- Two mutually perpendicular coordinate lines intersecting at point O - the origin of reference, form rectangular coordinate system, also called the Cartesian coordinate system.
- The plane on which the coordinate system is chosen is called coordinate plane. The coordinate lines are called coordinate axes. The horizontal axis is the abscissa axis (Ox), the vertical axis is the ordinate axis (Oy).
- Coordinate axes divide the coordinate plane into four parts - quarters. The serial numbers of the quarters are usually counted counterclockwise.
- Any point in the coordinate plane is specified by its coordinates - abscissa and ordinate. For example, A(3; 4). Read: point A with coordinates 3 and 4. Here 3 is the abscissa, 4 is the ordinate.
I. Construction of point A(3; 4).
Abscissa 3 shows that from the beginning of the countdown - points O need to be moved to the right 3 unit segment, and then put it up 4 unit segment and put a point.
This is the point A(3; 4).
Construction of point B(-2; 5).
From zero we move to the left 2 single segment and then up 5 single segments.
Let's put an end to it IN.
Usually a unit segment is taken 1 cell.
II. Construct points in the xOy coordinate plane:
A (-3; 1);B(-1;-2);
C(-2:4);D (2; 3);
F(6:4);K(4; 0)
III. Determine the coordinates of the constructed points: A, B, C, D, F, K.
A(-4; 3);IN 20);
C(3; 4);D (6; 5);
F (0; -3);K (5; -2).
Mathematics is a rather complex science. While studying it, you have to not only solve examples and problems, but also work with various shapes and even planes. One of the most used in mathematics is the coordinate system on a 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 can be performed with its help, and also find out its main characteristics and features.
Definition of the concept
A coordinate plane is a plane on which a specific coordinate system is specified. Such a plane is defined by two straight lines intersecting at right angles. At the point of intersection of these lines is the origin of coordinates. Each point on the coordinate plane is specified by a pair of numbers called coordinates.
In a school mathematics course, schoolchildren have to work quite closely with a coordinate system - construct figures and points on it, determine which plane a particular coordinate belongs to, as well as determine the coordinates of a point and write or name them. Therefore, let's talk in more detail about all the features of coordinates. But first, let’s touch on the history of creation, and then we’ll talk about how to work on the coordinate plane.
Historical reference
Ideas about creating a coordinate system existed back in the time of Ptolemy. Even then, astronomers and mathematicians were thinking about how to learn 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 specified points using latitude and longitude. For a long time, this was one of the most used methods of plotting this or that information on a map. But in 1637, Rene Descartes created his own coordinate system, later named after the “Cartesian” one.
Already at the end of the 17th 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.
Examples of a coordinate plane
Before we talk about the theory, we will give some visual 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 coordinate is alphabetic, the second is 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 ships, you specify points on the coordinate plane.
This coordinate system is widely used not only in mathematics and logic games, but also in military affairs, astronomy, physics and many other sciences.
Coordinate axes
As already mentioned, there are two axes in the coordinate system. Let's talk a little about them, as they are of considerable importance.
The first axis is abscissa - horizontal. It is denoted as ( Ox). The second axis is the ordinate, which runs 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 the value 0 . Only if the plane is formed by two axes intersecting perpendicularly and having 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 a 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 into four quarters by two axes. Each of them has its own number, and the planes are numbered counterclockwise.
Each of the quarters has its own characteristics. So, in the first quarter the abscissa and ordinate are positive, in the second quarter the abscissa is negative, the ordinate is positive, in the third both the abscissa and 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 understood 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 and 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 we work directly with points or shapes. Moreover, even when constructing figures, points are first drawn on the plane, and then the figures are 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 construct a coordinate plane, you only need a ruler and a pen or pencil. First, the horizontal x-axis is drawn, then the vertical axis is drawn. It is important to remember that the axes intersect at right angles.
The next mandatory item is applying markings. On each of the axes in both directions, unit segments are marked and labeled. This is done so that you can then work with the plane with maximum convenience.
Mark 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 a plane, and even mark equations.
When constructing points, you should remember how their coordinates are correctly written. So, usually when specifying 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 constructed in this way. First mark on the axis Ox specified point, then mark the point on the axis Oy. Next, draw imaginary lines from these designations and find the place where they intersect - 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 any special skills.
Place the figure
Now let's move on to the issue of constructing figures on a coordinate plane. In order to construct 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 according to them that we will apply the ones you have chosen to our coordinate system. Let us consider the application of a rectangle, a triangle and a circle.
Let's start with a rectangle. It's quite easy to apply. First, four points are marked on the plane, indicating the corners of the rectangle. Then all the points are sequentially connected to each other.
Drawing a triangle is no different. The only thing is that it has three angles, which means that three points are marked on the plane, indicating its vertices.
Regarding the circle, you should know the coordinates of two points. The first point is the center of the circle, the second is the point indicating its radius. These two points are plotted on the plane. Then take a compass and measure the distance between two points. The point of the compass is placed at the point marking the center, and a circle is described.
As you can see, there is nothing complicated here either, the main thing is that you always have a ruler and compass at hand.
Now you know how to plot the coordinates of figures. Doing this on the coordinate plane is not as difficult as it might seem at first glance.
conclusions
So, we have looked at one of the most interesting and basic concepts for mathematics that every schoolchild has to deal with.
We have found out that the coordinate plane is a plane formed by the intersection of two axes. With its help, you can set the coordinates of points and draw 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 a 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 specified.
We hope that the information we presented was accessible and understandable, and was also useful to you and helped you better understand this topic.
Let it be given equation with two variables F(x; y). You have already become familiar with ways to solve such equations analytically. Many solutions of such equations can be represented in graph form.
The graph of the equation F(x; y) is the set of points on the coordinate plane xOy whose coordinates satisfy the equation.
To graph equations in two variables, first express the y variable in the equation in terms of the x variable.
Surely you already know how to build various graphs of equations with two variables: ax + b = c – straight line, yx = k – hyperbola, (x – a) 2 + (y – b) 2 = R 2 – circle whose radius is equal to R, and the center is at point O(a; b).
Example 1.
Graph the equation x 2 – 9y 2 = 0.
Solution.
Let's factorize the left side of the equation.
(x – 3y)(x+ 3y) = 0, that is, y = x/3 or y = -x/3.
Answer: Figure 1.
A special place is occupied by defining figures on a plane with equations containing the sign of the absolute value, which we will dwell on in detail. Let's consider the stages of constructing graphs of equations of the form |y| = f(x) and |y| = |f(x)|.
The first equation is equivalent to the system
(f(x) ≥ 0,
(y = f(x) or y = -f(x).
That is, its graph consists of graphs of two functions: y = f(x) and y = -f(x), where f(x) ≥ 0.
To plot the second equation, plot two functions: y = f(x) and y = -f(x).
Example 2.
Graph the equation |y| = 2 + x.
Solution.
The given equation is equivalent to the system
(x + 2 ≥ 0,
(y = x + 2 or y = -x – 2.
We build many points.
Answer: Figure 2.
Example 3.
Plot the equation |y – x| = 1.
Solution.
If y ≥ x, then y = x + 1, if y ≤ x, then y = x – 1.
Answer: Figure 3.
When constructing graphs of equations containing a variable under the modulus sign, it is convenient and rational to use area method, based on dividing the coordinate plane into parts in which each submodular expression retains its sign.
Example 4.
Graph the equation x + |x| + y + |y| = 2.
Solution.
In this example, the sign of each submodular expression depends on the coordinate quadrant.
1) In the first coordinate quarter x ≥ 0 and y ≥ 0. After expanding the module, the given equation will look like:
2x + 2y = 2, and after simplification x + y = 1.
2) In the second quarter, where x< 0, а y ≥ 0, уравнение будет иметь вид: 0 + 2y = 2 или y = 1.
3) In the third quarter x< 0, y < 0 будем иметь: x – x + y – y = 2. Перепишем этот результат в виде уравнения 0 · x + 0 · y = 2.
4) In the fourth quarter, when x ≥ 0, and y< 0 получим, что x = 1.
We will plot this equation by quarters.
Answer: Figure 4.
Example 5.
Draw a set of points whose coordinates satisfy the equality |x – 1| + |y – 1| = 1.
Solution.
