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Now let's talk about how to plot the coordinates of points on the coordinate plane. This is the foundation you need to know in order to successfully place various shapes on the plane, and even mark equations.

When building points, you should remember how to correctly record their coordinates. So, usually asking a point, two numbers are written in brackets. The first digit indicates the coordinate of the point along the abscissa, the second - along the ordinate.

To build a point follows this way. First mark a given point on the Ox axis, then mark a point on the Oy axis. Next, draw imaginary lines from these signs and find the place of their intersection - this will be the given point.

You just have to mark it and sign it. As you can see, everything is quite simple and does not require special skills.

### Special locations for points

- If the point lies on the axis "Oy", then its abscissa is 0. For example,

point C (0, 2). - If the point lies on the Ox axis, then its ordinate is 0. For example,

point F (3, 0). - Origin - point O has coordinates equal to zero O (0,0).
- The points of any direct perpendicular abscissa axis have the same abscissas.
- Points of any straight perpendicular axis of ordinates have the same ordinates.
- The coordinates of any point lying on the abscissa axis are of the form (x, 0).
- The coordinates of any point lying on the ordinate axis have the form (0, y).

### First way

To determine the position of a point by its coordinates,

for example, points D (−4, 2), it is necessary:

- Mark on the axis “Ox”, the point with the coordinate “−4”, and draw a straight line perpendicular to the axis “Ox” through it.
- Mark the point with coordinate 2 on the Oy axis and draw a line perpendicular to the Oy axis through it.
- The intersection point of perpendiculars (·) D is the desired point. Her abscissa is “−4”, and the ordinate is 2.

### Second way

To find the point D (−4,2), you need:

- Shift along the x-axis to the left by 4 units, since we have

4 is before “-”. - To rise from this point parallel to the y axis up 2 units, since we have a “+” before 2.

To quickly and conveniently find the coordinates of points or build points by coordinates on an A4 sheet in a cell, you can download and use the ready-made coordinate system on our website.

## Projection, projection types

For convenience, consideration of spatial figures using drawings with the image of these figures.

**Projection of a figure onto a plane** - drawing of a spatial figure.

Obviously, there are a number of rules used to construct a projection.

**Projection** - the process of constructing a drawing of a spatial figure on a plane using the rules of construction.

**Projection plane** - This is the plane in which the image is built.

The use of certain rules determines the type of projection: **central** or **parallel**.

A special case of parallel projection is perpendicular or orthogonal projection: it is mainly used in geometry. For this reason, in speech, the adjective “perpendicular” is often omitted: in geometry they simply say “projection of a figure” and mean by this construction of a projection by the method of perpendicular projection. In particular cases, of course, otherwise may be agreed.

We note the fact that the projection of a figure onto a plane is essentially the projection of all points of this figure. Therefore, in order to be able to study the spatial figure in the drawing, it is necessary to get the basic skill of projecting a point on a plane. What we will talk about below.

## The projection of a point onto a plane

Recall that most often in geometry, speaking of projection onto a plane, they mean the use of perpendicular projection.

We make constructions that will enable us to obtain a definition of the projection of a point onto a plane.

Suppose a three-dimensional space is defined, and in it there is a plane α and a point M 1 that does not belong to the plane α. Draw a line through the given point M 1 *but* perpendicular to a given plane α. We denote the intersection point of the line a and the plane α as H 1; by construction, it will serve as the basis of the perpendicular dropped from the point M 1 onto the plane α.

If a point M 2 belonging to a given plane α is given, then M 2 will serve as a projection of itself onto the plane α.

**The projection of a point onto a plane** - this is either the point itself (if it belongs to a given plane), or the base of a perpendicular dropped from a given point to a given plane.

## Finding the coordinates of the projection of a point on a plane, examples

Let them be given in three-dimensional space: a rectangular coordinate system O x y z, plane α, point M 1 (x 1, y 1, z 1). It is necessary to find the coordinates of the projection of the point M 1 on a given plane.

The solution obviously follows from the above definition of the projection of a point onto a plane.

We denote the projection of the point M 1 on the plane α as H 1. By definition, H 1 is the intersection point of a given plane α and a line a drawn through a point M 1 (perpendicular to the plane). Those. the coordinates of the projection of the point M 1 we need are the coordinates of the point of intersection of the line a and the plane α.

Thus, to find the coordinates of the projection of a point on a plane, it is necessary:

- get the equation of the plane α (if it is not specified). An article about the types of plane equations

- determine the equation of a line a passing through a point M 1 and perpendicular to the plane α (study the topic of the equation of a line passing through a given point perpendicular to a given plane),

- find the coordinates of the point of intersection of the line a and the plane α (article - finding the coordinates of the point of intersection of the plane and the line). The obtained data will be the coordinates we need for the projection of the point M 1 onto the plane α.

Consider the theory with practical examples.

Determine the coordinates of the projection of the point M 1 (- 2, 4, 4) on the plane 2 x - 3 y + z - 2 = 0.

**Decision**

As we see, the equation of the plane is given to us, i.e. there is no need to compose it.

We write the canonical equations of the line a passing through the point M 1 and perpendicular to the given plane. For these purposes, we determine the coordinates of the directing vector of the line a. Since the line a is perpendicular to the given plane, the directing vector of the line a is the normal vector of the plane 2 x - 3 y + z - 2 = 0. Thus, a → = (2, - 3, 1) is the directing vector of the line a.

Now we compose the canonical equations of the line in the space passing through the point M 1 (- 2, 4, 4) and having the direction vector a → = (2, - 3, 1):

x + 2 2 = y - 4 - 3 = z - 4 1

To find the desired coordinates, the next step is to determine the coordinates of the point of intersection of the line x + 2 2 = y - 4 - 3 = z - 4 1 and the plane 2 x - 3 y + z - 2 = 0 *.* For these purposes, we pass from the canonical equations to the equations of two intersecting planes:

x + 2 2 = y - 4 - 3 = z - 4 1 ⇔ - 3 · (x + 2) = 2 · (y - 4) 1 · (x + 2) = 2 · (z - 4) 1 · ( y - 4) = - 3 · (z + 4) ⇔ 3 x + 2 y - 2 = 0 x - 2 z + 10 = 0

We compose a system of equations:

3 x + 2 y - 2 = 0 x - 2 z + 10 = 0 2 x - 3 y + z - 2 = 0 ⇔ 3 x + 2 y = 2 x - 2 z = - 10 2 x - 3 y + z = 2

And solve it using the Cramer method:

∆ = 3 2 0 1 0 - 2 2 - 3 1 = - 28 ∆ x = 2 2 0 - 10 0 - 2 2 - 3 1 = 0 ⇒ x = ∆ x ∆ = 0 - 28 = 0 ∆ y = 3 2 0 1 - 10 - 2 2 2 1 = - 28 ⇒ y = Δ y Δ = - 28 - 28 = 1 Δ z = 3 2 2 1 0 - 10 2 - 3 2 = - 140 ⇒ z = Δ z Δ = - 140 - 28 = 5

Thus, the desired coordinates of a given point M 1 on a given plane α will be: (0, 1, 5).

**Answer:** ( 0 , 1 , 5 ) .

In the rectangular coordinate system O x y z of three-dimensional space, points A (0, 0, 2), B (2, - 1, 0), C (4, 1, 1), and M are given_{1}(-1, -2, 5). It is necessary to find the coordinates of the projection M 1 onto the plane A B C

*Decision*

First of all, we write the equation of a plane passing through three given points:

x - 0 y - 0 z - 0 2 - 0 - 1 - 0 0 - 2 4 - 0 1 - 0 1 - 2 = 0 ⇔ xyz - 2 2 - 1 - 2 4 1 - 1 = 0 ⇔ ⇔ 3 x - 6 y + 6 z - 12 = 0 ⇔ x - 2 y + 2 z - 4 = 0

Next, consider another solution that is different from what we used in the first example.

We write the parametric equations of the line a, which will pass through the point M 1 perpendicular to the plane A B C. The plane x - 2 y + 2 z - 4 = 0 has a normal vector with coordinates (1, - 2, 2), i.e. the vector a → = (1, - 2, 2) is the directing vector of the line a.

Now, having the coordinates of the point of the line M 1 and the coordinates of the directing vector of this line, we write the parametric equations of the line in space:

x = - 1 + λ y = - 2 - 2 · λ z = 5 + 2 · λ

Then we determine the coordinates of the intersection point of the plane x - 2 y + 2 z - 4 = 0 and the line

x = - 1 + λ y = - 2 - 2 · λ z = 5 + 2 · λ

To do this, substitute in the equation of the plane:

x = - 1 + λ, y = - 2 - 2 · λ, z = 5 + 2 · λ

Now, using the parametric equations x = - 1 + λ y = - 2 - 2 · λ z = 5 + 2 · λ, we find the values of the variables x, y and z for λ = - 1: x = - 1 + (- 1) y = - 2 - 2 · (- 1) z = 5 + 2 · (- 1) ⇔ x = - 2 y = 0 z = 3

Thus, the projection of the point M 1 onto the plane A B C will have coordinates (- 2, 0, 3).

**Answer:** ( - 2 , 0 , 3 ) .

Let us separately dwell on the question of finding the coordinates of the projection of a point on coordinate planes and planes that are parallel to coordinate planes.

Let the points M 1 (x 1, y 1, z 1) and the coordinate planes O x y, O x z and O y z be given. The coordinates of the projection of this point on these planes will be, respectively: (x 1, y 1, 0), (x 1, 0, z 1) and (0, y 1, z 1). Consider also the planes parallel to the given coordinate planes:

C z + D = 0 ⇔ z = - D C, B y + D = 0 ⇔ y = - D B

And the projections of the given point M 1 on these planes will be the points with coordinates x 1, y 1, - D C, x 1, - D B, z 1 and - D A, y 1, z 1.

Let us demonstrate how this result was obtained.

As an example, we define the projection of the point M 1 (x 1, y 1, z 1) onto the plane A x + D = 0. Other cases - by analogy.

The given plane is parallel to the coordinate plane O y z and i → = (1, 0, 0) is its normal vector. The same vector serves as a directing vector of a line perpendicular to the plane O y z. Then the parametric equations of the line drawn through the point M 1 and perpendicular to the given plane will have the form:

x = x 1 + λ y = y 1 z = z 1

Find the coordinates of the intersection point of this straight line and the given plane. First, substitute in the equation A x + D = 0 the equalities: x = x 1 + λ, y = y 1, z = z 1 and get: A · (x 1 + λ) + D = 0 ⇒ λ = - DA - x one

Then we calculate the desired coordinates using the parametric equations of the line with λ = - D A - x 1:

x = x 1 + - D A - x 1 y = y 1 z = z 1 ⇔ x = - D A y = y 1 z = z 1

That is, the projection of the point M 1 (x 1, y 1, z 1) onto the plane will be the point with coordinates - D A, y 1, z 1.

It is necessary to determine the coordinates of the projection of the point M 1 (- 6, 0, 1 2) on the coordinate plane O x y and on the plane 2 y - 3 = 0.

**Decision**

The coordinate plane O x y will correspond to the incomplete general equation of the plane z = 0. The projection of the point M 1 on the plane z = 0 will have coordinates (- 6, 0, 0).

The equation of the plane 2 y - 3 = 0 can be written as y = 3 2 2. Now just write down the coordinates of the projection of the point M 1 (- 6, 0, 1 2) onto the plane y = 3 2 2:

**Answer:** (- 6, 0, 0) and - 6, 3 2 2, 1 2

## Coordinate plane basics

As you know, each house has its number and street name - this is the address of the house. On the ticket to any auditorium, the row number and seat number are written - this is the seat address. To determine the position of a point on the globe, you need to know longitude and latitude - this is the address of a geographic point (geographic coordinates). Each object has its own ordered address (coordinates). Thus, the address or coordinates is a numerical or letter designation of the place where the object is located.

Mathematicians have developed a model that, in particular, allows you to describe any auditorium (location of seats in the hall). This model is called the coordinate plane.

In order to get a coordinate plane from an ordinary plane, it is necessary to draw two perpendicular straight lines, marking with arrows “to the right” and “up” (see Fig. 1). The lines are marked as a ruler, and the point of intersection of the lines is the zero mark for both scales. The horizontal line is designated and called **ordinate axis**.

Two perpendicular axes with a marking are called a rectangular, or Cartesian, coordinate system. The name "Cartesian" comes from the name of the French philosopher and mathematician Rene Descartes, who invented it.

Fig. 1. The coordinate plane

## Point coordinates

For any point on the coordinate plane, you can specify two numbers (coordinates). Figure 2 shows the point (see Figure 2).

Fig. 2. Determination of the coordinates of points on the coordinate plane

You can do everything in the reverse order. That is, draw a point on the plane in known coordinates.

1. To build points on the given coordinates

To build a point.

To build a point. (see Fig. 3).

Fig. 3. Construction of points on the coordinate plane according to the specified coordinates

2. Draw points on the given coordinates

To plot a point (see Fig. 4).

To plot a point (see Fig. 4).

Fig. 4. Construction of points on the coordinate plane according to the specified coordinates

Thus, if the coordinate is equal to zero.

1. Write out the coordinates of the points (see. Fig. 5).

2. Draw dots.

Fig. 5. Illustration for the problem

1. To determine the coordinates of a point (see Fig. 6).

To determine the coordinates of a point.

Point .

Point .

Fig. 6. Illustration for the problem

2. To build a point.

Coordinate

To build a point.

Coordinate

The two coordinates of the point, that is, the point of intersection of the two axes (origin).

Fig. 7. Illustration for the problem

## Coordinate quarters

Coordinate axes break the coordinate plane into four parts - quarters. The ordinal numbers of the quarters are considered counterclockwise (see Fig. 8).

Fig. 8. Numbering of quarters of the coordinate plane

If the point has a positive coordinate, then it lies in the first quarter.

If the point has a negative coordinate, then it lies in the second quarter.

If the point has a negative coordinate, then it lies in the third quarter.

If the point has a positive coordinate, then it lies in the fourth quarter.

For example, the point is negative, therefore, this point is in the fourth quarter.

**Other coordinate systems**

To assign a point a numerical "address" (coordinates), other coordinate systems are used.

Reasons for using different coordinate systems:

In this lesson, we looked at a rectangular coordinate system on a plane. The dimension of such a space is 2, that is, the point was specified by two coordinates. However, space can have a different dimension, for example, equal to unity, when a point can change its position in only one direction (move back and forth or up and down). An example is the movement of a car on a flat road or the movement of an elevator. Only one coordinate is needed to indicate the location of a point. This coordinate will mean the distance that the car traveled (see. Fig. 9), or the floor on which the elevator is located (see. Fig. 10).

Fig. 9. The coordinate in this case is the distance the car has moved

Fig. 10. The coordinate in this case is the floor on which the elevator is located.

In mathematics, such a coordinate system is represented by a numerical or coordinate axis. In order to get the coordinate axis from any straight line, it is necessary to mark the origin, scale and direction (see Fig. 11). From one coordinate, you can clearly understand where the point is.

Fig. 11. Coordinate axis

The dimension of space can be equal to three (the space in which we live has three dimensions). In this case, three coordinates are needed to indicate the position of the point. For example, if there is a cinema in a high-rise building on each floor, then three coordinates must be indicated to indicate the place on the ticket - floor, row, seat number. In mathematics, such a coordinate system is constructed in the same way as on a plane, only the third axis is added (see Fig. 12).

Fig. 12 Cartesian coordinate system in space

2. Another method for setting the coordinates of a point (using a polar coordinate system on a plane).

An axis is drawn (see Fig. 13).

Fig. 13. Polar coordinate system on the plane

In three-dimensional space similar systems are constructed, for example, a spherical or cylindrical coordinate system.

Thus, a rectangular coordinate system is widely used in mathematics, but is not the only one.

**Bibliography**

1. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Schwarzburd S.I. Mathematics 6. - M.: Mnemosyne, 2012.

2. Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics grade 6. - Gymnasium. 2006.

3. Depman I.Ya., Vilenkin N.Ya. Behind the pages of a math textbook. - M .: Education, 1989.

4. Rurukin A.N., Tchaikovsky I.V. Tasks on the course of mathematics 5-6 grade. - M.: ZH MEPhI, 2011.

5. Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5–6. A manual for students of 6 classes of the correspondence school of MEPhI. - M.: ZH MEPhI, 2011.

6. Shevrin L.N., Gein A.G., Koryakov I.O., Volkov M.V. Mathematics: Interlocutor textbook for grades 5–6 of high school. - M .: Education, Library of the teacher of mathematics, 1989.

**Additional recommended links to Internet resources**

1. Website mathematics-repetition.com (Source)

2. Website youtube.com (Source)

3. Website exponenta.ru (Source)

**Homework**

1. Questions at the end of section 45 (§9), assignment 1393, 1394, 1396, 1398 (pp. 245-246) - Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Schwarzburd S.I. . Mathematics 6 (Source)

2. Select the points located above the abscissa axis:.

3. In the coordinate plane, construct the following points connecting them sequentially with the previous point by a segment (get a certain picture):.

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