Coordinate plane with coordinates. Coordinate planes and graphs
Mathematics is a rather complex science. Studying it, one has not only to solve examples and problems, but also to work with various figures, and even planes. One of the most used in mathematics is the coordinate system on the plane. Children have been taught how to work with it correctly for more than one year. Therefore, it is important to know what it is and how to work with it correctly.
Let's figure out what this system is, what actions you can perform with it, and also find out its main characteristics and features.
Concept definition
A coordinate plane is a plane on which a particular coordinate system is defined. Such a plane is defined by two straight lines intersecting at a right angle. The point of intersection of these lines is the origin of coordinates. Each point on the coordinate plane is given by a pair of numbers, which are called coordinates.
In a school mathematics course, students have to work quite closely with a coordinate system - build figures and points on it, determine which plane a particular coordinate belongs to, and also determine the coordinates of a point and write or name them. Therefore, let's talk in more detail about all the features of the coordinates. But first, let's touch on the history of creation, and then we'll talk about how to work on the coordinate plane.
History reference
Ideas about creating a coordinate system were in the days of Ptolemy. Even then, astronomers and mathematicians were thinking about how to learn how to set the position of a point on a plane. Unfortunately, at that time there was no coordinate system known to us, and scientists had to use other systems.
Initially, they set points by specifying latitude and longitude. For a long time it was one of the most used ways of mapping this or that information. But in 1637, Rene Descartes created his own coordinate system, later named after "Cartesian".
Already at the end of the XVII century. the concept of "coordinate plane" has become widely used in the world of mathematics. Despite the fact that several centuries have passed since the creation of this system, it is still widely used in mathematics and even in life.
Coordinate plane examples
Before talking about the theory, we will give some illustrative examples of the coordinate plane so that you can imagine it. The coordinate system is primarily used in chess. On the board, each square has its own coordinates - one letter coordinate, the second - digital. With its help, you can determine the position of a particular piece on the board.
The second most striking example is the beloved game "Battleship". Remember how, when playing, you name a coordinate, for example, B3, thus indicating exactly where you are aiming. At the same time, when placing the ships, you set points on the coordinate plane.
This coordinate system is widely used not only in mathematics, logic games, but also in military affairs, astronomy, physics and many other sciences.
Coordinate axes
As already mentioned, two axes are distinguished in the coordinate system. Let's talk a little about them, as they are of considerable importance.
The first axis - abscissa - is horizontal. It is denoted as ( Ox). The second axis is the ordinate, which passes vertically through the reference point and is denoted as ( Oy). It is these two axes that form the coordinate system, dividing the plane into four quarters. The origin is located at the intersection point of these two axes and takes on the value 0 . Only if the plane is formed by two axes that intersect perpendicularly and have a reference point, is it a coordinate plane.
Also note that each of the axes has its own direction. Usually, when constructing a coordinate system, it is customary to indicate the direction of the axis in the form of an arrow. In addition, when constructing the coordinate plane, each of the axes is signed.
quarters
Now let's say a few words about such a concept as quarters of the coordinate plane. The plane is divided by two axes into four quarters. Each of them has its own number, while the numbering of the planes is counterclockwise.
Each of the quarters has its own characteristics. So, in the first quarter, the abscissa and the ordinate are positive, in the second quarter, the abscissa is negative, the ordinate is positive, in the third, both the abscissa and the ordinate are negative, in the fourth, the abscissa is positive, and the ordinate is negative.
By remembering these features, you can easily determine which quarter a particular point belongs to. In addition, this information may be useful to you if you have to do calculations using the Cartesian system.
Working with the coordinate plane
When we have dealt with the concept of a plane and talked about its quarters, we can move on to such a problem as working with this system, and also talk about how to put points, coordinates of figures on it. On the coordinate plane, this is not as difficult as it might seem at first glance.
First of all, the system itself is built, all important designations are applied to it. Then there is work directly with points or figures. In this case, even when constructing figures, points are first applied to the plane, and then the figures are already drawn.
Rules for constructing a plane
If you decide to start marking shapes and points on paper, you will need a coordinate plane. The coordinates of the points are plotted on it. In order to build a coordinate plane, you only need a ruler and a pen or pencil. First, the horizontal abscissa is drawn, then the vertical - ordinate. It is important to remember that the axes intersect at right angles.
The next obligatory item is marking. Units-segments are marked and signed on each of the axes in both directions. This is done so that you can then work with the plane with maximum convenience.
Marking a point
Now let's talk about how to plot the coordinates of points on the coordinate plane. This is the basics you need to know to successfully place a variety of shapes on the plane, and even mark equations.
When constructing points, one should remember how their coordinates are correctly recorded. So, usually setting a point, two numbers are written in brackets. The first digit indicates the coordinate of the point along the abscissa axis, the second - along the ordinate axis.
The point should be built in this way. Mark on axis first Ox given point, then mark a point on the axis Oy. Next, draw imaginary lines from these designations and find the place of their intersection - this will be the given point.
All you have to do is mark it and sign it. As you can see, everything is quite simple and does not require special skills.
Placing a Shape
Now let's move on to such a question as the construction of figures on the coordinate plane. In order to build any figure on the coordinate plane, you should know how to place points on it. If you know how to do this, then placing a figure on a plane is not so difficult.
First of all, you will need the coordinates of the points of the figure. It is on them that we will apply the ones you have chosen to our coordinate system. Let's consider drawing a rectangle, triangle and circle.
Let's start with a rectangle. Applying it is pretty easy. First, four points are applied to the plane, indicating the corners of the rectangle. Then all points are sequentially connected to each other.
Drawing a triangle is no different. The only thing is that it has three corners, which means that three points are applied to the plane, denoting its vertices.
Regarding the circle, here you should know the coordinates of two points. The first point is the center of the circle, the second is the point denoting its radius. These two points are plotted on a plane. Then a compass is taken, the distance between two points is measured. The point of the compass is placed at a point denoting the center, and a circle is described.
As you can see, there is also nothing complicated here, the main thing is that there is always a ruler and a compass at hand.
Now you know how to plot shape coordinates. On the coordinate plane, this is not so difficult to do, as it might seem at first glance.
conclusions
So, we have considered with you one of the most interesting and basic concepts for mathematics that every student has to deal with.
We have found out that the coordinate plane is the plane formed by the intersection of two axes. With its help, you can set the coordinates of points, put shapes on it. The plane is divided into quarters, each of which has its own characteristics.
The main skill that should be developed when working with the coordinate plane is the ability to correctly plot given points on it. To do this, you should know the correct location of the axes, the features of the quarters, as well as the rules by which the coordinates of the points are set.
We hope that the information provided by us was accessible and understandable, and was also useful for you and helped to better understand this topic.
The topic of this video lesson: Coordinate plane.
Goals and objectives of the lesson:
Acquainted with rectangular coordinate system on the plane
- learn to freely navigate on the coordinate plane
- build points according to its given coordinates
- determine the coordinates of a point marked on the coordinate plane
- well perceive the coordinates by ear
- accurately and accurately perform geometric constructions
- development of creative abilities
- raising interest in the subject
The term " coordinates"Derived from the Latin word -" ordered "
To indicate the position of a point on a plane, two perpendicular lines X and Y are taken.
X axis - abscissa
Y-axis y-axis
Point O - origin
The plane on which the coordinate system is given is called coordinate plane.
Each point M on the coordinate plane corresponds to a pair of numbers: its abscissa and ordinate. On the contrary, each pair of numbers corresponds to one point of the plane for which these numbers are coordinates.
Examples considered:
- by constructing a point by its coordinates
- finding the coordinates of a point located on the coordinate plane
Some additional information:
The idea to set the position of a point on a plane originated in antiquity - primarily among astronomers. In the II century. The ancient Greek astronomer Claudius Ptolemy used latitude and longitude as coordinates. A description of the use of coordinates was given in the book "Geometry" in 1637.
The description of the use of coordinates was given in the book "Geometry" in 1637 by the French mathematician Rene Descartes, so the rectangular coordinate system is often called Cartesian.
The words " abscissa», « ordinate», « coordinates» first began to use at the end of XVII.
For a better understanding of the coordinate plane, let's imagine that we are given: a geographical globe, a chessboard, a theater ticket.
To determine the position of a point on the earth's surface, you need to know the longitude and latitude.
To determine the position of a piece on a chessboard, you need to know two coordinates, for example: e3.
Seats in the auditorium are determined by two coordinates: row and seat.
Additional task.
After studying the video lesson, to consolidate the material, I suggest you take a pen and a piece of paper in a box, draw a coordinate plane and build shapes according to the given coordinates:
Fungus
1) (6; 0), (6; 2), (5; 1,5), (4; 3), (2; 1), (0; 2,5), (- 1,5; 1,5), (- 2; 5), (- 3; 0,5), (- 4; 2), (- 4; 0).
2) (2; 1), (2,2; 2), (2,3; 4), (2,5; 6), (2,3; 8), (2; 10), (6; 10), (4,8; 12), (3; 13,3), (1; 14),
(0; 14), (- 2; 13,3), (- 3,8; 12), (- 5; 10), (2; 10).
3) (- 1; 10), (- 1,3; 8), (- 1,5; 6), (- 1,2; 4), (- 0,8;2).
little mouse 1) (3; - 4), (3; - 1), (2; 3), (2; 5), (3; 6), (3; 8), (2; 9), (1; 9), (- 1; 7), (- 1; 6),
(- 4; 4), (- 2; 3), (- 1; 3), (- 1; 1), (- 2; 1), (-2; - 1), (- 1; 0), (- 1; - 4), (- 2; - 4),
(- 2; - 6), (- 3; - 6), (- 3; - 7), (- 1; - 7), (- 1; - 5), (1; - 5), (1; - 6), (3; - 6), (3; - 7),
(4; - 7), (4; - 5), (2; - 5), (3; - 4).
2) Tail: (3; - 3), (5; - 3), (5; 3).
3) Eye: (- 1; 5).
Swan
1) (2; 7), (0; 5), (- 2; 7), (0; 8), (2; 7), (- 4; - 3), (4; 0), (11; - 2), (9; - 2), (11; - 3),
(9; - 3), (5; - 7), (- 4; - 3).
2) Beak: (- 4; 8), (- 2; 7), (- 4; 6).
3) Wing: (1; - 3), (4; - 2), (7; - 3), (4; - 5), (1; - 3).
4) Eye: (0; 7).
Camel
1) (- 9; 6), (- 5; 9), (- 5; 10), (- 4; 10), (- 4; 4), (- 3; 4), (0; 7), (2; 4), (4; 7), (7; 4),
(9; 3), (9; 1), (8; - 1), (8; 1), (7; 1), (7; - 7), (6; - 7), (6; - 2), (4; - 1), (- 5; - 1), (- 5; - 7),
(- 6; - 7), (- 6; 5), (- 7;5), (- 8; 4), (- 9; 4), (- 9; 6).
2) Eye: (- 6; 7).
Elephant
1) (2; - 3), (2; - 2), (4; - 2), (4; - 1), (3; 1), (2; 1), (1; 2), (0; 0), (- 3; 2), (- 4; 5),
(0; 8), (2; 7), (6; 7), (8; 8), (10; 6), (10; 2), (7; 0), (6; 2), (6; - 2), (5; - 3), (2; - 3).
2) (4; - 3), (4; - 5), (3; - 9), (0; - 8), (1; - 5), (1; - 4), (0; - 4), (0; - 9), (- 3; - 9),
(- 3; - 3), (- 7; - 3), (- 7; - 7), (- 8; - 7), (- 8; - 8), (- 11; - 8), (- 10; - 4), (- 11; - 1),
(- 14; - 3), (- 12; - 1), (- 11;2), (- 8;4), (- 4;5).
3) Eyes: (2; 4), (6; 4).
Horse
1) (14; - 3), (6,5; 0), (4; 7), (2; 9), (3; 11), (3; 13), (0; 10), (- 2; 10), (- 8; 5,5),
(- 8; 3), (- 7; 2), (- 5; 3), (- 5; 4,5), (0; 4), (- 2; 0), (- 2; - 3), (- 5; - 1), (- 7; - 2),
(- 5; - 10), (- 2; - 11), (- 2; - 8,5), (- 4; - 8), (- 4; - 4), (0; - 7,5), (3; - 5).
2) Eye: (- 2; 7).
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 of coordinates, a positive direction is chosen on each axis. The positive direction of the axes (in the 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 X'X and Y'Y coordinate axes 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 OB segment, the y-coordinate is the length of the OC segment in the selected units. 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. They write it like this: A (x, y).
If point A lies in coordinate angle I, then point A has 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 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 the point O, which is called the origin, on each axis the positive direction indicated by the arrows is chosen, and the unit of measurement of the segments on the axes. The units of measure 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 side of the positive direction of the OZ axis. Such a coordinate system is called right. 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 coordinate system is formed. Similar fingers of the left hand form the left coordinate system. The right and left coordinate systems cannot be combined 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 equal to the length of the segment OC, the z coordinate is the length of the segment OD in the selected units. 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. They write it like this: A (a, b, c).
Horts
A rectangular coordinate system (of any dimension) is also described by a set of orts , co-directed with the coordinate axes. The number of orts is equal to the dimension of the coordinate system, and they are all perpendicular to each other.
In the 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 the right coordinate system, the following formulas with the vector product of vectors are valid:
- [i j]=k ;
- [j k]=i ;
- [k i]=j .
Story
René Descartes was the first to introduce a rectangular coordinate system in his Discourse on the Method in 1637. Therefore, the rectangular coordinate system is also called - Cartesian coordinate system. The coordinate method for describing geometric objects laid the foundation for analytical geometry. Pierre Fermat also contributed to the development of the coordinate method, but his work was 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 applied by Leonhard Euler already in the 18th century.
see also
Links
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See what the "Coordinate plane" is in other dictionaries:
cutting plane- (Pn) Coordinate plane tangent to the cutting edge at the considered point and perpendicular to the base plane. […
In topography, a network of imaginary lines encircling the globe in the latitudinal and meridional directions, with which you can accurately determine the position of any point on the earth's surface. Latitudes are measured from the equator - a great circle, ... ... Geographic Encyclopedia
In topography, a network of imaginary lines encircling the globe in the latitudinal and meridional directions, with which you can accurately determine the position of any point on the earth's surface. Latitudes are measured from the equator of the great circle, ... ... Collier Encyclopedia
This term has other meanings, see Phase diagram. The phase plane is a coordinate plane in which any two variables (phase coordinates) are plotted along the coordinate axes, which uniquely determine the state of the system ... ... Wikipedia
principal cutting plane- (Pτ) Coordinate plane perpendicular to the line of intersection of the main plane and the cutting plane. [GOST 25762 83] Topics of cutting Generalizing terms systems of coordinate planes and coordinate planes … Technical Translator's Handbook
instrumental principal cutting plane- (Pτi) Coordinate plane perpendicular to the line of intersection of the instrumental main plane and the cutting plane. [GOST 25762 83] Topics of cutting Generalizing terms systems of coordinate planes and coordinate planes … Technical Translator's Handbook
tool cutting plane- (Pni) Coordinate plane tangent to the cutting edge at the point in question and perpendicular to the instrument base plane. [GOST 25762 83] Topics for cutting Generalizing terms for systems of coordinate planes and ... ... Technical Translator's Handbook
kinematic principal cutting plane- (Pτк) Coordinate plane perpendicular to the line of intersection of the kinematic main plane and the cutting plane ... Technical Translator's Handbook
kinematic cutting plane- (Pnk) Coordinate plane tangent to the cutting edge at the point under consideration and perpendicular to the kinematic base plane ... Technical Translator's Handbook
main plane- (Pv) A coordinate plane drawn through the considered point of the cutting edge perpendicular to the direction of the velocity of the main or net cutting motion at that point. Note In the instrumental coordinate system, the direction ... ... Technical Translator's Handbook
The points are “registered” - “residents”, each point has its own “house number” - its coordinate. If the point is taken in a plane, then for its “registration” it is necessary to indicate not only the “house number”, but also the “apartment number”. Recall how this is done.
Let us draw two mutually perpendicular coordinate lines and consider the point of their intersection, point O, to be the starting point on both lines. Thus, a rectangular coordinate system is set on the plane (Fig. 20), which transforms the usual plane to coordinate. The point O is called the origin of coordinates, the coordinate lines (x-axis and y-axis) are called coordinate axes, and the right angles formed by the coordinate axes are called coordinate angles. Coordinate rectangular corners are numbered as shown in Figure 20.
And now let's turn to Figure 21, which shows a rectangular coordinate system and marked the point M. Let's draw a straight line through it parallel to the y axis. The line intersects the x-axis at some point, this point has a coordinate - on the x-axis. For the point shown in Figure 21, this coordinate is -1.5, it is called the abscissa of the point M. Next, we draw a straight line through the point M parallel to the x axis. The line intersects the y-axis at some point, this point has a coordinate - on the y-axis.
For point M, shown in Figure 21, this coordinate is 2, it is called the ordinate of the point M. Briefly written like this: M (-1.5; 2). The abscissa is written in the first place, the ordinate - in the second. They use, if necessary, another form of notation: x = -1.5; y = 2.
Remark 1 . In practice, to find the coordinates of the point M, usually instead of straight lines parallel to the coordinate axes and passing through the point M, segments of these lines are built from the point M to the coordinate axes (Fig. 22).
Remark 2. In the previous section, we introduced different notations for numerical intervals. In particular, as we agreed, the notation (3, 5) means that an interval with ends at points 3 and 5 is considered on the coordinate line. In this section, we consider a pair of numbers as coordinates of a point; for example, (3; 5) is a point on coordinate plane with the abscissa 3 and the ordinate 5. How is it correct to determine from the symbolic notation what is at stake: about the interval or about the coordinates of the point? Most of the time this is clear from the text. What if it's not clear? Pay attention to one detail: we used a comma in the interval designation, and a semicolon in the coordinate designation. This, of course, is not very significant, but still the difference; we will apply it.
Given the introduced terms and notation, the horizontal coordinate line is called the abscissa, or x-axis, and the vertical coordinate line is called the y-axis, or y-axis. The designations x, y are usually used when specifying a rectangular coordinate system on the plane (see Fig. 20) and they often say this: the xOy coordinate system is given. However, there are other designations: for example, in Figure 23, the coordinate system tOs is given.
Algorithm for finding the coordinates of the point M, given in the rectangular coordinate system хОу
This is exactly how we acted, finding the coordinates of the point M in Figure 21. If the point M 1 (x; y) belongs to the first coordinate angle, then x\u003e 0, y\u003e 0; if the point M 2 (x; y) belongs to the second coordinate angle, then x< 0, у >0; if the point M 3 (x; y) belongs to the third coordinate angle, then x< О, у < 0; если точка М 4 (х; у) принадлежит четвертому координатному углу, то х >OU< 0 (рис. 24).
But what happens if the point whose coordinates need to be found lies on one of the coordinate axes? Let point A lie on the x-axis, and point B lie on the y-axis (Fig. 25). It does not make sense to draw a straight line parallel to the y-axis through point A and find the point of intersection of this line with the x-axis, since such an intersection point already exists - this is point A, its coordinate (abscissa) is 3. In the same way, you do not need to draw through the point And the line parallel to the x-axis - this line is the x-axis itself, which intersects the y-axis at point O with coordinate (ordinate) 0. As a result, for point A we get A (3; 0). Similarly, for point B we get B(0; - 1.5). And for the point O we have O(0; 0).
In general, any point on the x-axis has coordinates (x; 0), and any point on the y-axis has coordinates (0; y)
So, we discussed how to find the coordinates of a point in the coordinate plane. But how to solve the inverse problem, i.e., how, having given the coordinates, to construct the corresponding point? To develop an algorithm, we will carry out two auxiliary, but at the same time important arguments.
First discussion. Let I be drawn in the xOy coordinate system, parallel to the y axis and intersecting the x axis at a point with coordinate (abscissa) 4
(Fig. 26). Any point lying on this line has an abscissa 4. So, for points M 1, M 2, M 3 we have M 1 (4; 3), M 2 (4; 6), M 3 (4; - 2). In other words, the abscissa of any point M of the straight line satisfies the condition x \u003d 4. They say that x \u003d 4 - the equation line l or that line I satisfies the equation x = 4.
Figure 27 shows lines that satisfy the equations x = - 4 (line I 1), x = - 1
(straight line I 2) x = 3.5 (straight line I 3). And which line satisfies the equation x = 0? Guessed? y axis
Second discussion. Let a straight line I be drawn in the xOy coordinate system, parallel to the x-axis and intersecting the y-axis at a point with coordinate (ordinate) 3 (Fig. 28). Any point lying on this line has an ordinate of 3. So, for points M 1, M 2, M 3 we have: M 1 (0; 3), M 2 (4; 3), M 3 (- 2; 3) . In other words, the ordinate of any point M of the line I satisfies the condition y \u003d 3. They say that y \u003d 3 is the equation of line I or that line I satisfies the equation y \u003d 3.
Figure 29 shows lines that satisfy the equations y \u003d - 4 (line l 1), y \u003d - 1 (line I 2), y \u003d 3.5 (line I 3) - A which line satisfies the equation y \u003d 01 Guess? x axis.
Note that mathematicians, striving for brevity of speech, say "a straight line x = 4", and not "a straight line that satisfies the equation x = 4". Likewise, they say "line y = 3", not "line satisfying y = 3". We will do exactly the same. Let us now return to Figure 21. Please note that the point M (- 1.5; 2), which is shown there, is the point of intersection of the line x \u003d -1.5 and the line y \u003d 2. Now, apparently, the algorithm for constructing the point will be clear according to its given coordinates.
Algorithm for constructing a point M (a; b) in a rectangular coordinate system хОу
EXAMPLE In the xOy coordinate system, construct points: A (1; 3), B (- 2; 1), C (4; 0), D (0; - 3).
Solution. Point A is the point of intersection of the lines x = 1 and y = 3 (see Fig. 30).
Point B is the point of intersection of the lines x = - 2 and y = 1 (Fig. 30). Point C belongs to the x-axis, and point D belongs to the y-axis (see Fig. 30).
In conclusion of the section, we note that for the first time a rectangular coordinate system on the plane began to be actively used to replace algebraic models geometric French philosopher René Descartes (1596-1650). Therefore, sometimes they say "Cartesian coordinate system", "Cartesian coordinates".
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The term "coordinates" in translation from Latin means the word "ordered".
Suppose we need to designate the position of a point on a plane. To do this, we take 2 perpendicular lines, which are called coordinate axes, where X will be the abscissa axis, Y is the ordinate axis, and the origin will be point O. The right angles formed using the coordinate axes will be called coordinate angles.
So we came to the definition and now we know that the coordinate plane is a plane with a given coordinate system.
And now let's see the numbering of the coordinate angles:
Now let's display a rectangular coordinate system and mark the point M in it.
Next, we need to draw a straight line through the point M, which will be parallel to the Y axis. Now, let's see what we got. As you can see, the straight line intersects the X axis at the point where the coordinate will be equal to −2. This coordinate is the abscissa of point M.
Now we need to draw a straight line through the point M, which will be parallel to the X axis.
We can see that this line intersects the X axis at the point whose coordinate is three. This coordinate will be the ordinate of point M.
Recording the coordinates of the current M will look like this:
In such a record, the abscissa is always put in the first place, and the ordinate is in the second place. If we consider the example of the coordinates of the point M (-2; 3), then -2 acts as the abscissa of the point M, and the ordinate of this point will be the number 3.
From this it follows that on the coordinate plane, each point M corresponds to such a pair of numbers as its abscissa and ordinate. The opposite statement will also be true, that is, each such pair of numbers corresponds to one point of the plane for which these numbers are coordinates.
Exercise:
Coordinate plane in life
In your opinion, can knowledge of the coordinate plane be useful in everyday life? And have you ever heard such a phrase as “leave your coordinates” or “what coordinates can you find”? And have you thought about what these expressions can mean?
It turns out that everything is very simple and banal, and this means the location of this or that object, by which it is easy to find a person or a certain place. It can be confidently asserted that coordinate systems are necessary in the practical life of a person everywhere.
Such a coordinate system can be either a home address or a telephone number, place of work, etc.
After all, even when buying train tickets, you know not only its number and destination, but also the number of the car and seat must be indicated.
To visit a classmate, it is not enough to know only the house in which he lives, but you also need to know the apartment number.
Exercise
1. What information do you need to have in order to take a place in the theater?
2. What data do you need to have in order to determine points on the earth's surface?
3. By what coordinates can you determine the place in the cinema?
4. What do you need to know in order to determine the position of a piece on a chessboard?
5. What coordinates do you use when playing sea battle?
History reference
The idea of using coordinates appeared in ancient times. Initially, astronomers began to use them to determine the heavenly bodies and geographers - to determine the location and objects on the surface of the Earth.
Thanks to the works of the ancient Greek astronomer Claudius Plotomeus, already in the second century, scientists learned to determine longitude and latitude.
Do you know why in mathematics there is such a thing as "Cartesian coordinate system"? It turns out that the method of coordinates, which has general mathematical significance, was discovered by French mathematicians Pierre Fermat and Rene Descartes in the 17th century, and in 1637 Rene Descartes first described it in a book on geometry.
But the terms "abscissa", "ordinate" and "coordinates" were first introduced by Wilhelm Leibniz in the seventeenth century.
Homework:
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