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What is a coordinate plane definition. Coordinate plane: what is it? How to mark points and construct figures on a coordinate plane? Working with the coordinate plane

What is a coordinate plane definition. Coordinate plane: what is it? How to mark points and construct figures on a coordinate plane? Working with the coordinate plane

Understanding the Coordinate Plane

Each object (for example, a house, a place in the auditorium, a point on the map) has its own ordered address (coordinates), which has a numerical or letter designation.

Mathematicians have developed a model that allows you to determine the position of an object and is called coordinate plane.

To construct a coordinate plane, you need to draw $2$ perpendicular straight lines, at the end of which the directions “to the right” and “up” are indicated using arrows. Divisions are applied to the lines, and the point of intersection of the lines is the zero mark for both scales.

Definition 1

The horizontal line is called x-axis and is denoted by x, and the vertical line is called y-axis and is denoted by y.

Two perpendicular x and y axes with divisions make up rectangular, or Cartesian, coordinate system, which was proposed by the French philosopher and mathematician Rene Descartes.

Coordinate plane

Point coordinates

A point on a coordinate plane is defined by two coordinates.

To determine the coordinates of point $A$ on the coordinate plane, you need to draw straight lines through it that will be parallel to the coordinate axes (indicated by a dotted line in the figure). The intersection of the line with the x-axis gives the $x$ coordinate of point $A$, and the intersection with the y-axis gives the y-coordinate of point $A$. When writing the coordinates of a point, the $x$ coordinate is first written, and then the $y$ coordinate.

Point $A$ in the figure has coordinates $(3; 2)$, and point $B (–1; 4)$.

To plot a point on the coordinate plane, proceed in the reverse order.

Constructing a point at specified coordinates

Example 1

On the coordinate plane, construct points $A(2;5)$ and $B(3; –1).$

Solution.

Construction of point $A$:

put the number $2$ on the $x$ axis and draw a perpendicular line;
On the y-axis we plot the number $5$ and draw a straight line perpendicular to the $y$ axis. At the intersection of perpendicular lines we obtain point $A$ with coordinates $(2; 5)$.

Construction of point $B$:

Let us plot the number $3$ on the $x$ axis and draw a straight line perpendicular to the x axis;
On the $y$ axis we plot the number $(–1)$ and draw a straight line perpendicular to the $y$ axis. At the intersection of perpendicular lines we obtain point $B$ with coordinates $(3; –1)$.

Example 2

Construct points on the coordinate plane with given coordinates $C (3; 0)$ and $D(0; 2)$.

Solution.

Construction of point $C$:

put the number $3$ on the $x$ axis;
coordinate $y$ is equal to zero, which means point $C$ will lie on the $x$ axis.

Construction of point $D$:

put the number $2$ on the $y$ axis;
coordinate $x$ is equal to zero, which means point $D$ will lie on the $y$ axis.

Note 1

Therefore, at coordinate $x=0$ the point will lie on the $y$ axis, and at coordinate $y=0$ the point will lie on the $x$ axis.

Example 3

Determine the coordinates of points A, B, C, D.$

Solution.

Let's determine the coordinates of point $A$. To do this, we draw straight lines through this point $2$ that will be parallel to the coordinate axes. The intersection of the line with the x-axis gives the coordinate $x$, the intersection of the line with the y-axis gives the coordinate $y$. Thus, we obtain that the point $A (1; 3).$

Let's determine the coordinates of point $B$. To do this, we draw straight lines through this point $2$ that will be parallel to the coordinate axes. The intersection of the line with the x-axis gives the coordinate $x$, the intersection of the line with the y-axis gives the coordinate $y$. We find that point $B (–2; 4).$

Let's determine the coordinates of point $C$. Because it is located on the $y$ axis, then the $x$ coordinate of this point is zero. The y coordinate is $–2$. Thus, point $C (0; –2)$.

Let's determine the coordinates of point $D$. Because it is on the $x$ axis, then the $y$ coordinate is zero. The $x$ coordinate of this point is $–5$. Thus, point $D (5; 0).$

Example 4

Construct points $E(–3; –2), F(5; 0), G(3; 4), H(0; –4), O(0; 0).$

Solution.

Construction of point $E$:

put the number $(–3)$ on the $x$ axis and draw a perpendicular line;
on the $y$ axis we plot the number $(–2)$ and draw a perpendicular line to the $y$ axis;
at the intersection of perpendicular lines we obtain the point $E (–3; –2).$

Construction of point $F$:

coordinate $y=0$, which means the point lies on the $x$ axis;
Let us plot the number $5$ on the $x$ axis and obtain the point $F(5; 0).$

Construction of point $G$:

put the number $3$ on the $x$ axis and draw a perpendicular line to the $x$ axis;
on the $y$ axis we plot the number $4$ and draw a perpendicular line to the $y$ axis;
at the intersection of perpendicular lines we obtain the point $G(3; 4).$

Construction of point $H$:

coordinate $x=0$, which means the point lies on the $y$ axis;
Let us plot the number $(–4)$ on the $y$ axis and obtain the point $H(0;–4).$

Construction of point $O$:

both coordinates of the point are equal to zero, which means that the point lies simultaneously on both the $y$ axis and the $x$ axis, therefore it is the intersection point of both axes (the origin of coordinates).

points are “registered” - “residents”, each point has its own “house number” - its coordinate. If the point is taken in a plane, then to “register” it you need to indicate not only the “house number”, but also the “apartment number”. Let us remind you how this is done.

Let us draw two mutually perpendicular coordinate lines and consider the origin of reference on both lines as the point of their intersection - point O. Thus, a rectangular coordinate system is specified on the plane (Fig. 20), which turns the usual plane to coordinate. Point O is called the origin of coordinates, the coordinate lines (x-axis and y-axis) are called coordinate axes, and right angles formed by the coordinate axes are called coordinate angles. Coordinate rectangular angles are numbered as shown in Figure 20.

Now let’s turn to Figure 21, where a rectangular coordinate system is depicted and point M is marked. Let’s draw a straight line through it parallel to the y-axis. The straight line intersects the x-axis at a certain point, this point has a coordinate - on the x-axis. For the point shown in Figure 21, this coordinate is equal to -1.5, it is called the abscissa of point M. Next, we draw a straight line through point M, parallel to the x-axis. The straight line intersects the y-axis at a certain point, this point has a coordinate - on the y-axis.

For point M, shown in Figure 21, this coordinate is equal to 2, it is called the ordinate of point M. Briefly written as follows: M (-1.5; 2). The abscissa is written in first place, the ordinate in second. If necessary, use another form of notation: x = -1.5; y = 2.

Note 1 . In practice, to find the coordinates of point M, usually instead of straight lines parallel to the coordinate axes and passing through point M, segments of these straight lines from point M to the coordinate axes are constructed (Fig. 22).

Note 2. In the previous paragraph we introduced different notations for numerical intervals. In particular, as we agreed, the notation (3, 5) means that on the coordinate line we consider an interval with ends at points 3 and 5. In this section, we consider a pair of numbers as the coordinates of a point; for example, (3; 5) is a point on coordinate plane with abscissa 3 and ordinate 5. How can one correctly determine from symbolic notation what we are talking about: an interval or the coordinates of a point? Most often this is clear from the text. What if it's not clear? Pay attention to one detail: we used a comma to indicate the interval, and a semicolon to indicate the coordinates. This, of course, is not very significant, but still a difference; we will use it.

Taking into account the introduced terms and notations, the horizontal coordinate line is called the abscissa, or x-axis, and the vertical coordinate line is called the ordinate axis, or y-axis. The notation x, y is usually used when specifying a rectangular coordinate system on a plane (see Fig. 20) and is often said like this: given a coordinate system xOy. However, there are other notations: for example, in Figure 23 the tOs coordinate system is specified.
Algorithm for finding the coordinates of point M specified in the rectangular coordinate system xOy

This is exactly what we did when finding the coordinates of point M in Figure 21. If point M 1 (x; y) belongs to the first coordinate angle, then x > 0, y > 0; if point M 2 (x; y) belongs to the second coordinate angle, then x< 0, у >0; if 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).

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 on the y-axis (Fig. 25). Drawing a line parallel to the y-axis through point A and finding the point of intersection of this line with the x-axis does not make sense, since such an intersection point already exists - this is point A, its coordinate (abscissa) is 3. In the same way, there is no need to draw through the point And the straight line parallel to the x-axis is the x-axis itself, which intersects the y-axis at point O with coordinate (ordinate) 0. As a result, for point A we obtain A(3; 0). Similarly, for point B we obtain B(0; - 1.5). And for 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. How to solve the inverse problem, i.e. how, having given the coordinates, construct the corresponding point? To develop an algorithm, we will carry out two auxiliary, but at the same time important, reasoning.

First reasoning. Let I be drawn in the xOy coordinate system, parallel to the axis y and intersecting the x axis at the 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 on the line satisfies the condition x = 4. They say that x = 4 - the equation line l or that line I satisfies the equation x = 4.

Figure 27 shows straight lines satisfying the equations x = - 4 (line I 1), x = - 1
(straight I 2) x = 3.5 (straight I 3). Which line satisfies the equation x = 0? Did you guess it? Y axis

Second reasoning. Let a 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 line I satisfies the condition y = 3. They say that y = 3 is the equation of line I or that line I satisfies the equation y = 3.

