On surface. Let one be x and the other be y. And let these lines be mutually perpendicular (that is, they intersect at right angles). Moreover, the point of their intersection will be the origin for both lines, and the unit segment is the same (Fig. 1).

So we got rectangular coordinate system, and our plane became coordinate. Lines x and y are called coordinate axes. Moreover, the x axis is the abscissa axis, and the y axis is the ordinate axis. A similar plane is usually indicated by the name of the axes and the reference point - xOy. The rectangular coordinate system is also called cartesian coordinate systemsince the first time the French mathematician and philosopher Rene Descartes began to actively use it.

The rectangular angles formed by the straight lines x and y are called coordinate angles. Each corner has its own number as shown in fig. 2.

So, when we talked about the coordinate line, every point of this line had one coordinate. Now, when it comes to the coordinate plane, then every point on this plane will already have two coordinates. One corresponds to the line x (this coordinate is called abscissa), the other corresponds to the straight line y (this coordinate is called ordinate) It is written this way: M (x; y), where x is the abscissa and y is the ordinate. It reads as: "Point M with x, y coordinates."

How to determine the coordinates of a point on a plane?
Now we know that each point on the plane has two coordinates. In order to find out its coordinates, it is enough for us to draw two straight lines perpendicular to the coordinate axes through this point. The intersection points of these lines with the coordinate axes will be the desired coordinates. So, for example, in fig. 3 we determined that the coordinates of point M are 5 and 3.

How to build a point on a plane by its coordinates?
It also happens that we already know the coordinates of a point on a plane. And we need to find her location. Suppose we have the coordinates of the point (-2; 5). That is, the abscissa is -2, and the ordinate is 5. Take the point with the coordinate -2 on the x-axis (abscissa axis) and draw a line a parallel to the y axis through it. Note that any point on this line will have an abscissa equal to -2. Now we find on the y line (ordinate axis) a point with coordinate 5 and draw a straight line b parallel to the x axis through it. Note that any point on this line will have an ordinate equal to 5. At the intersection of lines a and b, there will be a point with coordinates (-2; 5). Denote it by the letter P (Fig. 4).

We also add that the line a, all of whose points have an abscissa of -2, is given by the equation
x \u003d -2 or that x \u003d -2 is the equation of the line a. For convenience, we can say not “line, which is given by the equation x \u003d -2”, but simply “line x \u003d -2”. Indeed, for any point of the line a, the equality x \u003d -2 is valid. And line b, all of whose points have ordinate 5, is in turn given by the equation y \u003d 5 or that y \u003d 5 is the equation of line b.

§ 1 Coordinate system: definition and construction method

In this lesson we will get acquainted with the concepts of “coordinate system”, “coordinate plane”, “coordinate axis”, learn how to build points on a plane by coordinates.

Take the coordinate line x with the origin point O, a positive direction and a unit segment.

Through the origin, point О of the coordinate line x, we draw another coordinate line y, perpendicular to x, we set the positive direction up, the unit segment is the same. Thus, we have constructed a coordinate system.

We give the definition:

Two mutually perpendicular coordinate lines intersecting at a point that is the origin of each of them form a coordinate system.

§ 2 Coordinate axis and coordinate plane

The straight lines that form the coordinate system are called coordinate axes, each of which has its own name: the coordinate line x is the abscissa axis, the coordinate line y is the ordinate axis.

The plane on which the coordinate system is selected is called the coordinate plane.

The described coordinate system is called rectangular. Often it is called the Cartesian coordinate system in honor of the French philosopher and mathematician Rene Descartes.

Each point of the coordinate plane has two coordinates that can be determined by lowering the perpendiculars from the point on the coordinate axis. The coordinates of a point on a plane are a pair of numbers, of which the first number is the abscissa, the second number is the ordinate. The abscissa shows the perpendicular to the x axis, the ordinate shows the perpendicular to the y axis.

We mark point A on the coordinate plane, draw perpendiculars from it to the axes of the coordinate system.

On the perpendicular to the abscissa axis (x axis) we determine the abscissa of point A, it is 4, the ordinate of point A - on the perpendicular to the ordinate axis (y axis) - this is 3. The coordinates of our points are 4 and 3. A (4; 3). Thus, the coordinates can be found for any point on the coordinate plane.

§ 3 Construction of a point on a plane

And how to build a point on a plane with given coordinates, i.e. determine the position of the point of the plane? In this case, the actions are performed in the reverse order. On the coordinate axes we find the points corresponding to the given coordinates, through which we draw straight lines perpendicular to the x and y axes. The intersection point of the perpendiculars will be the desired one, i.e. point with given coordinates.

We carry out the task: to build a point M (2; -3) on the coordinate plane.

To do this, on the abscissa axis we find a point with coordinate 2, draw a line perpendicular to the x axis through this point. On the ordinate axis, we find the point with the coordinate -3, through it we draw a straight line perpendicular to the y axis. The intersection point of the perpendicular lines will be the given point M.

Now consider a few special cases.

We note on the coordinate plane the points A (0; 2), B (0; -3), C (0; 4).

The abscissas of these points are 0. The figure shows that all points are on the ordinate axis.

Therefore, the points whose abscissas are equal to zero lie on the ordinate axis.

Change the coordinates of these points in places.

It turns out A (2; 0), B (-3; 0) C (4; 0). In this case, all ordinates are 0 and the points are on the abscissa.

Therefore, the points whose ordinates are equal to zero lie on the abscissa axis.

Let us examine two more cases.

On the coordinate plane, we mark the points M (3; 2), N (3; -1), P (3; -4).

It is easy to see that all the abscissas of the points are the same. If you connect these points, you get a straight line parallel to the ordinate axis and perpendicular to the abscissa axis.

The conclusion suggests itself: points having the same abscissa lie on one straight line that is parallel to the ordinate axis and perpendicular to the abscissa axis.

If you change the coordinates of the points M, N, P in places, you get M (2; 3), N (-1; 3), P (-4; 3). The ordinates of the points will be the same. In this case, if we connect these points, we get a straight line parallel to the abscissa axis and perpendicular to the ordinate axis.

Thus, points having the same ordinate lie on one straight line parallel to the abscissa axis and perpendicular to the ordinate axis.

In this lesson you got acquainted with the concepts of “coordinate system”, “coordinate plane”, “coordinate axes - abscissa axis and ordinate axis”. We learned how to find the coordinates of a point on a coordinate plane and learned how to build points on a plane by its coordinates.

List of used literature:

1. Maths. Grade 6: lesson plans for the textbook I.I. Zubareva, A.G. Mordkovich // author-compiler L.A. Topilina. - Mnemosyne, 2009.
2. Maths. Grade 6: a textbook for students of educational institutions. I.I. Zubareva, A.G. Mordkovich. - M.: Mnemozina, 2013.
3. Maths. Grade 6: a textbook for educational institutions / G.V. Dorofeev, I.F. Sharygin, S.B. Suvorov et al. / Edited by G.V. Dorofeeva, I.F. Sharygin; Ros.akad.nauk, Ros.akad.obrazovaniya. - M .: "Education", 2010
4. Math reference - http://lyudmilanik.com.ua
5. Reference for students in high school http://shkolo.ru

Rectangular coordinate system on the plane

The rectangular coordinate system on the plane is formed by two mutually perpendicular coordinate axes X’X and Y’Y. The coordinate axes intersect at the point O, which is called the origin, and a positive direction is chosen on each axis. The positive direction of the axes (in the right-handed coordinate system) is chosen so that when the X'X axis is rotated counter-clockwise by 90 °, its positive direction coincides with the positive the direction of the y'y axis. The four angles (I, II, III, IV) formed by the X’X and Y’Y coordinate axes are called coordinate angles (see Figure 1).

The position of point A on the plane is determined by the two coordinates x and y. The x coordinate is equal to the length of the segment OB, the y coordinate is the length of the segment OC in the selected units. The OB and OC segments are defined by lines drawn from point A parallel to the Y’Y and X’X axes, respectively. The x coordinate is called the abscissa of point A, the y coordinate is called the ordinate of point A. They are written like this: A (x, y).

If point A lies in coordinate angle I, then point A has positive abscissa and ordinate. If point A lies in coordinate angle II, then point A has a negative abscissa and a positive ordinate. If point A lies in coordinate angle III, then point A has negative abscissa and ordinate. If point A lies in coordinate angle IV, then point A has a positive abscissa and a negative ordinate.

Rectangular coordinate system in space  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 indicated by arrows and a unit of measurement for the segments on the axes are selected. Units are the same for all axes. OX is the abscissa axis, OY is the ordinate axis, OZ is the applicate axis. The positive direction of the axes is chosen so that when the OX axis is rotated 90 ° counterclockwise, 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. If the thumb of the right hand is taken for the X direction, the index for the Y direction, and the middle for the Z direction, then the right coordinate system is formed. Similar fingers of the left hand form the left coordinate system. The right and left coordinate systems cannot be combined so that the corresponding axes coincide (see Fig. 2).

The position of point A in space is determined by the 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 OC segment, the z coordinate is the length of the OD segment in the selected units. The segments OB, OC, and OD are defined by planes drawn from point A parallel to the planes YOZ, XOZ, and XOY, respectively. The x coordinate is called the abscissa of the point A, the y coordinate is the ordinate of the point A, the z coordinate is the applicate of the point A. It is written as: A (a, b, c).

Horta

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. Moreover, in the case of the right coordinate system, the following formulas with the vector product of vectors are valid:

• [i j]=k ;
• [j k]=i ;
• [k i]=j .

History

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

The coordinate method for three-dimensional space was first applied by Leonard Euler already in the XVIII century.

see also

References

Wikimedia Foundation. 2010.

See what the "Coordinate plane" is in other dictionaries:

cutting plane  - (Pn) The coordinate plane tangent to the cutting edge at the point in question and perpendicular to the main plane. [...

In topography, a network of imaginary lines encircling the globe in the latitudinal and meridional directions, with which you can accurately determine the position of any point on the earth's surface. Latitude is counted from the equator - a large circle, ... ... Geographic Encyclopedia

In topography, a network of imaginary lines encircling the globe in the latitudinal and meridional directions, with which you can accurately determine the position of any point on the earth's surface. Latitude is counted from the equator of a large circle, ... ... Collier Encyclopedia

This term has other meanings, see Phase diagram. The phase plane is the coordinate plane in which any two variables (phase coordinates) are laid out along the coordinate axes, which uniquely determine the state of the system ... ... Wikipedia

main secant plane  - (Pτ) Coordinate plane perpendicular to the line of intersection of the main plane and the cutting plane. [GOST 25762 83] Topics machining General terms of a coordinate plane system and coordinate planes ... Technical Translator Reference

instrumental major secant plane  - (Pτи) The coordinate plane perpendicular to the line of intersection of the instrumental main plane and the cutting plane. [GOST 25762 83] Topics machining General terms of a coordinate plane system and coordinate planes ... Technical Translator Reference

cutting tool plane  - (Pnи) The coordinate plane tangent to the cutting edge at the point in question and perpendicular to the instrumental main plane. [GOST 25762 83] Topics machining General terms of the coordinate plane system and ... ... Technical Translator Reference

Instruction manual

Construct three coordinate planes in order to have a reference point at point O. On the drawing, the projection plane is in the form of three axes — oh, oy and oz, with the oz axis pointing up, the oy axis pointing to the right. To build the last axis oX, divide the angle between the axes oy and oz in half (if you draw on a sheet in a cage, just draw this axis).

Note that if the coordinates of point A are written in three in brackets (a, b, c), then the first number a is from the x plane, the second b from y, and the third c from z. First, take the first coordinate a and mark it on the axis ox, left and down if the number a is positive, right and up if it is negative. Name the resulting letter B.

Then postpone the last number with up along the z axis if it is positive, and down along the same axis if it is negative. Mark received a point  letter D.

From the obtained points, draw the projections of the desired point on the planes. That is, at point B draw two straight lines that will be parallel to the axes oy and oz, at point C draw straight lines parallel to the axes oh and oz, and at point D draw straight lines parallel to oh and oy.

If one of the coordinates of the point is zero, the point lies in one of the projection planes. In this case, just mark the known coordinates on the plane and find a point  intersection of their projections. Be careful when building points with coordinates  (a, 0, c) and (a, b, 0), do not forget that the projection onto the x axis is carried out at an angle of 45 °.

Related videos

Sources:

• to build coordinates

Tip 2: How to verify that the points do not lie on the same line

Based on the axiom describing the properties straight: whatever the line is, there is pointsbelonging and not belonging to her. Therefore, it is logical that not all points  will lie on one straight  lines.

You will need

• - a pencil;
• - ruler;
• - a pen;
• - notebook;
• - calculator.

Instruction manual

In the event that (x - x1) * (y2 - y1) - (x2 - x1) * (y - y1) is less than zero, point K is located above or to the left of the line. In other words, only if an equation of the form (x - x1) * (y2 - y1) - (x2 - x1) * (y - y1) \u003d 0 is true, points  A, B and K will be located on one straight.

In other cases, only two points  (A and B), which, by the terms of the assignment, lie on straightwill belong to it: the line will not pass through the third point (point K).

Consider the second option of ownership points  prima: this time we need to check whether the point C (x, y) belongs to the segment with the end points B (x1, y1) and A (x2, y2), which is part straight  z.

The points of the considered segment are described by the equation pOB + (1-p) OА \u003d z, provided that 0≤p≤1. OB and OA are vectors. If there is a number p that is greater than or equal to 0, but less than or equal to 1, then pOB + (1-p) OА \u003d С, and, point C will lie on the segment AB. Otherwise, this point will not belong to this segment.

Write down the equality pOB + (1-p) OА \u003d С coordinatewise: px1 + (1-p) x2 \u003d x and py1 + (1-p) y2 \u003d y.

Find the number p from the first and substitute its value in the second equality. If the equality will meet the conditions 0≤p≤1, then the point C belongs to the segment AB.

note

Make sure the calculations are correct!

Useful advice

To find the k - slope of the line, you need (y2 - y1) / (x2 - x1).

Sources:

• Algorithm for checking whether a point belongs to a polygon. Ray Tracing Method in 2019

Three-dimensional space consists of three basic concepts that you gradually study in the school curriculum: point, line, plane. When working with some mathematical quantities, you may need to combine these elements, for example, to build a plane in space along a point and a straight line.

Instruction manual

To understand the algorithm for constructing planes in space, pay attention to some axioms that describe the properties of a plane or planes. First: a plane passes through three points that do not lie on one straight line, with only one. Therefore, to build a plane, you only need three points that satisfy the axiom in position.

Second: a line passes through any two points, with only one. Accordingly, you can build a plane through a straight line and a point that does not lie on it. If the opposite is true: any line contains at least two points through which it passes, if another point is known, not on this line, through these three points you can build a line, as in the first paragraph. Each point of this line will belong to a plane.

Third: a plane passes through two intersecting lines, with only one. Intersecting lines can form only one common point. If in space, they will have an infinite number of common points, and therefore be one straight line. When you know two lines that have a point of intersection, you can build no more than one plane passing through these lines.

Fourth: a plane can be drawn through two parallel lines, with only one. Accordingly, if you know that the lines are parallel, you can draw a plane through them.

Fifth: an infinite number of planes can be drawn through a straight line. All these planes can be considered as the rotation of one plane around a given line, or as an infinite number of planes having one intersection line.

So, you can build a plane if you find all the elements that determine its position in space: three points that do not lie on a line, a line and a point that does not belong to a line, two intersecting or two parallel lines.

Related videos

Do you know that the human body is a mini-power plant? Each of us produces a small amount of electricity. This happens both in motion and at rest - then the generation of electricity occurs in the internal organs, one of which is the heart.

One of the medical studies that can determine the condition of the heart is an ECG. A cardiologist takes an electrocardiogram to find out how the atria, valves and ventricles are located in the chest, their shape and whether there are any functional changes. One of the most important ECG indicators is the orientation of the electrical axis of the heart.

What is the axis of the heart and how to find it?

The heart axis (like the earth axis) cannot be seen or touched. It is determined only with the help of an electrocardiograph, because it records the electrical activity of the heart. When the cells of the heart muscle tighten and relax, obeying the impulses coming from the nervous system, they form an electric field, the center of which is the EOS (electrical axis of the heart).

But if you look into the anatomical atlas, you can draw a vertical line that will divide the heart into two equal parts - this is how the axis of the heart is located. From this we can conclude that the EOS coincides with the so-called anatomical axis. Of course, each person is individual, therefore, the electric axis in different people can be located in a different way (for example, if we start from the heart-rate value, then for a thin person the EOS is located vertically, and for a fat person - horizontally).

When does the cardiac axis change position?

Having removed the ECG and found out how the EOS is located, the cardiologist can tell you how in the chest, whether the myocardium (cardiac) is healthy, how nerve impulses pass to different parts of the heart.

If the electrocardiogram shows that the electric axis is right or left, this will indicate to the doctor any pathological process. A deviation to the right can lead to suspicions of an incorrect position of the heart (its displacement may be congenital or occur due to aortic expansion, the occurrence of neoplasms and other pathologies). In addition, the deviation of EOS is a sign of life-threatening conditions: dextrocardia, blockade of the His bundle, myocardial infarction (its anterior wall).

If EOS is significantly deviated to the left, this may be a sign of cardiomyopathy, hypertrophy of some parts of the heart, apical infarction, or congenital malformation.

A number of heart diseases can be asymptomatic for the time being. Therefore, it is so important to periodically undergo a physical examination, one of the components of which is an ECG. After all, the disease is easier to prevent. And heart disease is a must, because they are a direct threat to life.

Coordinate plane basics

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 build a coordinate plane, you need to draw $ 2 $ perpendicular lines, at the end of which are indicated with the direction arrows "right" and "up". The lines are marked with divisions, and the point of intersection of the lines is a zero mark for both scales.

Definition 1

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

Two perpendicular axes x and y with divisions make up rectangular, or cartesian, coordinate systemproposed by the French philosopher and mathematician Rene Descartes.

Coordinate plane

  Point coordinates

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

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

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

For drawing points on the coordinate plane, they act in the reverse order.

Building a point by given coordinates

Example 1

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

Decision.

Building a point $ A $:

• put the number $ 2 $ on the axis $ x $ and draw a perpendicular line;
• set the number $ 5 $ on the y axis and draw a line perpendicular to the $ y $ axis. At the intersection of the perpendicular lines we get the point $ A $ with the coordinates $ (2; 5) $.

Building a point $ B $:

• set $ 3 $ on the $ x $ axis and draw a line perpendicular to the x axis;
• on the $ y $ axis, postpone the number $ (- 1) $ and draw a line perpendicular to the $ y $ axis. At the intersection of the perpendicular lines we get the point $ B $ with the coordinates $ (3; –1) $.

Example 2

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

Decision.

Building the point $ C $:

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

Building the point $ D $:

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

Remark 1

Therefore, at the coordinate $ x \u003d 0 $ the point will lie on the $ y $ axis, and at the coordinate $ y \u003d 0 $ the point will lie on the $ x $ axis.

Example 3

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

Decision.

Define the coordinates of the point $ A $. To do this, draw through this point $ 2 $ lines that will be parallel to the coordinate axes. The intersection of the line with the abscissa gives the coordinate $ x $, the intersection of the line with the ordinate gives the coordinate $ y $. Thus, we obtain that the point $ A (1; 3). $

Define the coordinates of the point $ B $. To do this, draw through this point $ 2 $ lines that will be parallel to the coordinate axes. The intersection of the line with the abscissa gives the coordinate $ x $, the intersection of the line with the ordinate gives the coordinate $ y $. We get that the point $ B (–2; 4). $

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

Define the coordinates of the point $ D $. Because it is located on the axis $ x $, then the coordinate $ y $ is equal to zero. The coordinate $ x $ of this point is $ –5 $. Thus, the point $ D (5; 0). $

Example 4

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

Decision.

Building point $ E $:

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

Building point $ F $:

• the coordinate is $ y \u003d 0 $, which means that the point lies on the axis $ x $;
• put the number $ 5 $ on the axis $ x $ and get the point $ F (5; 0). $

Building the point $ G $:

• set the number $ 3 $ on the $ x $ axis and draw a perpendicular line to the $ x $ axis;
• on the $ y $ axis, postpone the number $ 4 $ and draw a perpendicular line to the $ y $ axis;
• at the intersection of perpendicular lines we get the point $ G (3; 4). $

Building point $ H $:

• the coordinate is $ x \u003d 0 $, which means that the point lies on the $ y $ axis;
• put the number $ (- 4) $ on the axis $ y $ and get the point $ H (0; –4). $

Building point $ O $:

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