The zeros of the submodular expressions x = 1 and y = 1 divide the coordinate plane into four regions. Let's break down the modules by region. Let's arrange this in the form of a table.
| Region |
Submodular expression sign |
The resulting equation after expanding the module |
| I | x ≥ 1 and y ≥ 1 | x + y = 3 |
| II | x< 1 и y ≥ 1 | -x + y = 1 |
| III | x< 1 и y < 1 | x + y = 1 |
| IV | x ≥ 1 and y< 1 | x – y = 1 |
Answer: Figure 5.
On the coordinate plane, figures can be specified and inequalities.
Inequality graph with two variables is the set of all points of the coordinate plane whose coordinates are solutions to this inequality.
Let's consider algorithm for constructing a model for solving inequalities with two variables:
- Write down the equation corresponding to the inequality.
- Graph the equation from step 1.
- Select an arbitrary point in one of the half-planes. Check whether the coordinates of the selected point satisfy this inequality.
- Draw graphically the set of all solutions to the inequality.
Let us first consider the inequality ax + bx + c > 0. The equation ax + bx + c = 0 defines a straight line dividing the plane into two half-planes. In each of them, the function f(x) = ax + bx + c retains its sign. To determine this sign, it is enough to take any point belonging to the half-plane and calculate the value of the function at this point. If the sign of the function coincides with the sign of the inequality, then this half-plane will be the solution to the inequality.
Let's look at examples of graphical solutions to the most common inequalities with two variables.
1) ax + bx + c ≥ 0. Figure 6.
2)
|x| ≤ a, a > 0. Figure 7.
3) x 2 + y 2 ≤ a, a > 0. Figure 8.
4) y ≥ x 2 . Figure 9.
5)
xy ≤ 1. Figure 10. 
If you have questions or want to practice drawing on a plane model the sets of all solutions to inequalities in two variables using mathematical modeling, you can conduct free 25-minute lesson with an online tutor after . To further work with a teacher, you will have the opportunity to choose the one that suits you
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An ordered system of two or three intersecting axes perpendicular to each other with a common origin (origin of coordinates) and a common unit of 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, to look for solutions to 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. Thus, the coordinates of a point on a circle with a center at a 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 the given point, and an ordinate equal in absolute value to the ordinate of the given point, and opposite in 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) on a 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.
A rectangular coordinate system on a plane is defined by two mutually perpendicular straight lines. Straight lines are called coordinate axes (or coordinate axes). The point of intersection of these lines is called the origin and is designated by the letter O.
Usually one of the lines is horizontal, the other is vertical. The horizontal line is designated as the x-axis (or Ox) and is called the abscissa axis, the vertical line is the y-axis (Oy), called the ordinate axis. The entire coordinate system is designated xOy.
Point O divides each of the axes into two semi-axes, one of which is considered positive (denoted by an arrow), the other - negative.
Each point F of the plane is assigned a pair of numbers (x;y) - its coordinates.
The x coordinate is called the abscissa. It is equal to Ox, taken with the appropriate sign.
The y coordinate is called the ordinate and is equal to the distance from point F to the Oy axis (with the appropriate sign).
Axle distances are usually (but not always) measured in the same unit of length.
Points located to the right of the y-axis have positive abscissas. Points that lie to the left of the ordinate axis have negative abscissas. For any point lying on the Oy axis, its x coordinate is zero.
Points with a positive ordinate lie above the x-axis, and points with a negative ordinate lie below. If a point lies on the Ox axis, its y coordinate is zero.
Coordinate axes divide the plane into four parts, which are called coordinate quarters (or coordinate angles or quadrants).
1 coordinate quarter located in the upper right corner of the xOy coordinate plane. Both coordinates of points located in the first quarter are positive.
The transition from one quarter to another is carried out counterclockwise.
2 coordinate quarter is located in the upper left corner. Points lying in the second quarter have a negative abscissa and a positive ordinate.
3 coordinate quarter lies in the lower left quadrant of the xOy plane. Both coordinates of the points belonging to the III coordinate angle are negative.
4 coordinate quarter is the lower right corner of the coordinate plane. Any point from the IV quarter has a positive first coordinate and a negative second.
An example of the location of points in a rectangular coordinate system:
