkey.

28. Fundamental Donkey Decision System

29. Inhomogeneous systems of linear equations. Properties of their solutions. Construction of a general solution to nslu.

30. Linear spaces. Definition. Examples, consequences from axioms.

31. Linear dependence of vectors of linear space. Properties

32. Basis of linear space. Dimension

33. The uniqueness of the expansion of vectors in the basis. Coordinates. Actions on vectors in coordinate form.

34. Changing the coordinates of the vector when moving to a new basis. Transition matrix.

35. Euclidean space. Definition, examples. Vector module. Angle between vectors. Cauchy-Bunyakovsky inequality.

36. Linear operator. Linear operator matrix. Changing the matrix of a linear operator when passing to a new basis.

37. Image and kernel of a linear operator. The rank of the linear operator.

38 in a separate file.

39. Eigenvectors and eigenvalues ​​of a linear operator. Their properties

40. Consistency. Sequence limit. Bounded, unlimited, infinitesimal and infinitely large sequences. Definition

[Edit] Examples

[Edit] Operations on sequences

[Edit] Subsequences

[Edit] Examples

[Edit] Properties

[Edit] End point of sequence

[Edit] Limit of sequence

[Edit] Some kinds of sequences

[Edit] Restricted and Unlimited Sequences

[Edit] Criterion for the boundedness of a numerical sequence

[Edit] Restricted Sequence Properties

[Edit] Infinitely large and infinitely small sequences

[Edit] Properties of infinitesimal sequences

[Edit] Converging and diverging sequences

[Edit] Properties of converging sequences

41. The concept of function. Methods for setting the function.

42. The limit of a function at a point, at infinity. Geometric interpretation. Definitions and examples.

43. Limit theorems:

44. Continuous functions and their properties:

Properties Local

Global

Sign preservation theorem for a continuous function

Proof

45. The first remarkable limit. Consequences. Theorem on the limit of the sum, product and quotient.

46. ​​Limited functions and their properties. A necessary condition for the existence of the limit of a function at a point.

47. Infinitesimal functions, their properties. Lemmas

Lemmas on infinitesimal

48. Criterion for the existence of the limit of a function at a point.

49. Infinitely large functions, connection with infinitely small functions.

50. Disclosure of uncertainties. The second is a remarkable limit.

51. Equivalent infinitesimal functions. Table of equivalent infinitesimal functions.

52. A theorem on the application of equivalent infinitesimal to the calculation of limits.

3.2. Basic formulas for the equivalence of infinitesimal.

53. One-sided limits of a function at a point. One-sided continuity of a function at a point.

54. Breakpoints of a function and their classification.

55. Properties of functions that are continuous on a segment.

56. Tasks leading to the concept of a derivative. Derivative concept. The geometric and physical meaning of the derivative.

1.1 Problems leading to the concept of a derivative

, If.

57. Differentiability of the function. A criterion for the differentiability of a function at a point.

57. Differentiability of the function. A criterion for the differentiability of a function at a point.

58. Derivative of a complex function.

59. Differential function. The invariance of the notation for the first differential.

60. Inverse function and its derivative.

60. Inverse function and its derivative.

61. Rules of differentiation.

63. Logarithmic differentiation. Derivative of the exponential function.

5.4. The derivative of the exponential function

64. See separate file.

65. Average theorems - Fermat, Rolle.

66. Average theorems - Lagrange, Cauchy.

67. Differentials of higher orders. Non-invariance of the notation form.

68. L'Hôpital's rule. Disclosure of Uncertainties Using L'Hôpital's Rule.

69. Taylor's formula. Expansion of a function according to the Taylor formula.

70. Monotonicity of the function. Monotonicity conditions.

71. Extrema of function. A necessary condition for the existence of an extremum.

72. Sufficient conditions for an extremum.

73. Convexity and concavity of the graph of the function. Inflection points.

74. Asymptotes of the graph.

[Edit] Types of graph asymptotes [edit] Vertical

[Edit] Horizontal

[Edit] Oblique

[Edit] Finding the asymptotes

76. The method of changing variables in the indefinite integral.

77. Integration by parts in the indefinite integral. Classes of functions that are integrable by parts.

78. Rational fractions. Decomposition of rational fractions into the sum of elementary fractions.

79. Integration of the simplest rational fractions.

80. Integration of trigonometric functions.

81. Integration of irrationalities of the form ...

82. Integration of irrationalities of the kind ...

83. The concept of a definite integral, its geometric meaning and properties. Average theorem.

84. Integral with variable upper limit. Newton-Leibniz formula.

85. Polar coordinate system. Equations of curves in a polar coordinate system.

Equation of curves in polar coordinates

Circle

Polar rose

Archimedes spiral

Conical sections

86. Calculation of a definite integral. Its application to the calculation of the areas of flat figures, the length of the arc of a curve.

87. Calculation of volumes of bodies, volumes of bodies of revolution.

88. Application of a definite integral to problems of physics.

89. Improper integrals of the first kind.

89. Improper integrals of the first kind.

Improper integrals of the first kind

The geometric meaning of an improper integral of the first kind

Examples of

90. Improper integrals of the second kind.

Geometric meaning of improper integrals of the second kind

Normal equation of the plane.

The general equation of the plane of the view is called normal equation of the plane if the length of the vector is equal to one, that is, , and .

It can often be seen that the normal equation of the plane is written in the form. Here are the direction cosines of the normal vector of a given plane of unit length, that is, and p- a non-negative number equal to the distance from the origin to the plane.

Normal equation of a plane in a rectangular coordinate system Oxyz defines a plane that is a distance from the origin p in the positive direction of the normal vector of this plane ... If p = 0, then the plane passes through the origin.

Let us give an example of the normal equation of the plane.

Let the plane be given in a rectangular coordinate system Oxyz general equation of the plane of the form ... This general equation of the plane is the normal equation of the plane. Indeed, the normal vector of this plane has a length equal to one, since .

Plane Equation in Normal View allows you to find the distance from a point to a plane.

Distance from point to plane.

The distance from a point to a plane is the smallest of the distances between that point and points on the plane. It is known that distance from point to plane is equal to the length of the perpendicular dropped from this point to the plane.

If and the origin lie on opposite sides of the plane, in the opposite case. The distance from a point to a plane is

Mutual arrangement of planes. Conditions for parallelism and perpendicularity of planes.

Distance between parallel planes

Related concepts

The planes are parallel , if

or (Vector product)

The planes are perpendicular, if

Or ... (Scalar product)

Straight in space. Different types of straight line equations.

Equations of a straight line in space - initial information.

Equation of a straight line on a plane Oxy is a linear equation in two variables x and y, which is satisfied by the coordinates of any point of the line and not satisfied by the coordinates of any other points. With a straight line in three-dimensional space, the situation is a little different - there is no linear equation in three variables. x, y and z, which would be satisfied only by the coordinates of the points of a straight line given in a rectangular coordinate system Oxyz... Indeed, an equation of the form, where x, y and z- variables, and A, B, C and D- some real numbers, and A, V and WITH at the same time are not equal to zero, is general equation of the plane... Then the question arises: "How can a straight line be described in a rectangular coordinate system Oxyz»?

The answer to it is contained in the following paragraphs of the article.

The equations of a straight line in space are the equations of two intersecting planes.

Let us recall one axiom: if two planes in space have a common point, then they have a common straight line on which all common points of these planes are located. Thus, a straight line in space can be specified by specifying two planes that intersect along this straight line.

Let us translate the last statement into the language of algebra.

Let a rectangular coordinate system be fixed in three-dimensional space Oxyz and it is known that the straight line a is the line of intersection of two planes and, which correspond to the general equations of the plane of the view, respectively. Since straight a is the set of all common points of the planes and, then the coordinates of any point of the straight line a will simultaneously satisfy the equation and the equation, the coordinates of no other points will not simultaneously satisfy both equations of the planes. Therefore, the coordinates of any point of the straight line a in a rectangular coordinate system Oxyz represent particular solution of a system of linear equations of the kind , and the general solution of the system of equations defines the coordinates of each point of the straight line a, that is, defines a straight line a.

So, a straight line in space in a rectangular coordinate system Oxyz can be given by a system of equations of two intersecting planes .

Here is an example of defining a straight line in space using a system of two equations - .

Describing a straight line by equations of two intersecting planes is great for finding the coordinates of the point of intersection of a straight line and a plane and also for finding the coordinates of the point of intersection of two straight lines in space.

We recommend that you continue your study of this topic by referring to the article equations of a straight line in space - equations of two intersecting planes... It provides more detailed information, analyzes in detail the solutions of typical examples and problems, and also shows a way to go to equations of a straight line in space of a different kind.

It should be noted that there are various ways to define a straight line in space, and in practice, a straight line is often specified not by two intersecting planes, but by a direction vector of a straight line and a point lying on this straight line. In these cases, it is easier to obtain the canonical and parametric equations of a straight line in space. We will talk about them in the next paragraphs.

Parametric equations of a straight line in space.

Parametric equations of a straight line in space have the form ,

where x 1 ,y 1 and z 1 - coordinates of some point of a straight line, a x , a y and a z (a x , a y and a z are not equal to zero at the same time) - the corresponding coordinates of the direction vector of the straight line, and - some parameter that can take any valid values.

For any value of the parameter, according to the parametric equations of a straight line in space, we can calculate a triple of numbers,

it will correspond to some point of a straight line (hence the name of this type of equations of a straight line). For example, for

from the parametric equations of a straight line in space, we obtain the coordinates x 1 , y 1 and z 1 : .

As an example, consider a straight line that is defined by parametric equations of the form ... This line passes through a point, and the direction vector of this line has coordinates.

We recommend that you continue to study the topic by referring to the article material. parametric equations of a straight line in space... It shows the derivation of the parametric equations of a straight line in space, disassembles particular cases of parametric equations of a straight line in space, gives graphic illustrations, gives detailed solutions to typical problems, and indicates the connection of parametric equations of a straight line with other types of equations of a straight line.

Canonical equations of a straight line in space.

Solving each of the parametric equations of the straight line relative to the parameter, it is easy to go to canonical equations of a straight line in space of the kind .

The canonical equations of a straight line in space define a straight line passing through a point , and the direction vector of the straight line is the vector ... For example, the equations of the straight line in the canonical form correspond to a straight line passing through a point in space with coordinates, the direction vector of this straight line has coordinates.

It should be noted that one or two of the numbers in the canonical equations of the line can be zero (all three numbers cannot be zero at the same time, since the direction vector of the line cannot be zero). Then an entry of the form is considered formal (since there will be zeros in the denominators of one or two fractions) and should be understood as , where.

If one of the numbers in the canonical equations of the straight line is zero, then the straight line lies in one of the coordinate planes, or in a plane parallel to it. If two of the numbers are equal to zero, then the line either coincides with one of the coordinate axes, or is parallel to it. For example, a straight line corresponding to the canonical equations of a straight line in a space of the form , lies in the plane z = -2 which is parallel to the coordinate plane Oxy, and the coordinate axis Oy is defined by canonical equations.

For graphic illustrations of these cases, the derivation of the canonical equations of a straight line in space, detailed solutions of typical examples and problems, as well as the transition from canonical equations of a straight line to other equations of a straight line in space, see the article canonical equations of a straight line in space.

General equation of the straight line. The transition from the general to the canonical equation.

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