Let's consider three-dimensional space.
Definition 8.1. Under affine coordinate system in three-dimensional space we will understand a geometric image consisting of a fixed point O and an affine basis.
We will denote an affine coordinate system by . Dot ABOUT called origin, and the vectors are coordinate vectors.
Similarly under rectangular Cartesian coordinate system we will understand a geometric image consisting of a fixed point ABOUT- the origin of coordinates and a rectangular Cartesian basis.
Directed lines passing through the origin and parallel to the coordinate vectors are called coordinate axes. Axes parallel to vectors (or vectors) are called respectively abscissa axes, ordinate And applicate and are designated Ox, Oy, Oz. Planes defined by axes Oh And Oh, Ox And Oz, Oy And Oz, are called coordinate planes and are denoted accordingly by Oxy, Oxz, Oyz. The coordinate system (or) is also denoted Oxyz.
In the future, all arguments will be carried out in a rectangular Cartesian coordinate system.
Let be a rectangular Cartesian coordinate system. Consider an arbitrary point A three-dimensional space.
Definition 8.2. The directed segment is called radius vector points A.
Note that there is a one-to-one correspondence between points in space and their radius vectors.
Definition 8.3.Coordinates (rectangular Cartesian coordinates) of point A three-dimensional space is called a triple of numbers ( x, y, z), Where x, y, z- coordinates of the radius vector in the orthonormal basis, i.e.
Similar to the name of the coordinate axes, the first coordinate is called abscissa, second - ordinate and the third - applicate point.
To plot a point A in a rectangular Cartesian coordinate system we use formula (8.1). Let's postpone from the point O vectors , , . Let's construct a rectangular parallelepiped so that its three dimensions are equal 
, then the vector coincides with the diagonal of the parallelepiped. It is easy to verify the validity of the above by alternately adding the vectors and then the vectors 
according to the parallelogram rule. The end of the vector is the desired point (see Fig. 9).
Solution. From Figure 10 it is clear that 
. Taking into account (8.1), we have: 
, 
. Using Corollary 7.1, we obtain:
Thus, in order to find the coordinates of a vector with known coordinates of its beginning and end, you need to subtract the coordinates of the beginning from the end coordinates.
Problem 2 ( on dividing a segment in a given ratio)
. Consider the segment , and 
. Let this segment be a point M is divided in the ratio . Find the coordinates of the point M.
Solution. From Figure 11 it is clear that the vector equality is true
.
Let's assume that the point M has coordinates. Finding the coordinates of the vectors using formula (8.2) 
and taking into account Theorem 7.1, we obtain the equalities:
Expressing from the first equality x, from the second - y, and from the third - z, find the coordinates of the point M:
In case, i.e. 
, we obtain the formula for the coordinates of the middle of the segment
Comment. On a plane (in two-dimensional space) you can also define a rectangular coordinate system Oxy. Using the introduced coordinate system, any point or its radius vector can be represented by a pair of numbers ( x, y). All the relationships that we obtained earlier for the coordinates of vectors and points of three-dimensional space will be valid on the plane with the only difference that we need to remove the third coordinate from them everywhere z. Similar reasoning can be repeated for an arbitrary line (one-dimensional space).
Projection of a vector onto an axis
Definition 9.1.Axis is a straight line with a unit vector (ort) lying on it, specifying a positive direction on the line.
In the figure, we will depict the axis as a directed straight line.
Let an axis be given in space l and period A, not belonging to the axis.
Definition 9.2. The base of a perpendicular dropped from a point A directly l, point, is called projection (orthogonal projection) of a point onto an axis.
In case the point A belongs to the axis l, then the projection of the point onto the axis coincides with the point itself A.
Let some vector be given. Finding the projections of the beginning and end of the vector onto the axis l, we obtain the vector , where - respectively, the projections of points A, IN per axis l.
Definition 9.3.Projection of the vector onto the l axis we will call a positive number equal to if the vector and axis l have the same directions (see Fig. 12) and a negative number if the vector and axis l are directed in the opposite direction (see Fig. 13).
