Which quarter is each point in: A(-2;5), B(4;2), C(3;-6), A(-2;5), B(4;2), C(3;- 6), D(7;1), E(-5;-3), M(-5;4), D(7;1), E(-5;-3), M(-5;4) , K(-8;-2), P(1;-7), N(1;3), K(-8;-2), P(1;-7), N(1;3), R (-7;-1). R(-7;-1). I I IIIV I III III IV III II Card 1.
Self-test: 1. Two straight lines forming right angles when intersecting... 2. The plane on which the coordinate system is selected... 3. The coordinate line y Two perpendicular coordinate lines x and y, which intersect at the origin - point O,... 5. The coordinate line straight line x ... ... are called perpendicular. ... called the coordinate plane. ...is called the y-axis. ...is called a coordinate system on a plane. ... called the abscissa axis. Card 3.
Excursion to the zoo. Excursion to the zoo. Construct a figure at given coordinates. Construct a figure at given coordinates. Find the riddle about who you saw at the Zoo. Find the riddle about who you saw at the Zoo. Simulator "Catch a Fish" Simulator "Catch a Fish"
If you are at some zero point and are wondering how many units of distance you need to go straight ahead and then straight to the right to get to some other point, then you are already using a rectangular Cartesian coordinate system on the plane. And if the point is located above the plane on which you stand, and to your calculations you add an ascent to the point along the stairs strictly upward also by a certain number of distance units, then you are already using a rectangular Cartesian coordinate system in space.
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 .
The name of the French mathematician René Descartes (1596-1662) is associated primarily with a coordinate system in which a common unit of length is measured on all axes and the axes are straight. In addition to the rectangular one, there is general Cartesian coordinate system (affine coordinate system). It may also include axes that are not necessarily perpendicular. If the axes are perpendicular, then the coordinate system is rectangular.
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) 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 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.
What is an abscissa and what is an ordinate? and got the best answer
Answer from Lisa[expert]
abscissa is x
y ordinate
Answer from Nikolay Katkov[guru]
Drawing
Answer from Arseny Rodin[active]
y-axis
Answer from Murad Khalidov[active]
I studied this topic in the 6th grade and you probably did too, but judging by the fact that this issue was resolved 5 years ago, I concluded that in the 11th grade. Thank you for such a simple and clear answer (the best)!
Answer from Dasha Kazina[newbie]
The abscissa point (according to the coordinates it comes first) lies horizontally on the X axis, and the ordinate (according to the coordinates it comes second) lies vertically on the Y axis
Answer from Dimon Dimon[newbie]
The abscissa (lat. 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 "abscissa 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 Y’Y 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 Y'O, then the ordinate has a negative value. If point A lies on the X’X axis, then its ordinate is zero.
In a rectangular coordinate system, the Y'Y axis is called the "y-axis".
When plotting functions, the y-axis is usually used as the range of the function.
Drawing here
Answer from Vadix[active]
Short and clear and no need to read, just watch and listen! 🙂
What is an ordinate?
What is an abscissa?
Answer from Bai Pazylov[newbie]
abscissa-x
ordinate-y
Answer from No show off.[active]
It’s easy to remember if it’s difficult: “Ah” and “Oh” :)
Answer from Vsevolod Yablonovsky[active]
abscissa is x
Answer from Yoanseth Shimmer[newbie]
abscissa is x
y ordinate
Answer from Vlad Chubinsky[newbie]
abscissa is x
y ordinate
Answer from Dmitry Kornev[newbie]
x-axis
y-axis
Answer from 3 answers[guru]
Hello! Here is a selection of topics with answers to your question: What is an abscissa and what is an ordinate?
In everyday life you can often hear the phrase: “Leave me your coordinates.” In response, a person usually leaves his address or phone number, that is, data by which he can be found.
Coordinates can be indicated by a variety of sets of numbers or letters.
For example, a car number is coordinates, because by the car number you can determine what city it is from and who its owner is.
Important!
Coordinates is a set of data from which the position of an object is determined.
Examples of coordinates are: car and seat number on a train, latitude and longitude on a geographic map, recording the position of a piece on a chessboard, position of a point on a number line, etc.
Whenever, according to certain rules, we unambiguously designate an object with a set of letters, numbers or other symbols, we specify the coordinates of the object.
Cartesian coordinate system
The French mathematician Rene Descartes (1596-1650) proposed specifying the position of a point on a plane using two coordinates.
To find coordinates, you need landmarks from which to count.
- On a plane, two numerical axes will serve as such reference points. In the drawing, the first axis is usually drawn horizontally, it is called the ABSCISS axis and is designated by the letter “X”, the axis is written “Ox”. The positive direction on the x-axis is chosen from left to right and shown with an arrow.
- The second axis is drawn vertically, it is called the ORDINATE axis and is designated by the letter “Y”, the axis is written “Oy”. The positive direction on the ordinate axis is chosen from bottom to top and is shown with an arrow.
The axes are mutually perpendicular (i.e. the angle between them is 90°) and intersect at a point designated “O”. Point “O” is the origin for each of the axes.
Remember!
Coordinate system- these are two mutually perpendicular coordinate lines intersecting at a point, which is the origin of reference for each of them.
Coordinate axes are straight lines that form a coordinate system.
Abscissa axis"Ox" - horizontal axis.
Y axis"Oy" - vertical axis.
Coordinate plane is the plane in which the coordinate system is constructed. The plane is designated as “x0y”.
We draw your attention to the choice of the length of single segments along the axes.
The numbers indicating numerical values on the axes can be placed either to the right or to the left of the “Oy” axis. The numbers on the “Ox” axis are usually written below the axis.
Typically, a unit segment on the “0y” axis is equal to a unit segment on the “0x” axis. But there are times when they are not equal to each other.
The coordinate axes divide the plane into 4 angles, which are called coordinate quarters. The quarter formed by the positive semi-axes (upper right corner) is considered the first I.
We count the quarters (or coordinate angles) counterclockwise.
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".
Spelling
Please note the spelling: Ab With cissa, but not abscissa and not abscissa.
see also
Wikimedia Foundation. 2010.
See what “X-axis” is in other dictionaries:
abscissa axis- Horizontal axis in the Cartesian coordinate system. Topics information technology in general EN abscise axishorizontal axisX axis … Technical Translator's Guide
abscissa axis- abscisių ašis statusas T sritis automatika atitikmenys: engl. abscissa axis vok. Abszissenachse, f rus. abscissa axis, f pranc. ax d abscisses, m … Automatikos terminų žodynas
abscissa axis- abscisių ašis statusas T sritis fizika atitikmenys: engl. abscissa axis vok. Abszissenachse, f rus. abscissa axis, f pranc. ax d'abscisses, m ... Fizikos terminų žodynas
Axis (the word “axis” comes from the Old Russian “awn” - a long tendril on the chaff of each grain of spiked plants or hair in a fur product) the concept of a certain central straight line, including an imaginary straight line (line): In technology: ... ... Wikipedia
AXIS- (1) in applied mechanics, a rod resting on supports and supporting rotating parts of machines (car wheels) or mechanisms (clock gears). Unlike (see) O. does not transmit useful torque (see (5)), but works in ... ... Big Polytechnic Encyclopedia
definition- 2.7 definition: The process of performing a series of operations, regulated in a test method document, as a result of which a single value is obtained. Source … Dictionary-reference book of terms of normative and technical documentation
- (from the Greek στροφή rotation) algebraic curve of the 3rd order. It is built like this (see Fig. 1): Fig. 1 ... Wikipedia
A branch of geometry that studies the simplest geometric objects using elementary algebra based on the coordinate method. The creation of analytical geometry is usually attributed to R. Descartes, who outlined its foundations in the last chapter of his... ... Collier's Encyclopedia
Rice. 1. Construction of a cissoid. Blue and red lines of the cissoid branch. Diocles' cissoid is a plane algebraic curve of the third order. In a Cartesian coordinate system, where the x-axis is directed along ... Wikipedia
Diocles' cissoid is a plane algebraic curve of the third order. In the Cartesian coordinate system, where the abscissa axis is directed along OX and the ordinate axis along OY, on the segment OA = 2a, as on a diameter, an auxiliary circle is constructed. At point A is carried out... ... Wikipedia