Figure 29 shows straight lines that satisfy the equations y = - 4 (straight line l 1), y = - 1 (straight line I 2), y = 3.5 (straight line I 3) - And which straight line satisfies the equation y = 01 Did you guess it? x axis

Note that mathematicians, striving for brevity, say “the line x = 4”, and not “the line satisfying the equation x = 4”. Likewise, they say "the line y = 3" rather than "the line satisfying the equation y = 3." We will do the same. Let us now return to Figure 21. Please note that the point M (- 1.5; 2), which is depicted there, is the intersection point of the straight line x = -1.5 and the straight line y = 2. Now, apparently, the algorithm for constructing the point will be clear according to its given coordinates.

Algorithm for constructing point M (a; b) in a rectangular coordinate system xOy

EXAMPLE In the xOy coordinate system, construct the points: A (1; 3), B (- 2; 1), C (4; 0), D (0; - 3).

Solution. Point A is the intersection point of the lines x = 1 and y = 3 (see Fig. 30).

Point B is the intersection point 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).

At the end of the section, we note that for the first time, the rectangular coordinate system on a plane began to be actively used to replace algebraic models geometric French philosopher René Descartes (1596-1650). Therefore, they sometimes say “Cartesian coordinate system”, “ Cartesian coordinates».

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To indicate relative position some objects under study are used:

coordinate ray, when their placement or movement occurs along a straight line on one side of a given object, taken as the origin;
coordinate line, when their placement or movement occurs along a straight line on opposite sides of a given object, taken as the origin;
coordinate plane, when their placement or movement occurs along an arbitrary indirect line.

Elements of the coordinate plane

The coordinate plane differs from the ordinary plane in that a coordinate system is applied to it. An example would be an image of any continent with parallels and meridians marked on it, which define the system geographical coordinates, allowing you to find or set the position of any object on the map.

The coordinate system consists of two coordinate lines mutually intersecting at right angles at the origin points. The horizontal coordinate line is usually called the abscissa axis (abscissa in Latin means segment). The vertical line is the ordinate axis (ordinate from Latin means alignment in order).

Similarly, the coordinate line differs from an ordinary straight line in that some point on it is chosen as the origin; choose the scale of a single segment depending on what distances are to be depicted; positive direction of reference, indicated on the coordinate straight arrow.

The position of an object on such a plane is indicated by a point with two numbers - coordinates: abscissa and ordinate.

Using coordinate planes

Coordinate planes are widely used to solve geometric and physical problems. Moreover, in physics the time axis is often taken as the abscissa axis. Then the ordinate axis specifies the coordinate of the body on a coordinate line located along the rectilinear trajectory of the body.

Understanding the Coordinate Plane

Each object (for example, a house, a place in the auditorium, a point on the map) has its own ordered address (coordinates), which has a numerical or letter designation.

Mathematicians have developed a model that allows you to determine the position of an object and is called coordinate plane.

Definition 1

The horizontal line is called x-axis and is denoted by x, and the vertical line is called y-axis and is denoted by y.

Two perpendicular x and y axes with divisions make up rectangular, or Cartesian, coordinate system, which was proposed by the French philosopher and mathematician Rene Descartes.

Coordinate plane

Point coordinates

A point on a coordinate plane is defined by two coordinates.

Point $A$ in the figure has coordinates $(3; 2)$, and point $B (–1; 4)$.

To plot a point on the coordinate plane, proceed in the reverse order.

Constructing a point at specified coordinates

Example 1

On the coordinate plane, construct points $A(2;5)$ and $B(3; –1).$

Solution.

Construction of point $A$:

put the number $2$ on the $x$ axis and draw a perpendicular line;
On the y-axis we plot the number $5$ and draw a straight line perpendicular to the $y$ axis. At the intersection of perpendicular lines we obtain point $A$ with coordinates $(2; 5)$.

Construction of point $B$:

Let us plot the number $3$ on the $x$ axis and draw a straight line perpendicular to the x axis;
On the $y$ axis we plot the number $(–1)$ and draw a straight line perpendicular to the $y$ axis. At the intersection of perpendicular lines we obtain point $B$ with coordinates $(3; –1)$.

Example 2

Construct points on the coordinate plane with given coordinates $C (3; 0)$ and $D(0; 2)$.

Solution.

Construction of point $C$:

put the number $3$ on the $x$ axis;
coordinate $y$ is equal to zero, which means point $C$ will lie on the $x$ axis.

Construction of point $D$:

put the number $2$ on the $y$ axis;
coordinate $x$ is equal to zero, which means point $D$ will lie on the $y$ axis.

Note 1

Therefore, at coordinate $x=0$ the point will lie on the $y$ axis, and at coordinate $y=0$ the point will lie on the $x$ axis.

Example 3

Determine the coordinates of points A, B, C, D.$

Solution.

Let's determine the coordinates of point $C$. Because it is located on the $y$ axis, then the $x$ coordinate of this point is zero. The y coordinate is $–2$. Thus, point $C (0; –2)$.

