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 algebra 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.
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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.
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 the general case, on a plane, a Cartesian coordinate system (affine coordinate system) is specified by the point O (the origin) and an ordered pair of vectors e 1 and e 2 (basic 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 the values of the orthogonal projections of point M on the Ox and Oy axes, 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 appended to Geometry, in which for the first time both positive and negative coordinate values were allowed. 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.
Ministry of Education and Science of the Russian Federation
FSBEI HPE "Mari" State University»
Department of Pedagogy
abstract
Discipline: methods of teaching mathematics
on the topic of: "Cartesian coordinate system"
Performed:
Viktorova O.K.
Checked:
Ph.D. ped. sciences, professor
Borodina M.V.
Yoshkar-Ola
2015
- Rene Descartes. Biography…………………………………………………………….3
- Descartes' contribution to the development of mathematics as a science…………………….6
- Possible method studying the Cartesian coordinate system using the example of the legend of its discovery………………………………………………………8
- Conclusion……………………………………………………………15
- List of references………………………………………………………..16
- Biography
Rene Descartes French philosopher, mathematician, mechanic, physicist and physiologist, creator of analytical geometry and modern algebraic symbolism, author of the method of radical doubt in philosophy, mechanism in physics, forerunner of reflexology.
Descartes came from an old but impoverished noble family of de Cartes; from here his Latinized name Cartesius and the direction in philosophy - Cartesianism - subsequently arose; and was the youngest (third) son in the family. He was born on March 31, 1596 in Lae, France. His mother died when he was 1 year old. Descartes' father was a judge in the city of Rennes and rarely appeared in Lae; The boy was raised by his maternal grandmother. As a child, Rene was distinguished by fragile health and incredible curiosity.
Elementary education Descartes received his studies at the Jesuit college La Flèche, where his teacher was Jean François. At college, Descartes met Marin Mersenne (then a student, later a priest), the future coordinator scientific life France. Religious education only strengthened the young Descartes's skeptical attitude towards the philosophical authorities of that time. Later he formulated his method of cognition: deductive (mathematical) reasoning over the results of reproducible experiments.
In 1612, Descartes graduated from college, studied law for some time in Poitiers, then went to Paris, where for several years he alternated between an absent-minded life and mathematical studies. Then he entered military service(1617) first in revolutionary Holland (in those years an ally of France), then in Germany, where he participated in the short battle for Prague (Thirty Years' War). In Holland in 1618, Descartes met the outstanding physicist and natural philosopher Isaac Beckmann, who had a significant influence on his formation as a scientist. Descartes spent several years in Paris, indulging in scientific work, where, among other things, he discovered the principle of virtual speeds, which at that time no one was yet ready to appreciate.
Then several more years of participation in the war (the siege of La Rochelle). Upon returning to France, it turned out that Descartes' freethinking became known to the Jesuits, and they accused him of heresy. Therefore, Descartes moved to Holland (1628), where he spent 20 years in solitary scientific studies.
He maintains extensive correspondence with the best scientists in Europe (through the faithful Mersenne) and studies a variety of sciences, from medicine to meteorology. Finally, in 1634, he completed his first programmatic book entitled “The World” (Le Monde), consisting of two parts: “Treatise on Light” and “Treatise on Man”. But the moment for publication was unfortunate: a year earlier, the Inquisition almost tortured Galileo. Therefore, Descartes decided not to publish this work during his lifetime. He wrote to Mersenne about Galileo's condemnation:
“This struck me so much that I decided to burn all my papers, or at least not show them to anyone; for I was not able to imagine that he, an Italian, who enjoyed the favor of even the Pope, could be condemned for, without a doubt, wanting to prove the movement of the Earth... I confess, if the movement of the Earth is a lie, then all the foundations of my philosophy are lies, for they clearly lead to the same conclusion.”
Soon, however, one after another, other books of Descartes appear:
“Discourse on Method...” (1637)
"Reflections on First Philosophy..." (1641)
"Principles of Philosophy" (1644)
Descartes’ main theses are formulated in the “Principles of Philosophy”:
“God created the world and the laws of nature, and then the Universe acts as an independent mechanism.”
“There is nothing in the world except moving matter various types. Matter consists of elementary particles, the local interaction of which produces all natural phenomena.”
"Mathematics powerful and universal method knowledge of nature, a model for other sciences."
Cardinal Richelieu reacted favorably to the works of Descartes and allowed their publication in France, but the Protestant theologians of Holland placed a curse on them (1642); Without the support of the Prince of Orange, the scientist would have had a hard time.
In 1649, Descartes, exhausted by many years of persecution for freethinking, succumbed to the persuasion of the Swedish Queen Christina (with whom he actively corresponded for many years) and moved to Stockholm. Almost immediately after moving, he caught a serious cold and soon died. The suspected cause of death was pneumonia. There is also a hypothesis about his poisoning, since the symptoms of Descartes' disease were similar to those arising from acute arsenic poisoning. This hypothesis was put forward by Ikey Pease, a German scientist, and then supported by Theodor Ebert. The reason for the poisoning, according to this version, was the fear of Catholic agents that Descartes' freethinking might interfere with their efforts to convert Queen Christina to Catholicism (this conversion actually occurred in 1654).
Towards the end of Descartes' life, the church's attitude towards his teachings became sharply hostile. Soon after his death, the main works of Descartes were included in the notorious "Index", and Louis XIV, by a special decree, banned the teaching of Descartes' philosophy ("Cartesianism") in all educational institutions France.
- Descartes' contribution to the development of mathematics as a science
In 1637, Descartes’s main philosophical and mathematical work, “Discourse on Method” (full title: “Discourse on a method that allows you to direct your mind and find truth in the sciences”) was published.
This book presented analytical geometry, and in its applications numerous results in algebra, geometry, optics (including correct wording law of light refraction) and much more.
Of particular note is the mathematical symbolism of Vieta, which he reworked, which from that moment was close to modern. He denoted the coefficients as a, b, c..., and the unknowns as x, y, z. The natural exponent is taken modern look(fractional and negative ones were established thanks to Newton). A line appears over the radical expression. The equations are reduced to canonical form (zero on the right side).
Descartes called symbolic algebra “Universal Mathematics,” and wrote that it should explain “everything pertaining to order and measure.”
The creation of analytical geometry made it possible to translate the study of the geometric properties of curves and bodies into algebraic language, that is, to analyze the equation of a curve in a certain coordinate system. This translation had the disadvantage that now it was necessary to carefully determine the true geometric properties that do not depend on the coordinate system (invariants). However, the advantages of the new method were exceptionally great, and Descartes demonstrated them in the same book, discovering many provisions unknown to ancient and contemporary mathematicians.
The “Geometry” appendix provided methods for solving algebraic equations (including geometric and mechanical) and classification of algebraic curves. New way defining a curve using the equation was a decisive step towards the concept of function. Descartes formulates a precise "rule of signs" for determining the number of positive roots of an equation, although he does not prove it.
Descartes studied algebraic functions (polynomials), as well as a number of “mechanical” ones (spirals, cycloids). For transcendental functions, according to Descartes, general method research does not exist.
Complex numbers were not yet considered by Descartes on equal terms with real ones, but he formulated (although did not prove) the fundamental theorem of algebra: total number real and complex roots of a polynomial is equal to its degree. Descartes traditionally called negative roots false, but combined them with positive ones under the term real numbers, separating them from imaginary (complex) ones. This term entered mathematics. However, Descartes showed some inconsistency: the coefficients a, b, c... were considered positive for him, and the case of an unknown sign was specially marked with an ellipsis on the left.
All non-negative real numbers, not excluding irrational ones, are considered by Descartes as equal; they are defined as the ratio of the length of a certain segment to a length standard. Later, Newton and Euler adopted a similar definition of number. Descartes does not yet separate algebra from geometry, although he changes their priorities; he understands solving an equation as constructing a segment with a length equal to the root of the equation. This anachronism was soon discarded by his students, primarily the English ones, for whom geometric constructions are a purely auxiliary device.
The book “Method” immediately made Descartes a recognized authority in mathematics and optics. It is noteworthy that it was published in French and not in Latin. The application “Geometry” was, however, immediately translated into Latin and was published separately several times, growing from comments and becoming reference book European scientists. The works of mathematicians of the second half of the 17th century reflect the strong influence of Descartes.
- A possible method for studying the Cartesian coordinate system using the example of the legend of its discovery
There are several legends about the invention of the coordinate system, which bears the name of Descartes.
One day, Rene Descartes lay in bed all day, thinking about something, and a fly buzzed around and did not allow him to concentrate. He began to think about how to describe the position of a fly at any given time mathematically in order to be able to swat it without missing. And... came up with Cartesian coordinates, one of greatest inventions in the history of mankind. Let's follow the path of opening the coordinate system according to this legend in pictures.
Opening time: 1637.
Scene: Rene Descartes's "office".
The figure roughly shows three walls of the office:
wall with doorway
Profile plane
floor - horizontal plane
wall with window openings
Frontal plane;
Note!Every two planes intersect in a straight line
lines.
- A fly lands on the frontal plane
- Let's pretend that
Rene Descartes looks at
frontal plane in
perpendicular to it
direction.
We see that fly
is located
frontal plane.
But how to accurately determine
her position?
- Eureka!
You need to take two mutually perpendicular number lines. We denote the point of intersection of the lines as O - the origin of the coordinate system. Let's call one of the lines the X axis, the other the Y axis.
In our figure, the distance between divisions on the number lines
equals one.
Attention! You can select the origin and direction of the axes
in a way that is convenient for a specific task.
- Let us determine the exact position of the “co-author” - the fly.
Let's draw two straight lines through the point where the fly is located:
- Parallel to the X axis. The straight line intersects the Y axis at a point with a numerical
value equal to 4. Let’s call this value the “y” coordinate of our
- Parallel to the Y axis. The straight line intersects the X axis at a point with a numerical
value equal to (-2). Let's call this value the "x" coordinate of our object.
It is customary to write the coordinates of an object, usually a point, in the form (x, y). For our fly, we can say that it is located at the point with coordinates (-2, 4).
The problem of accurately determining the position of the fly is solved!
The novelty of the idea is that the position of a point or object on
The plane is defined using two intersecting axes.
The same can be done to determine the position of the fly on
ceiling.
Determine the position of the beetle and butterfly on the coordinate plane.
All these examples demonstrate the advantages of the coordinate method of determining the position of a fly, beetle and butterfly on a plane using the Descartes coordinate system. How can we determine the coordinates of the same insects if they fly, because in this case they do not crawl along the surface of a wall or ceiling.
To measure the position of objects in space in the early 19th century
a Z axis has been added, which is directed perpendicular to the X and Y axes.
In the figure, the Z axis is directed upward.
Imagine that an Amur cat is sitting on a tree branch.
If the cat fell onto a horizontal plane - the XOY plane, point
its fall had coordinates (X1, Y1). The cat sits at a height Z1 from the horizontal plane. So, the position of the Amur cat in space
can be described by three coordinates (X1, Y1 Z1), it is located at some
height above the ground.
Coordinates can have different numerical values, including
zero, this means that the object is located on some coordinate axis.
If all three coordinates have zero values, the object is at the origin of the coordinate system.
Let's determine the coordinates of various objects in the following
drawing.
The parrot is at the point with coordinates(0, 0, Z1) .
The beaver on the left is (X1 0 0) . Beaver on the right - (0 Y1 0) .
Mouse - (X1 Y1 0) . Amur cat - (X1 Y1 Z1).
Answer the question:
“Where should this chameleon sit?”
- Conclusion
The Cartesian coordinate system pushed the science of mathematics and brought it to a completely new level. Geometry began to develop more rapidly. This work examines the coordinate system at the 5th-6th grade level so that children become interested and, most importantly, understand how to work with the coordinate system. Of course, in the future the study of the Cartesian coordinate system will be more in-depth. In higher grades we will talk about three-dimensional space. About the construction of three-dimensional figures, etc. The study of the Cartesian coordinate system is one of the most important aspects mathematics as a science, and every teacher must convey his knowledge to every student so that this knowledge is learned for life.
- Bibliography
- Lyubimov N.A. Philosophy of Descartes. St. Petersburg, 1886
- Lyat-ker Ya.A. Descartes. M., 1975
- Fischer K. Descartes: his life, writings and teachings. St. Petersburg, 1994
- Mamardashvili M.K. Cartesian reflections. M., 1995
- Sites used: https://ru.wikipedia.org
To determine the position of a point in space, we will use Cartesian rectangular coordinates (Fig. 2).
The Cartesian rectangular coordinate system in space is formed by three mutually perpendicular coordinate axes OX, OY, 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 usually (not necessarily) the same for all axes. The OX axis is called the abscissa axis (or simply abscissa), the OY axis is the ordinate axis, and the OZ axis is the applicate axis.
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 a point 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, and the z coordinate is called the applicate of point A.
Symbolically it is written like this:
or link a coordinate record to a specific point using an index:
x A , y A , z A ,
Each axis is considered as a number line, that is, it has a positive direction, and points lying on a negative ray are assigned negative coordinate values (the distance is taken with a minus sign). That is, if, for example, point B lay not as in the figure - on the ray OX, but on its continuation in reverse side from point O (on the negative part of the axis OX), then the x abscissa of point A would be negative (minus the distance OB). Likewise for the other two axes.
Coordinate axes OX, OY, OZ, shown in Fig. 2, form a right-handed coordinate system. This means that if you look at the YOZ plane along the positive direction of the OX axis, then the movement of the OY axis towards the OZ axis will be clockwise. This situation can be described using the gimlet rule: if the gimlet (screw with a right-hand thread) is rotated in the direction from the OY axis to the OZ axis, then it will move along the positive direction of the OX axis.
Vectors of unit length directed along the coordinate axes are called coordinate unit vectors. They are usually designated as 
(Fig. 3). There is also the designation 
The unit vectors form the basis of the coordinate system.
In the case of a right-handed coordinate system, the following formulas with vector products of unit vectors are valid:
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 a fixed reference system. 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. a modern understanding of the coordinate system was formed, which received the name of 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 into two mutually perpendicular coordinate axes.
In a two-dimensional coordinate system, the horizontal axis is called the x-axis (axis OX), the vertical axis is the ordinate axis (OY 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 in three-dimensional space, the 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 are called (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).
Unit vectors can be specified on the coordinate axes 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 to the 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.
encyclopedic Dictionary. 2009 .
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
The rectangular or Cartesian coordinate system is the most common coordinate system on the plane and in space. Contents 1 Rectangular coordinate system on a plane ... Wikipedia
Cartesian coordinate system
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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
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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
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