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Cartesian coordinate system: basic concepts and examples. Great encyclopedia of oil and gas

This point on the axis X'X in a rectangular coordinate system. Abscissa value of a point A equal to the length of the segment O.B.(see picture). If the point B belongs to the positive semi-axis OX, then the abscissa has positive value. If the point B belongs to the negative semi-axis X'O, then the abscissa has negative value. If the point A lies on the axis Y'Y, then its abscissa is zero.

In a rectangular coordinate system, a ray (straight line) X'X called the "abscissa axis". When plotting functions, the x-axis is usually used as the domain of definition of the function.

Etymology

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Excerpt characterizing Abscissa

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“No, not at all,” said Boris. And if you are tired, let’s go to my room and lie down and rest.
- Indeed...
They entered the small room where Boris was sleeping. Rostov, without sitting down, immediately with irritation - as if Boris was guilty of something in front of him - began to tell him Denisov’s case, asking if he wanted and could ask about Denisov through his general from the sovereign and through him deliver a letter. When they were left alone, Rostov became convinced for the first time that he was embarrassed to look Boris in the eyes. Boris crossed his legs and stroked his thin fingers with his left hand right hand, listened to Rostov, as a general listens to the report of a subordinate, now looking to the side, now with the same clouded gaze, looking directly into Rostov’s eyes. Each time Rostov felt awkward and lowered his eyes.
“I have heard about this kind of thing and I know that the Emperor is very strict in these cases. I think we should not bring it to His Majesty. In my opinion, it would be better to directly ask the corps commander... But in general I think...
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At this time, Zhilinsky’s voice was heard at the door, calling Boris.
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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), this coordinate system is named in which the common unit of length is measured on all axes and the axes are straight.

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

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

The coordinate method, which arose in the works of Rene Descartes, marked a revolutionary restructuring of all mathematics. It became possible to interpret algebraic equations(or inequalities) in the form of geometric images (graphs) and, conversely, look for 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 pass through the point M Ox. This straight line intersects the axis Ox at the point Mx. Let's pass 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 pass through the point M plane perpendicular to the axis Oy. This plane intersects the axis Oy at the point My. Let's pass 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) to the plane Oyz ;

4) on the abscissa axis;

5) on the ordinate axis;

6) on the applicate axis.

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

Axy (4; 3; 0);

Bxy (-3; 2; 0);

Cxy(2;-3;0).

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

Axz (4; 0; 5);

Bxz (-3; 0; 1);

Cxz (2; 0; 0).

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

Ayz(0; 3; 5);

Byz (0; 2; 1);

Cyz (0; -3; 0).

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

Ax (4; 0; 0);

Bx (-3; 0; 0);

Cx(2;0;0).

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

Ay(0; 3; 0);

By (0; 2; 0);

Cy(0;-3;0).

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

Az (0; 0; 5);

Bz (0; 0; 1);

Cz (0; 0; 0).

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

A(2; 3; 1) ;

B(5; -3; 2) ;

C(-3; 2; -1) .

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

1) plane Oxy ;

2) planes Oxz ;

3) planes Oyz ;

4) abscissa axes;

5) ordinate axes;

6) applicate axes;

7) origin of coordinates.

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

A"(2; 3; -1) ;

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

C"(-3; 2; 1) .

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

A"(2; -3; 1) ;

B"(5; 3; 2) ;

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

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

A"(-2; 3; 1) ;

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

C"(3; 2; -1) .

By analogy with symmetrical points on the plane and points in space symmetrical to the data relative to the planes, we note that in the case of symmetry relative 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 terms the same 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 in sign. So, we obtain the following coordinates of points symmetrical to the data relative to the origin.

abscissa is x

abscissa is x
y ordinate

I studied this topic in the 6th grade and you probably did too, but judging by the fact that this question was solved 5 years ago, I concluded that it was in the 11th grade. Thank you for such a simple and clear answer (the best)!

In a rectangular coordinate system, the YY axis is called the y-axis.

When plotting functions, the y-axis is usually used as the range of the function.

Drawing here

It’s easy to remember if it’s difficult: “Ah” and “Oh” :)

Short and clear and no need to read, just watch and listen! 🙂
What is an ordinate? https://www.youtube.com/watch?v=M9v-9dwIUuY
What is an abscissa? https://www.youtube.com/watch?v=dPZ6QUtveH8

The abscissa (lat. abscissa segment) of point A is the coordinate of this point on the XX axis in a rectangular coordinate system. The abscissa of point A is equal to the length of the segment OB (see Fig. 1). If point B belongs to the positive semi-axis OX, then the abscissa has a positive value. If point B belongs to the negative semi-axis XO, then the abscissa has a negative value. If point A lies on the YY axis, then the abscissa is zero.

In a rectangular coordinate system, the XX axis is called the x-axis.

When plotting functions, the x-axis is usually used as the domain of the function.

The ordinate (from the Latin ordinatus - located in order) of point A is the coordinate of this point on the YY axis in a rectangular coordinate system. The ordinate value of point A is equal to the length of the segment OC (see Fig. 1). If point C belongs to the positive semi-axis OY, then the ordinate has a positive value. If point C belongs to the negative semi-axis YO, then the ordinate has a negative value. If point A lies on the XX axis, then the e ordinate is zero.

Abscissa is a common term in mathematics that many people do not understand. The concept of abscissa will help in understanding many mathematical problems. The topic of this article is dedicated to it.

What is an abscissa

Before you understand what an abscissa is, you need to learn about the essence of several more terms, namely:

Rectangular coordinate system. A rectangular coordinate system is a system where there are only two directions. Such a system is usually called two-dimensional. One direction is in the form of a horizontal straight line and is indicated by the letter x, the second direction is a vertical straight line, which is designated by the letter y. The intersection of these two directions is called the origin. The coordinate report starts from this point. Those values of the horizontal line that are to the right of the origin are positive. Those to the left are negative. Accordingly, those y values of the line that are above the origin are positive, and those below are negative.
Ordinate. The coordinate of any point that corresponds to the axis y(in a coordinate system) is called an ordinate.

Based on the last condition, you can easily guess that if the ordinate is the coordinate on the axis y, which corresponds to any point, then the abscissa is the coordinate of the same point, but which is located on the axis x.

Point A is given, with coordinates (4; 6). What is the abscissa and what is the ordinate?

Remember that when the coordinates of a point are written, the coordinates on the axis are indicated first x, and on the second - the axes y. Thus, the abscissa of point A is 4 and the ordinate is 6.

Now you know what an abscissa is and you can, without hesitation, delve into the meaning of the problem when you see this word. Good to study this topic, because coordinates are used in many areas - from mathematics to programming.

abscissa- segment) of point A is the coordinate of this point on the X’X axis in a rectangular coordinate system. The abscissa of point A is equal to the length of the segment OB (see Fig. 1). If point B belongs to the positive semi-axis OX, then the abscissa has a positive value. If point B belongs to the negative semi-axis X'O, then the abscissa has a negative value. If point A lies on the Y’Y axis, then its abscissa is zero.

In a rectangular coordinate system, the X'X axis is called the "x-axis".