Let's determine the coordinates of point $D$. Because it is on the $x$ axis, then the $y$ coordinate is zero. The $x$ coordinate of this point is $–5$. Thus, point $D (5; 0).$

Example 4

Construct points $E(–3; –2), F(5; 0), G(3; 4), H(0; –4), O(0; 0).$

Solution.

Construction of point $E$:

put the number $(–3)$ on the $x$ axis and draw a perpendicular line;
on the $y$ axis we plot the number $(–2)$ and draw a perpendicular line to the $y$ axis;
at the intersection of perpendicular lines we obtain the point $E (–3; –2).$

Construction of point $F$:

coordinate $y=0$, which means the point lies on the $x$ axis;
Let us plot the number $5$ on the $x$ axis and obtain the point $F(5; 0).$

Construction of point $G$:

put the number $3$ on the $x$ axis and draw a perpendicular line to the $x$ axis;
on the $y$ axis we plot the number $4$ and draw a perpendicular line to the $y$ axis;
at the intersection of perpendicular lines we obtain the point $G(3; 4).$

Construction of point $H$:

coordinate $x=0$, which means the point lies on the $y$ axis;
Let us plot the number $(–4)$ on the $y$ axis and obtain the point $H(0;–4).$

Construction of point $O$:

both coordinates of the point are equal to zero, which means that the point lies simultaneously on both the $y$ axis and the $x$ axis, therefore it is the intersection point of both axes (the origin of coordinates).

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, a positive direction is selected on each axis. The positive direction of the axes (in a 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 coordinate axes X'X and Y'Y 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 segment OB, the y coordinate is equal to the length of the segment OC in the selected units of measurement. 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. It is written as follows: A(x, y).

If point A lies in coordinate angle I, then point A has a 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 a 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 point O, which is called the origin, on each axis a positive direction is selected, indicated by arrows, and a unit of measurement for the segments on the axes. The units of measurement are 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 positive direction of the OZ axis. Such a coordinate system is called right-handed. 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-handed coordinate system is formed. Similar fingers of the left hand form the left coordinate system. It is impossible to combine the right and left coordinate systems 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 the length of the segment OC, the z coordinate is the length of the segment OD in the selected units of measurement. The segments OB, OC and OD are defined by planes drawn from 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. It is written as follows: A(a, b, c).

Orty

A rectangular coordinate system (of any dimension) is also described by a set of unit vectors aligned with the coordinate axes. The number of unit vectors is equal to the dimension of the coordinate system and they are all perpendicular to each other.

In the three-dimensional case, such unit vectors are usually denoted i j k or e x e y e z. In this case, in the case of a right-handed coordinate system, the following formulas with the vector product of vectors are valid:

[i j]=k ;
[j k]=i ;
[k i]=j .

Story

The rectangular coordinate system was first introduced by Rene Descartes in his work “Discourse on Method” in 1637. Therefore, the rectangular coordinate system is also called - Cartesian coordinate system. The coordinate method of describing geometric objects marked the beginning of analytical geometry. Pierre Fermat also contributed to the development of the coordinate method, but his works were first published after his death. Descartes and Fermat used coordinate method only on a plane.

Coordinate method for three-dimensional space was first used by Leonhard Euler in the 18th century.

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See what “Coordinate plane” is in other dictionaries:

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main cutting plane- (Pτ) Coordinate plane perpendicular to the intersection of the main plane and the cutting plane. [GOST 25762 83] Topics: cutting processing General terms: coordinate plane systems and coordinate planes... Technical Translator's Guide

instrumental main cutting plane- (Pτi) Coordinate plane perpendicular to the line of intersection of the instrumental main plane and the cutting plane. [GOST 25762 83] Topics: cutting processing General terms: coordinate plane systems and coordinate planes... Technical Translator's Guide

tool cutting plane- (Pni) Coordinate plane tangent to the cutting edge at the point under consideration and perpendicular to the instrumental main plane. [GOST 25762 83] Subjects of cutting processing General terms of coordinate plane systems and... ... Technical Translator's Guide

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 Guide

kinematic cutting plane- (Pnк) Coordinate plane tangent to the cutting edge at the point under consideration and perpendicular to the kinematic main plane ... Technical Translator's Guide

main plane- (Pv) The coordinate plane drawn through the point of interest on the cutting edge perpendicular to the direction of the speed of the main or resulting cutting movement at this point. Note In the instrumental coordinate system, the direction... ... Technical Translator's Guide

What is a coordinate plane definition. Coordinate plane: what is it? How to mark points and construct figures on a coordinate plane? Working with the coordinate plane

Point coordinates

Constructing a point at specified coordinates

Elements of the coordinate plane

Using coordinate planes

Point coordinates

Constructing a point at specified coordinates

Orty

Story

see also

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See what “Coordinate plane” is in other dictionaries: