Decorative coordinate system. Cartesian coordinates

Rectangular system coordinates on the plane is formed by two mutually perpendicular coordinate axes X’X and Y’Y. The coordinate axes intersect at point O, which is called the origin, a positive direction is selected on each axis. The positive direction of the axes (in a right-handed coordinate system) is chosen so that when the X'X axis is rotated counterclockwise by 90°, its positive direction coincides with the positive direction of the Y'Y axis. The four angles (I, II, III, IV) formed by the coordinate axes X'X and Y'Y are called coordinate angles (see Fig. 1).

The position of point A on the plane is determined by two coordinates x and y. The x coordinate is equal to the length of the segment OB, the y coordinate is equal to the length of the segment OC in the selected units of measurement. Segments OB and OC are defined by lines drawn from point A parallel to the Y'Y and X'X axes, respectively. The x coordinate is called the abscissa of point A, the y coordinate is called the ordinate of point A. It is written as follows: A(x, y).

If point A lies in coordinate angle I, then point A has a positive abscissa and ordinate. If point A lies in coordinate angle II, then point A has a negative abscissa and a positive ordinate. If point A lies in coordinate angle III, then point A has a negative abscissa and ordinate. If point A lies in coordinate angle IV, then point A has a positive abscissa and a negative ordinate.

Rectangular coordinate system in space is formed by three mutually perpendicular coordinate axes OX, OY and OZ. The coordinate axes intersect at point O, which is called the origin, on each axis a positive direction is selected, indicated by arrows, and a unit of measurement for the segments on the axes. The units of measurement are the same for all axes. OX - abscissa axis, OY - ordinate axis, OZ - applicate axis. The positive direction of the axes is chosen so that when the OX axis is rotated counterclockwise by 90°, its positive direction coincides with the positive direction of the OY axis, if this rotation is observed from the positive direction of the OZ axis. Such a coordinate system is called right-handed. If thumb right hand take the X direction as the X direction, the index one as the Y direction, and the middle one as the Z direction, then a right-handed coordinate system is formed. Similar fingers of the left hand form the left coordinate system. It is impossible to combine the right and left coordinate systems so that the corresponding axes coincide (see Fig. 2).

The position of point A in space is determined by three coordinates x, y and z. The x coordinate is equal to the length of the segment OB, the y coordinate is the length of the segment OC, the z coordinate is the length of the segment OD in the selected units of measurement. The segments OB, OC and OD are defined by planes drawn from point A parallel to the planes YOZ, XOZ and XOY, respectively. The x coordinate is called the abscissa of point A, the y coordinate is called the ordinate of point A, the z coordinate is called the applicate of point A. It is written as follows: A(a, b, c).

Orty

A rectangular coordinate system (of any dimension) is also described by a set of unit vectors aligned with the coordinate axes. The number of unit vectors is equal to the dimension of the coordinate system and they are all perpendicular to each other.

IN three-dimensional case such vectors are usually denoted i j k or e x e y e z. In this case, in the case of a right-handed coordinate system, the following formulas with the vector product of vectors are valid:

[i j]=k ;
[j k]=i ;
[k i]=j .

Story

The rectangular coordinate system was first introduced by Rene Descartes in his work “Discourse on Method” in 1637. Therefore, the rectangular coordinate system is also called - Cartesian coordinate system. The coordinate method of describing geometric objects marked the beginning of analytical geometry. Pierre Fermat also contributed to the development of the coordinate method, but his works were first published after his death. Descartes and Fermat used the coordinate method only on the plane.

The coordinate method for three-dimensional space was first used by Leonhard Euler already in the 18th century.

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See what “Cartesian coordinate system” is in other dictionaries:

CARTESIAN COORDINATE SYSTEM- a rectangular coordinate system on a plane or in space, in which the scales along the axes are the same and the coordinate axes are mutually perpendicular. D. s. K. is denoted by the letters x:, y for a point on a plane or x, y, z for a point in space. (Cm.… …

CARTESIAN COORDINATE SYSTEM, a system introduced by Rene DESCARTES, in which the position of a point is determined by the distance from it to mutually intersecting lines (axes). In the simplest version of the system, the axes (denoted x and y) are perpendicular.... ... Scientific and technical encyclopedic dictionary

Cartesian coordinate system

A rectilinear coordinate system (See Coordinates) on a plane or in space (usually with equal scales along the axes). R. Descartes himself in “Geometry” (1637) used only a coordinate system on a plane (in general, oblique). Often… … Great Soviet Encyclopedia

A set of definitions that implements the coordinate method, that is, a way to determine the position of a point or body using numbers or other symbols. The set of numbers that determine the position of a specific point is called the coordinates of this point. In... ... Wikipedia

cartesian system- Dekarto koordinačių sistema statusas T sritis fizika atitikmenys: engl. Cartesian system; Cartesian system of coordinates vok. cartesisches Koordinatensystem, n; kartesisches Koordinatensystem, n rus. Cartesian system, f; Cartesian system... ... Fizikos terminų žodynas

COORDINATE SYSTEM- a set of conditions that determine the position of a point on a straight line, on a plane, in space. There are various spherical shapes: Cartesian, oblique, cylindrical, spherical, curvilinear, etc. Linear and angular quantities that determine the position... ... Big Polytechnic Encyclopedia

Orthonormal rectilinear coordinate system in Euclidean space. D.p.s. on a plane is specified by two mutually perpendicular straight coordinate axes, on each of which a positive direction is chosen and a segment of the unit ... Mathematical Encyclopedia

Rectangular coordinate system is a rectilinear coordinate system with mutually perpendicular axes on a plane or in space. The simplest and therefore most commonly used coordinate system. Very easily and directly summarized for... ... Wikipedia

Books

Computational fluid dynamics. Theoretical basis. Textbook, Valery Alekseevich Pavlovsky, Dmitry Vladimirovich Nikushchenko. The book is devoted to a systematic presentation theoretical foundations for setting tasks mathematical modeling flows of liquids and gases. Special attention devoted to issues of construction...

An ordered system of two or three intersecting axes perpendicular to each other with common beginning reference (origin) and common unit length is called rectangular Cartesian coordinate system .

General Cartesian coordinate system (affine coordinate system) may not necessarily include perpendicular axes. In honor of the French mathematician Rene Descartes (1596-1662), just such a coordinate system is named in which a common unit of length is measured on all axes and the axes are straight.

Rectangular Cartesian coordinate system on a plane has two axes and rectangular Cartesian coordinate system in space - three axes. Each point on a plane or in space is defined by an ordered set of coordinates - numbers corresponding to the unit of length of the coordinate system.

Note that, as follows from the definition, there is a Cartesian coordinate system on a straight line, that is, in one dimension. The introduction of Cartesian coordinates on a line is one of the ways by which any point on a line is associated with a well-defined real number, that is, a coordinate.

The coordinate method, which arose in the works of Rene Descartes, marked a revolutionary restructuring of all mathematics. It became possible to interpret algebraic equations(or inequalities) in the form of geometric images (graphs) and, conversely, look for a solution geometric problems using analytical formulas and systems of equations. Yes, inequality z < 3 геометрически означает полупространство, лежащее ниже плоскости, параллельной координатной плоскости xOy and located above this plane by 3 units.

Using the Cartesian coordinate system, the membership of a point on a given curve corresponds to the fact that the numbers x And y satisfy some equation. So, the coordinates of a point on a circle with center at given point (a; b) satisfy the equation (x - a)² + ( y - b)² = R² .

Rectangular Cartesian coordinate system on a plane

Two perpendicular axes on a plane with a common origin and the same scale unit form Cartesian rectangular coordinate system on the plane . One of these axes is called the axis Ox, or x-axis , the other - the axis Oy, or y-axis . These axes are also called coordinate axes. Let us denote by Mx And My respectively, the projection of an arbitrary point M on the axis Ox And Oy. How to get projections? Let's go through the point M Ox. This straight line intersects the axis Ox at the point Mx. Let's go through the point M straight line perpendicular to the axis Oy. This straight line intersects the axis Oy at the point My. This is shown in the picture below.

x And y points M we will call the values of the directed segments accordingly OMx And OMy. The values of these directed segments are calculated accordingly as x = x0 - 0 And y = y0 - 0 . Cartesian coordinates x And y points M abscissa And ordinate . The fact that the point M has coordinates x And y, is denoted as follows: M(x, y) .

Coordinate axes divide the plane into four quadrant , the numbering of which is shown in the figure below. It also shows the arrangement of signs for the coordinates of points depending on their location in a particular quadrant.

In addition to Cartesian rectangular coordinates on a plane, the polar coordinate system is also often considered. About the method of transition from one coordinate system to another - in the lesson polar coordinate system .

Rectangular Cartesian coordinate system in space

Cartesian coordinates in space are introduced in complete analogy with Cartesian coordinates in the plane.

Three mutually perpendicular axes in space (coordinate axes) with a common origin O and with the same scale unit they form Cartesian rectangular coordinate system in space .

One of these axes is called an axis Ox, or x-axis , the other - the axis Oy, or y-axis , the third - axis Oz, or axis applicate . Let Mx, My Mz- projections of an arbitrary point M space on the axis Ox , Oy And Oz respectively.

Let's go through the point M OxOx at the point Mx. Let's go through the point M plane perpendicular to the axis Oy. This plane intersects the axis Oy at the point My. Let's go through the point M plane perpendicular to the axis Oz. This plane intersects the axis Oz at the point Mz.

Cartesian rectangular coordinates x , y And z points M we will call the values of the directed segments accordingly OMx, OMy And OMz. The values of these directed segments are calculated accordingly as x = x0 - 0 , y = y0 - 0 And z = z0 - 0 .

Cartesian coordinates x , y And z points M are called accordingly abscissa , ordinate And applicate .

Coordinate axes taken in pairs are located in coordinate planes xOy , yOz And zOx .

Problems about points in a Cartesian coordinate system

Example 1.

A(2; -3) ;

B(3; -1) ;

C(-5; 1) .

Find the coordinates of the projections of these points onto the abscissa axis.

Solution. As follows from the theoretical part of this lesson, the projection of a point onto the abscissa axis is located on the abscissa axis itself, that is, the axis Ox, and therefore has an abscissa equal to the abscissa of the point itself, and an ordinate (coordinate on the axis Oy, which the x-axis intersects at point 0), which is equal to zero. So we get the following coordinates of these points on the x-axis:

Ax(2;0);

Bx(3;0);

Cx (-5; 0).

Example 2. In the Cartesian coordinate system, points are given on the plane

A(-3; 2) ;

B(-5; 1) ;

C(3; -2) .

Find the coordinates of the projections of these points onto the ordinate axis.

Solution. As follows from the theoretical part of this lesson, the projection of a point onto the ordinate axis is located on the ordinate axis itself, that is, the axis Oy, and therefore has an ordinate equal to the ordinate of the point itself, and an abscissa (coordinate on the axis Ox, which the ordinate axis intersects at point 0), which is equal to zero. So we get the following coordinates of these points on the ordinate axis:

Ay(0;2);

By(0;1);

Cy(0;-2).

Example 3. In the Cartesian coordinate system, points are given on the plane

A(2; 3) ;

B(-3; 2) ;

C(-1; -1) .

Ox .

Ox Ox Ox, will have the same abscissa as given point, and ordinate equal to absolute value ordinate of a given point, and its opposite sign. So we get the following coordinates of points symmetrical to these points relative to the axis Ox :

A"(2; -3) ;

B"(-3; -2) ;

C"(-1; 1) .

Solve problems using the Cartesian coordinate system yourself, and then look at the solutions

Example 4. Determine in which quadrants (quarters, drawing with quadrants - at the end of the paragraph “Rectangular Cartesian coordinate system on a plane”) a point can be located M(x; y) , If

1) xy > 0 ;

2) xy < 0 ;

3) x − y = 0 ;

4) x + y = 0 ;

5) x + y > 0 ;

6) x + y < 0 ;

7) x − y > 0 ;

8) x − y < 0 .

Example 5. In the Cartesian coordinate system, points are given on the plane

A(-2; 5) ;

B(3; -5) ;

C(a; b) .

Find the coordinates of points symmetrical to these points relative to the axis Oy .

Let's continue to solve problems together

Example 6. In the Cartesian coordinate system, points are given on the plane

A(-1; 2) ;

B(3; -1) ;

C(-2; -2) .

Find the coordinates of points symmetrical to these points relative to the axis Oy .

Solution. Rotate 180 degrees around the axis Oy directional segment from the axis Oy up to this point. In the figure, where the quadrants of the plane are indicated, we see that the point symmetrical to the given one relative to the axis Oy, will have the same ordinate as the given point, and an abscissa equal in absolute value to the abscissa of the given point and opposite in sign. So we get the following coordinates of points symmetrical to these points relative to the axis Oy :

A"(1; 2) ;

B"(-3; -1) ;

C"(2; -2) .

Example 7. In the Cartesian coordinate system, points are given on the plane

A(3; 3) ;

B(2; -4) ;

C(-2; 1) .

Find the coordinates of points symmetrical to these points relative to the origin.

Solution. We rotate the directed segment going from the origin to the given point by 180 degrees around the origin. In the figure, where the quadrants of the plane are indicated, we see that a point symmetrical to the given point relative to the origin of coordinates will have an abscissa and ordinate equal in absolute value to the abscissa and ordinate of the given point, but opposite in sign. So we get the following coordinates of points symmetrical to these points relative to the origin:

A"(-3; -3) ;

B"(-2; 4) ;

C(2; -1) .

Example 8.

A(4; 3; 5) ;

B(-3; 2; 1) ;

C(2; -3; 0) .

Find the coordinates of the projections of these points:

1) on a plane Oxy ;

2) on a plane Oxz ;

3) to the plane Oyz ;

4) on the abscissa axis;

5) on the ordinate axis;

6) on the applicate axis.

1) Projection of a point onto a plane Oxy is located on this plane itself, and therefore has an abscissa and ordinate equal to the abscissa and ordinate of a given point, and an applicate equal to zero. So we get the following coordinates of the projections of these points onto Oxy :

Axy (4; 3; 0);

Bxy (-3; 2; 0);

Cxy(2;-3;0).

2) Projection of a point onto a plane Oxz is located on this plane itself, and therefore has an abscissa and applicate equal to the abscissa and applicate of a given point, and an ordinate equal to zero. So we get the following coordinates of the projections of these points onto Oxz :

Axz (4; 0; 5);

Bxz (-3; 0; 1);

Cxz (2; 0; 0).

3) Projection of a point onto a plane Oyz is located on this plane itself, and therefore has an ordinate and applicate equal to the ordinate and applicate of a given point, and an abscissa equal to zero. So we get the following coordinates of the projections of these points onto Oyz :

Ayz(0; 3; 5);

Byz (0; 2; 1);

Cyz (0; -3; 0).

4) As follows from the theoretical part of this lesson, the projection of a point onto the abscissa axis is located on the abscissa axis itself, that is, the axis Ox, and therefore has an abscissa equal to the abscissa of the point itself, and the ordinate and applicate of the projection are equal to zero (since the ordinate and applicate axes intersect the abscissa at point 0). We obtain the following coordinates of the projections of these points onto the abscissa axis:

Ax(4;0;0);

Bx (-3; 0; 0);

Cx(2;0;0).

5) The projection of a point onto the ordinate axis is located on the ordinate axis itself, that is, the axis Oy, and therefore has an ordinate equal to the ordinate of the point itself, and the abscissa and applicate of the projection are equal to zero (since the abscissa and applicate axes intersect the ordinate axis at point 0). We obtain the following coordinates of the projections of these points onto the ordinate axis:

Ay(0; 3; 0);

By (0; 2; 0);

Cy(0;-3;0).

6) The projection of a point onto the applicate axis is located on the applicate axis itself, that is, the axis Oz, and therefore has an applicate equal to the applicate of the point itself, and the abscissa and ordinate of the projection are equal to zero (since the abscissa and ordinate axes intersect the applicate axis at point 0). We obtain the following coordinates of the projections of these points onto the applicate axis:

Az (0; 0; 5);

Bz (0; 0; 1);

Cz(0; 0; 0).

Example 9. In the Cartesian coordinate system, points are given in space

A(2; 3; 1) ;

B(5; -3; 2) ;

C(-3; 2; -1) .

Find the coordinates of points symmetrical to these points with respect to:

1) plane Oxy ;

2) planes Oxz ;

3) planes Oyz ;

4) abscissa axes;

5) ordinate axes;

6) applicate axes;

7) origin of coordinates.

1) “Move” the point on the other side of the axis Oxy Oxy, will have an abscissa and ordinate equal to the abscissa and ordinate of a given point, and an applicate equal in magnitude to the aplicate of a given point, but opposite in sign. So, we get the following coordinates of points symmetrical to the data relative to the plane Oxy :

A"(2; 3; -1) ;

B"(5; -3; -2) ;

C"(-3; 2; 1) .

2) “Move” the point on the other side of the axis Oxz to the same distance. From the figure displaying the coordinate space, we see that a point symmetrical to a given one relative to the axis Oxz, will have an abscissa and applicate equal to the abscissa and applicate of a given point, and an ordinate equal in magnitude to the ordinate of a given point, but opposite in sign. So, we get the following coordinates of points symmetrical to the data relative to the plane Oxz :

A"(2; -3; 1) ;

B"(5; 3; 2) ;

C"(-3; -2; -1) .

3) “Move” the point on the other side of the axis Oyz to the same distance. From the figure displaying the coordinate space, we see that a point symmetrical to a given one relative to the axis Oyz, will have an ordinate and an aplicate equal to the ordinate and an aplicate of a given point, and an abscissa equal in value to the abscissa of a given point, but opposite in sign. So, we get the following coordinates of points symmetrical to the data relative to the plane Oyz :

A"(-2; 3; 1) ;

B"(-5; -3; 2) ;

C"(3; 2; -1) .

By analogy with symmetrical points on a plane and points in space that are symmetrical to data relative to planes, we note that in the case of symmetry with respect to some axis of the Cartesian coordinate system in space, the coordinate on the axis with respect to which the symmetry is given will retain its sign, and the coordinates on the other two axes will be the same in absolute value as the coordinates of a given point, but opposite in sign.

4) The abscissa will retain its sign, but the ordinate and applicate will change signs. So, we obtain the following coordinates of points symmetrical to the data relative to the abscissa axis:

A"(2; -3; -1) ;

B"(5; 3; -2) ;

C"(-3; -2; 1) .

5) The ordinate will retain its sign, but the abscissa and applicate will change signs. So, we obtain the following coordinates of points symmetrical to the data relative to the ordinate axis:

A"(-2; 3; -1) ;

B"(-5; -3; -2) ;

C"(3; 2; 1) .

6) The applicate will retain its sign, but the abscissa and ordinate will change signs. So, we obtain the following coordinates of points symmetrical to the data relative to the applicate axis:

A"(-2; -3; 1) ;

B"(-5; 3; 2) ;

C"(3; -2; -1) .

7) By analogy with symmetry in the case of points on a plane, in the case of symmetry about the origin of coordinates, all coordinates of a point symmetrical to a given one will be equal in absolute value to the coordinates of a given point, but opposite to them in sign. So, we obtain the following coordinates of points symmetrical to the data relative to the origin.

In the 2nd century BC. Greek scientist Hipparchus proposed to encircle the map Earth parallels and meridians, covering it with a kind of conditional grid, and introduce geographical coordinates- latitude and longitude.

True, even before this, astronomers used this technique when studying the vault of heaven.

In the 2nd century AD. The famous ancient Greek astronomer and mathematician Claudius Ptolemy actively used longitude and latitude as geographical coordinates.
But these concepts were systematized in the 17th century by Rene Descartes.

René Descartes (1596 - 1650) - French mathematician, philosopher, physicist and physiologist.
It was he who came up with the coordinate system in 1637, which is used all over the world and is known to every schoolchild. It is also called the “Cartesian coordinate system”.

What kind of person was Descartes?

Descartes came from noble family and was the youngest (third) son in the family. He was born in 1596 in France. His mother died when he was 1 year old. Rene got great elementary education at the prestigious College La Flèche. Here he studied with Jesuit priests.

During his ten years at college, Descartes acquired writing skills, studied the arts of music and drama, and even mastered such noble pursuits as horse riding and fencing.
After spending two more years at the University of Poitiers, he received academic degree in law, but abandoned a career in law.
Rene entered military service and began to travel a lot around Europe.

Descartes then lived in the Netherlands for about twenty years. The tolerant Dutch in the 17th century got along quietly without such things as the Inquisition, heresy, the rack and burning at the stake, which threatened all European original thinkers. Here, unlike other countries, there was no need to pay for your ideas.
Descartes maintains extensive correspondence with the best scientists in Europe, studies the most various sciences, writes books. He studied astronomy and medicine.

The great physiologist Ivan Petrovich Pavlov considered Descartes the forerunner

their research. Rene Descartes was the first to propose the concept of reflex.

(Monument to R. Descartes. Sculptor: I.F. Bezpalov. Address: Alley of Busts of Great Scientists in Koltushi.)

He owns famous phrase: "Cogito, ergo sum",
which translated from Latin means:
“I think, therefore I exist.”

Cartesian coordinate system

To define a Cartesian rectangular coordinate system on a plane, mutually perpendicular straight lines, called axes, are selected.
The point where the axes intersect – “O” is called the origin of coordinates.
On each axis (ОX and ОY), a positive direction is specified and a scale unit (unit segment) is selected.

The position of point A on the plane is determined by two coordinates x and y.
The x coordinate is equal to the length of the segment OB, the y coordinate is equal to the length of the segment OC in the selected units of measurement.
The x coordinate is called the abscissa of point A, the y coordinate is called the ordinate of point A.
Each point on the coordinate plane corresponds to a pair of numbers: its abscissa and ordinate: (x; y). And vice versa: each pair of numbers corresponds to a single point on the coordinate plane.

In a space in which the position of a point can be defined as its projection onto fixed lines intersecting at a single point, called the origin. These projections are called point coordinates, and the straight lines are called coordinate axes.

IN general case on the plane, the Cartesian coordinate system ( affine system coordinates) is given by point O (origin of coordinates) and an ordered pair of vectors e 1 and e 2 (basis vectors) attached to it that do not lie on the same line. Straight lines passing through the origin in the direction of the basis vectors are called the coordinate axes of a given Cartesian coordinate system. The first, determined by the vector e 1, is called the abscissa axis (or Ox axis), the second is the ordinate axis (or Oy axis). The Cartesian coordinate system itself is denoted Oe 1 e 2 or Oxy. The Cartesian coordinates of point M (Figure 1) in the Cartesian coordinate system Oe 1 e 2 are called an ordered pair of numbers (x, y), which are the coefficients of the expansion of the vector OM along the basis (e 1, e 2), that is, x and y are such that OM = xe 1 + ue 2. Number x, -∞< x < ∞, называется абсциссой, чис-ло у, - ∞ < у < ∞, - ординатой точки М. Если (x, у) - координаты точки М, то пишут М(х, у).

If two Cartesian coordinate systems Oe 1 e 2 and 0'e' 1 e' 2 are introduced on the plane so that the basis vectors (e' 1, e' 2) are expressed through the basis vectors (e 1, e 2) by the formulas

e’ 1 = a 11 e 1 + a 12 e 2, e’ 2 = a 21 e 1 + a 22 e 2

and the point O' has coordinates (x 0, y 0) in the Cartesian coordinate system Oe 1 e 2, then the coordinates (x, y) of the point M in the Cartesian coordinate system Oe 1 e2 and the coordinates (x', y') of the same point in the Cartesian coordinate system O'e 1 e' 2 are related by the relations

x = a 11 x’ + a 21 y’ + x 0, y = a 12 x’+ a 22 y’+ y 0.

A Cartesian coordinate system is called rectangular if the basis (e 1, e 2) is orthonormal, that is, the vectors e 1 and e 2 are mutually perpendicular and have lengths equal to one(vectors e 1 and e 2 are called vectors in this case). In a rectangular Cartesian coordinate system, the x and y coordinates of point M are quantities orthogonal projections points M on the Ox and Oy axis, respectively. In the rectangular Cartesian coordinate system Oxy, the distance between points M 1 (x 1, y 1) and M 2 (x 2, y 2) is equal to √(x 2 - x 1) 2 + (y 2 -y 1) 2

Formulas for transition from one rectangular Cartesian coordinate system Oxy to another rectangular Cartesian coordinate system O'x'y', the beginning of which O' of the Cartesian coordinate system Oxy is O'(x0, y0), have the form

x = x’cosα - y’sinα + x 0, y = x’sin α + y’cosα + y 0

x = x'cosα + y'sinα + x 0, y = x'sinα - y'cosα + y 0.

In the first case, the O'x'y' system is formed by rotating the basis vectors e 1 ; e 2 by angle α and subsequent transfer of the origin of coordinates O to point O’ (Figure 2),

and in the second case - by rotating the basis vectors e 1, e 2 by an angle α, subsequent reflection of the axis containing the vector e 2 relative to the straight line carrying the vector e 1, and transferring the origin O to point O’ (Figure 3).

Sometimes oblique Cartesian coordinate systems are used, which differ from the rectangular one in that the angle between the unit basis vectors is not right.

The general Cartesian coordinate system (affine coordinate system) in space is defined similarly: a point O is specified - the origin of coordinates and an ordered triple of vectors е 1 , е 2 , е 3 (basis vectors) attached to it and not lying in the same plane. As in the case of a plane, coordinate axes are determined - the abscissa axis (Ox axis), the ordinate axis (Oy axis) and the applicate axis (Oz axis) (Figure 4).

The Cartesian coordinate system in space is denoted Oe 1 e 2 e 3 (or Oxyz). Planes passing through pairs of coordinate axes are called coordinate planes. A Cartesian coordinate system in space is called right-handed if the rotation from the Ox axis to the Oy axis is made in the direction opposite to the clockwise movement when looking at the Oxy plane from some point on the positive semi-axis Oz; otherwise, the Cartesian coordinate system is called left-handed. If the basis vectors e 1, e 2, e 3 have lengths equal to one and are pairwise perpendicular, then the Cartesian coordinate system is called rectangular. The position of one rectangular Cartesian coordinate system in space relative to another rectangular Cartesian coordinate system with the same orientation is determined by three Euler angles.

The Cartesian coordinate system is named after R. Descartes, although in his work “Geometry” (1637) an oblique coordinate system was considered, in which the coordinates of points could only be positive. In the edition of 1659-61, the work of the Dutch mathematician I. Gudde was added to Geometry, in which for the first time both positive and negative values coordinates The spatial Cartesian coordinate system was introduced by the French mathematician F. Lahire (1679). At the beginning of the 18th century, the notations x, y, z for Cartesian coordinates were established.

CARTESIAN COORDINATE SYSTEM CARTESIAN COORDINATE SYSTEM

CARTESIAN COORDINATE SYSTEM, a rectilinear coordinate system on a plane or in space (usually with mutually perpendicular axes and equal scales along the axes). Named after R. Descartes (cm. DESCARTES Rene).
Descartes was the first to introduce a coordinate system, which was significantly different from the generally accepted one today. He used an oblique coordinate system on a plane, considering a curve relative to some straight line with fixed system countdown. The position of the curve points was specified using a system of parallel segments, inclined or perpendicular to the original straight line. Descartes did not introduce a second coordinate axis and did not fix the direction of reference from the origin of coordinates. Only in the 18th century. formed modern understanding coordinate system, named after Descartes.
***
To define a Cartesian rectangular coordinate system, mutually perpendicular straight lines, called axes, are selected. Axial intersection point O called the origin. On each axis, a positive direction is specified and a scale unit is selected. Point coordinates P are considered positive or negative depending on which semi-axis the projection of the point falls on P.
2D coordinate system
P on a plane in a two-dimensional coordinate system, the distances taken with a certain sign (expressed in scale units) of this point to two mutually perpendicular lines - coordinate axes or projections of the radius vector - are called r points P on two mutually perpendicular coordinate axes.
In a two-dimensional coordinate system, the horizontal axis is called the x-axis (axis OX), vertical axis- ordinate axis (ОY axis). Positive directions are chosen on the axis OX- to the right, on the axis OY- up. Coordinates x And y are called the abscissa and ordinate of the point, respectively. The notation P(a,b) means that a point P on the plane has an abscissa a and an ordinate b.
Three-dimensional coordinate system
Cartesian rectangular coordinates of a point P V three-dimensional space are called distances taken with a certain sign (expressed in scale units) of this point to three mutually perpendicular coordinate planes or projections of the radius vector (cm. RADIUS VECTOR) r points P into three mutually perpendicular coordinate axes.
Through an arbitrary point in space O- origin of coordinates - three pairs of perpendicular straight lines are drawn: axis OX(x axis), axis OY(y-axis), axis OZ(applicate axis).
On the coordinate axes can be specified unit vectors i, j, k along the axes OX,OY, OZ respectively.
Depending on the relative position positive directions of coordinate axes, right and left coordinate systems are possible. As a rule, a right-handed coordinate system is used. In the right coordinate system, positive directions are chosen as follows: along the axis OX- on the observer; along the OY axis - to the right; along the OZ axis - up. In a right-handed coordinate system, the shortest rotation from the X-axis to the Y-axis is counterclockwise; if simultaneously with such a rotation we move along the positive direction of the axis Z, then the result will be movement according to the rule of the right screw.
The notation P(a,b,c) means that point P has an abscissa a, an ordinate b and an applicate c.
Each triple of numbers (a,b,c) defines a single point P. Consequently, the rectangular Cartesian coordinate system establishes a one-to-one correspondence between the set of points in space and the set of ordered triplets of real numbers.
In addition to coordinate axes, there are also coordinate planes. Coordinate surfaces for which one of the coordinates remains constant are planes parallel to the coordinate planes, and coordinate lines along which only one coordinate changes are straight lines parallel coordinate axes. Coordinate surfaces intersect along coordinate lines.
Coordinate plane XOY contains axes OX And OY, coordinate plane YOZ contains axes OY And OZ, coordinate plane XOZ contains axes OX And OZ.

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