Corollary 9.2. Projections of equal vectors onto the same axis are equal to each other.
Two-dimensional coordinate system
Dot P has coordinates (5,2).
The modern Cartesian coordinate system in two dimensions (also known as rectangular coordinate system) is given by two axes located at right angles to each other. The plane in which the axes are located is sometimes called xy-plane. The horizontal axis is denoted as x(x axis), vertical as y(ordinate axis). In three-dimensional space, up to two, a third axis is added, perpendicular to xy-plane- axis z. All points in the Cartesian coordinate system make up the so-called Cartesian space.
The intersection point where the axes meet is called origin and is denoted as O. Accordingly, the axis x can be designated as Ox, and the y axis is like Oy. Straight lines drawn parallel to each axis at a distance of a unit segment (unit of length measurement) starting from the origin of coordinates form coordinate grid.
A point in a two-dimensional coordinate system is specified by two numbers that determine the distance from the axis Oy(abscissa or x-coordinate) and from the axis Oh(ordinate or y-coordinate) respectively. Thus, the coordinates form an ordered pair (tuple) of numbers (x, y). In three-dimensional space, another z-coordinate is added (the distance of the point from the xy-plane), and an ordered triple of coordinates is formed (x, y, z).
The choice of letters x, y, z comes from the general rule of naming unknown quantities with the second half of the Latin alphabet. The letters of its first half are used to name known quantities.
The arrows on the axes reflect that they extend to infinity in that direction.
The intersection of the two axes creates four quadrants on the coordinate plane, which are designated by the Roman numerals I, II, III, and IV. Typically, the order of quadrant numbering is counterclockwise, starting from the upper right (i.e., where the abscissa and ordinate are positive numbers). The meanings of the abscissa and ordinate in each quadrant can be summarized in the following table:
| Quadrant | x | y |
| I | > 0 | > 0 |
| II | <0 | > 0 |
| III | <0 | <0 |
| IV | > 0 | <0 |
3D and n-dimensional coordinate system
In this figure, point P has coordinates (5,0,2) and point Q has coordinates (-5, -5,10)
Coordinates in three-dimensional space form a triple (x, y, z).
The x, y, z coordinates for a three-dimensional Cartesian system can be understood as the distances from a point to the corresponding planes: yz, xz, and xy.
The three-dimensional Cartesian coordinate system is very popular, as it corresponds to the usual ideas about spatial dimensions - height, width and length (that is, three dimensions). But depending on the area of application and the characteristics of the mathematical apparatus, the meaning of these three axes can be completely different.
Higher dimensional coordinate systems are also used (for example, the 4-dimensional system for depicting space-time in the special theory of relativity).
Cartesian coordinate system in the abstract n-dimensional space is a generalization of the above provisions and has n axes (each per dimension), which are mutually perpendicular. Accordingly, the position of a point in such a space will be determined by a tuple of n coordinates, or n-koy.
Equation of a line in (planimetry) in the canonical
form, parametric and general form.
These equations are called canonical equations of the line in space.
may be equal to zero, this means that the numerator of the corresponding fraction is also equal to zero.
If in (1) we enter the parameter t
|
then the equations of the line can be written in the form
If we introduce a coordinate system on a plane or in three-dimensional space, we will be able to describe geometric figures and their properties using equations and inequalities, that is, we will be able to use algebraic methods. Therefore, the concept of a coordinate system is very important.
In this article we will show how a rectangular Cartesian coordinate system is defined on a plane and in three-dimensional space and find out how the coordinates of points are determined. For clarity, we provide graphic illustrations.
Page navigation.
Rectangular Cartesian coordinate system on a plane.
Let us introduce a rectangular coordinate system on the plane.
To do this, draw two mutually perpendicular lines on the plane and select on each of them positive direction, indicating it with an arrow, and select on each of them scale(unit of length). Let us denote the point of intersection of these lines by the letter O and consider it starting point. So we got rectangular coordinate system on surface.
Each of the straight lines with a selected origin O, direction and scale is called coordinate line or coordinate axis.
A rectangular coordinate system on a plane is usually denoted by Oxy, where Ox and Oy are its coordinate axes. The Ox axis is called x-axis, and the Oy axis – y-axis.
Now let's agree on the image of a rectangular coordinate system on a plane.
Typically, the unit of measurement of length on the Ox and Oy axes is chosen to be the same and is plotted from the origin on each coordinate axis in the positive direction (marked with a dash on the coordinate axes and the unit is written next to it), the abscissa axis is directed to the right, and the ordinate axis is directed upward. All other options for the direction of the coordinate axes are reduced to the voiced one (Ox axis - to the right, Oy axis - up) by rotating the coordinate system at a certain angle relative to the origin and looking at it from the other side of the plane (if necessary).
The rectangular coordinate system is often called Cartesian, since it was first introduced on the plane by Rene Descartes. Even more commonly, a rectangular coordinate system is called a rectangular Cartesian coordinate system, putting it all together.
Rectangular coordinate system in three-dimensional space.
The rectangular coordinate system Oxyz is set in a similar way in three-dimensional Euclidean space, only not two, but three mutually perpendicular lines are taken. In other words, a coordinate axis Oz is added to the coordinate axes Ox and Oy, which is called axis applicate.
Depending on the direction of the coordinate axes, right and left rectangular coordinate systems in three-dimensional space are distinguished.
If viewed from the positive direction of the Oz axis and the shortest rotation from the positive direction of the Ox axis to the positive direction of the Oy axis occurs counterclockwise, then the coordinate system is called right.
If viewed from the positive direction of the Oz axis and the shortest rotation from the positive direction of the Ox axis to the positive direction of the Oy axis occurs clockwise, then the coordinate system is called left.
Coordinates of a point in a Cartesian coordinate system on a plane.
First, consider the coordinate line Ox and take some point M on it.
Each real number corresponds to a single point M on this coordinate line. For example, a point located on a coordinate line at a distance from the origin in the positive direction corresponds to the number , and the number -3 corresponds to a point located at a distance of 3 from the origin in the negative direction. The number 0 corresponds to the starting point.
On the other hand, each point M on the coordinate line Ox corresponds to a real number. This real number is zero if point M coincides with the origin (point O). This real number is positive and equal to the length of the segment OM on a given scale if point M is removed from the origin in the positive direction. This real number is negative and equal to the length of the segment OM with a minus sign if point M is removed from the origin in the negative direction.
The number is called coordinate points M on the coordinate line.
Now consider a plane with the introduced rectangular Cartesian coordinate system. Let us mark an arbitrary point M on this plane.
Let be the projection of point M onto the line Ox, and let be the projection of point M onto the coordinate line Oy (if necessary, see the article). That is, if through the point M we draw lines perpendicular to the coordinate axes Ox and Oy, then the points of intersection of these lines with the lines Ox and Oy are points and, respectively.
Let the number correspond to a point on the Ox coordinate axis, and the number to a point on the Oy axis.
Each point M of the plane in a given rectangular Cartesian coordinate system corresponds to a unique ordered pair of real numbers, called coordinates of point M on surface. The coordinate is called abscissa of point M, A - ordinate of point M.
The converse statement is also true: each ordered pair of real numbers corresponds to a point M on the plane in a given coordinate system.
Coordinates of a point in a rectangular coordinate system in three-dimensional space.
Let us show how the coordinates of point M are determined in a rectangular coordinate system defined in three-dimensional space.
Let and be the projections of point M onto the coordinate axes Ox, Oy and Oz, respectively. Let these points on the coordinate axes Ox, Oy and Oz correspond to real numbers and.
When introducing a coordinate system on a plane or in three-dimensional space, a unique opportunity appears to describe geometric figures and their properties using equations and inequalities. This has another name - algebra methods.
This article will help you understand the definition of a rectangular Cartesian coordinate system and the determination of the coordinates of points. A more clear and detailed image is available in graphic illustrations.
To introduce a coordinate system on a plane, you need to draw two perpendicular lines on the plane. Choose positive direction, indicated by an arrow. Must select scale. Let's call the point of intersection of the lines the letter O. She is considered starting point. This is called rectangular coordinate system on surface.
Lines with origin O having direction and scale are called coordinate line or coordinate axis.
The rectangular coordinate system is denoted O x y. The coordinate axes are called O x and O y, called respectively abscissa axis And ordinate axis.
Image of a rectangular coordinate system on a plane.
The abscissa and ordinate axes have the same unit of change and scale, which is shown as a prime at the origin of the coordinate axes. The standard direction of O x is from left to right, and O y is from bottom to top. Sometimes an alternative rotation at the required angle is used.
The rectangular coordinate system was called Cartesian in honor of its discoverer Rene Descartes. You can often find the name as a rectangular Cartesian coordinate system.
Three-dimensional Euclidean space has a similar system, only it consists not of two, but of three Ox, Oy, Oz axes. These are three mutually perpendicular lines, where O z is called applicator axis
According to the direction of the coordinate axes, they are divided into right and left rectangular coordinate systems of three-dimensional space.
The coordinate axes intersect at a point O, called the origin. Each axis has a positive direction, which is indicated by arrows on the axes. If, when O x is rotated counterclockwise by 90°, its positive direction coincides with positive O y, then this applies to the positive direction of O z. Such a system is considered right. In other words, if you compare the direction of X with the thumb, then the index finger is responsible for Y, and the middle finger for Z.
The left coordinate system is formed in a similar way. It is impossible to combine both systems, since the corresponding axes will not coincide.
To begin with, let's plot point M on the O x coordinate axis. Any real number x M is equal to the only point M located on a given line. If a point is located on the coordinate line at a distance of 2 from the origin in the positive direction, then it is equal to 2, if - 3, then the corresponding distance is 3. Zero is the origin of the coordinate lines.
In other words, each point M located on O x is equal to the real number x M . This real number is zero if point M is located at the origin, that is, at the intersection of O x and O y. The length number of a segment is always positive if the point is removed in the positive direction and vice versa.
The available number x M is called coordinate point M on a given coordinate line.
Let's take the point as a projection of the point M x onto O x, and as a projection of the point M y onto O y. This means that through the point M we can draw straight lines perpendicular to the O x and O y axes, where we obtain the corresponding intersection points M x and M y.
Then the point M x on the O x axis has the corresponding number x M, and M y on O y - y M. On the coordinate axes it looks like this:
Each point M on a given plane in a rectangular Cartesian coordinate system has one corresponding pair of numbers (x M, y M), called its coordinates. Abscissa M– this is x M, ordinate M– this is y M .
The converse is also true: every ordered pair (x M, y M) has a corresponding point defined in the plane.
Determination of point M in three-dimensional space. Let there be M x, M y, M z, which are projections of the point M onto the corresponding axes O x, O y, O z. Then the values of these points on the O x, O y, O z axes will take on the values x M, y M, z M. Let's depict this on coordinate lines.
To obtain the projections of point M, it is necessary to add perpendicular straight lines O x, O y, O z, continue and depict them in the form of planes that pass through M. Thus, the planes will intersect at M x , M y , M z
Each point in three-dimensional space has its own data (x M, y M, z M), which are called coordinates of the point M, x M, y M, z M - these are numbers called abscissa, ordinate And applicate given point M. For this judgment, the converse statement is also true: each ordered triple of real numbers (x M, y M, z M) in a given rectangular coordinate system has one corresponding point M of three-dimensional space.
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A rectangular coordinate system on a 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 the thumb of the right hand is taken as the X direction, the index finger as the Y direction, and the middle finger 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 the three-dimensional case, such unit 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.
see also
Links
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See what “Cartesian coordinate system” is in other dictionaries:
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 (see DESCARTES Rene). Descartes first introduced... encyclopedic Dictionary
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
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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
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- Computational fluid dynamics. Theoretical basis. Textbook, Pavlovsky Valery Alekseevich, Nikushchenko Dmitry Vladimirovich. The book is devoted to a systematic presentation of the theoretical foundations for posing problems of mathematical modeling of flows of liquids and gases. Particular attention is paid to the issues of constructing...