To determine the position of a point in space, we will use Cartesian rectangular coordinates (Fig. 2).
The Cartesian rectangular coordinate system in space is formed by three mutually perpendicular coordinate axes OX, OY, 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 usually (not necessarily) the same for all axes. The OX axis is called the abscissa axis (or simply the abscissa), the OY axis is the ordinate axis, and the OZ axis is the applicate axis.
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 a point 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, and the z coordinate is called the applicate of point A.
Symbolically it is written like this:
or link the coordinate record to specific point using index:
x A , y A , z A ,
Each axis is considered as a number line, i.e. has a positive direction, and points lying on negative ray, attributed negative values coordinates (distance is taken with a minus sign). That is, if, for example, point B lay not as in the figure - on the ray OX, but on its continuation in reverse side from point O (on the negative part of the axis OX), then the x abscissa of point A would be negative (minus the distance OB). Likewise for the other two axes.
Coordinate axes OX, OY, OZ, shown in Fig. 2, form a right-handed coordinate system. This means that if you look at the YOZ plane along the positive direction of the OX axis, then the movement of the OY axis towards the OZ axis will be clockwise. This situation can be described using the gimlet rule: if the gimlet (screw with a right-hand thread) is rotated in the direction from the OY axis to the OZ axis, then it will move along the positive direction of the OX axis.
Vectors of unit length directed along the coordinate axes are called coordinate unit vectors. They are usually designated as 
(Fig. 3). There is also the designation 
The unit vectors form the basis of the coordinate system.
In the case of a right-handed coordinate system, valid following formulas with vector products of vectors:
1. Rectangular coordinate system on a plane
A rectangular coordinate system on a plane is formed by two mutually perpendicular coordinate axes X"X And Y"Y O, which is called the origin, the positive direction is chosen on each axis. IN right-sided coordinate system, the positive direction of the axes is chosen so that when the axis is directed Y"Y up, axis X"X looked to the right.
Four corners (I, II, III, IV) formed by the coordinate axes X"X And Y"Y, are called coordinate angles or quadrants (see Fig. 1).
Point position A on the plane is determined by two coordinates x And y. Coordinate x equal to the length of the segment O.B., coordinate y- length of the segment O.C. in selected units of measurement. Segments O.B. And O.C. are determined by lines drawn from the point A parallel to the axes Y"Y And X"X respectively. Coordinate x called abscissa points A, coordinate y - ordinate points A. Write it down like this: A ( x, y)
If the point A lies in coordinate angle I then point A has a positive abscissa and ordinate. If the point A lies in coordinate angle II, then the point A has a negative abscissa and a positive ordinate. If the point A lies in coordinate angle III, then the point A has negative abscissa and ordinate. If the point A lies in coordinate angle IV, then the point A has a positive abscissa and a negative ordinate.
2. Polar coordinates.
A polar grid with several angles marked in degrees.
Polar system coordinates- a two-dimensional coordinate system in which each point on the plane is defined by two numbers - angle and distance. The polar coordinate system is especially useful in cases where relationships between points are more easily represented in terms of distances and angles; in the more common Cartesian or rectangular coordinate system, such relationships can only be established by applying trigonometric equations.
The polar coordinate system is defined by a ray, which is called the zero or polar axis. The point from which this ray emerges is called the origin or pole. Any point on the plane is defined by two polar coordinates: radial and angular. Radial coordinate (usually denoted r) corresponds to the distance from the point to the origin. The angular coordinate, also called the polar angle or azimuth and denoted φ, is equal to the angle by which the polar axis must be rotated counterclockwise to get to that point.
The radial coordinate defined in this way can take values from zero to infinity, and the angular coordinate varies from 0° to 360°. However, for convenience, the range polar coordinate can be expanded beyond full angle, and also allow it to take negative values, which corresponds to a clockwise rotation of the polar axis.
3. Dividing segments into in this regard.
It is required to divide the segment AB connecting points A(x1;y1) and B(x2;y2) in a given ratio λ > 0, i.e..jpg" align="left" width="84 height=84" height=" 84">
Solution: Let's introduce vectors https://pandia.ru/text/78/214/images/image006_41.gif" width="18" height="13 src=">..gif" width="79" height=" 15 src=">, i.e. and i.e..
Equation (9.1) takes the form
Considering that equal vectors have equal coordinates, we get:
https://pandia.ru/text/78/214/images/image014_27.gif" width="56 height=28" height="28"> (9.2) and
https://pandia.ru/text/78/214/images/image016_26.gif" width="60 height=29" height="29"> (9.3)
Formulas (9.2) and (9.3) are called formulas for dividing a segment in this respect. In particular, for λ = 1, i.e..gif" width="54" height="29 src=">. In this case, the point M(x;y) is midpoint of the segment AB.
Comment:
If λ = 0, then this means that points A and M coincide if λ< 0, то точка Μ лежит вне отрезка АВ - говорят, что точка M делит отрезок АВ externally, because otherwise, i.e. AM + MB = 0, i.e. AB = 0).
4. Distance between points.
It is required to find the distance d between points A(x1;y1) and B(x2;y2) of the plane.
Solution: The required distance d is equal to the length of the vector, i.e. 
5. Equation of a line passing through two points.
If we mark two arbitrary points M1(x1, y1, z1) and M2(x2, y2, z2) on a straight line in space, then the coordinates of these points must satisfy the equation of the straight line obtained above:
.
In addition, for point M1 we can write:
.
Solving these equations together, we get:
.
This is the equation of a line passing through two points in space.
6. Determinants of 2nd order.
The value of the 2nd order determinant is easily calculated by definition using the formula.
7. Determinants of 3rd order.
https://pandia.ru/text/78/214/images/image030_15.gif" width="120" height="61 src="> scheme for calculating the determinant using the triangle method, i.e.:
https://pandia.ru/text/78/214/images/image034_15.gif" width="72" height="51 src=">DIV_ADBLOCK251">
9. Solving SLEs using Cramer's method.
Cramer's theorem: A system of N equations with N unknowns, the Determinant of which is nonzero, always has a solution, and a unique one. It is found as follows: the value of each of the unknowns is equal to a fraction, the denominator of which is the determinant of the system, and the numerator is obtained from the determinant of the system by replacing the column of coefficients for the unknown unknown with the column of the required terms.
This system of equations will have the only solution only when the determinant made up of the coefficients of X1 - n will not be equal to zero. Let us denote this determinant by the sign - Δ. If this determinant is not equal to zero, then we solve further. Then each Xi = Δi / Δ, where Δi is a determinant made up of coefficients for X1 - n, only the values of the coefficients in the i -th column are replaced with values after the equal sign in the system of equations, and Δ is the main determinant
Nth order system https://pandia.ru/text/78/214/images/image037_14.gif" width="112" height="46"> .gif" width="79" height="46">.gif" width="264" height="48">.gif" width="120" height="29">DIV_ADBLOCK252">
10. Solving SLEs using the matrix method.
Matrices make it possible to briefly write down the system linear equations. Let a system of 3 equations with three unknowns be given:
https://pandia.ru/text/78/214/images/image046_13.gif" width="75" height="41"> and matrices columns of unknown and free members
Let's find the work
https://pandia.ru/text/78/214/images/image049_13.gif" width="108" height="41"> or shorter A∙X=B.
Here are the matrices A And B are known, and the matrix X unknown. It is necessary to find it, because its elements are the solution to this system. This equation is called matrix equation.
Let the determinant of the matrix be different from zero | A| ≠ 0. Then matrix equation is solved as follows. Multiply both sides of the equation on the left by the matrix A-1, inverse of the matrix A: https://pandia.ru/text/78/214/images/image051_13.gif" width="168" height="59"> 
Decide matrix method the following system equations:
Attention: Zeros appear if one variable is missing, i.e., for example, if X3 is not given in the condition, then it is automatically equal to zero. Same with X1 and X2
https://pandia.ru/text/78/214/images/image057_9.gif" width="56 height=54" height="54">
https://pandia.ru/text/78/214/images/image065_8.gif" width="160 height=51" height="51">
Answer:
# a) Given:
https://pandia.ru/text/78/214/images/image074_5.gif" width="59 height=16" height="16"> 
Answer:
https://pandia.ru/text/78/214/images/image081_5.gif" width="106" height="50 src=">
Let's find the inverse matrix.
Subtract the 1st line from all the lines below it. This action does not contradict elementary transformations matrices.
https://pandia.ru/text/78/214/images/image083_4.gif" width="172" height="52 src=">
Subtract the 3rd line from all the lines above it. This action does not contradict elementary matrix transformations.
https://pandia.ru/text/78/214/images/image085_5.gif" width="187" height="53 src=">
Let's reduce all coefficients on the main diagonal of the matrix to 1. Divide each row of the matrix by the coefficient of this row located on the main diagonal, if it is not equal to 1. The square matrix that turns out to the right of the unit one is the inverse of the main one.
https://pandia.ru/text/78/214/images/image087_4.gif" width="172" height="52 src=">
11. Vectors. Vector addition.
http://www. bigpi. *****/encicl/articles/15/1001553/1001553A. htm
Vector name the quantity characterized numerical value, direction in space and adding up to another, similar quantity geometrically.
Graphically, vectors are depicted as directed straight segments of a certain length, like https://pandia.ru/text/78/214/images/image089_5.gif" width="17" height="17 src="> or DIV_ADBLOCK254">
Vector addition: The sum of the vectors a(a1; a2) and b(b1; b2) is the vector c(a1+b1; a2+b2). For any vectors a(a1; a2), b(b1; b2), c(с1; с2) the equalities are valid:
Theorem: Whatever the three points A, B and C, there is a vector equality https://pandia.ru/text/78/214/images/image094_4.gif" width="126" height="49">
When adding two vectors often use the so-called “ parallelogram rule" In this case, a parallelogram is constructed using the summand vectors as its adjacent sides. The diagonal of the parallelogram drawn from the point where the beginnings of the vectors connect is the required sum (Fig. 4, left).
It is easy to see (Fig. 4, right) that this rule leads to the same result as the above method. When adding more than two vectors " parallelogram rule» is practically not used due to the cumbersome construction. Vector addition is commutative, that is,
A + b = b + A.
And also, the amount a certain number vectors does not depend on the order in which they are added, that is, ( A + b) + d = a + (b + d). In this case, they say that the addition of vectors is associative, that is, the combination law is satisfied for it.
12. Dot product of vectors.
http://www. dpva. info/Guide/GuideMathematics/linearAlgebra/ScalarVectorsMultiplication/
The dot product of vectors is an operation on two vectors that results in a number (not a vector).
https://pandia.ru/text/78/214/images/image097_5.gif" width="86" height="23">
In other words, the scalar product of vectors is equal to the product of the lengths of these vectors and the cosine of the angle between them. It should be noted that the angle between two vectors is the angle that they form if they are set aside from one point, that is, the origins of the vectors must coincide.
The following simplest properties follow directly from the definition:
1. The scalar product of an arbitrary vector a and itself (scalar square of vector a) is always non-negative, and equal to the square of the length of this vector. Moreover, the scalar square of a vector is equal to zero if and only if the given vector is zero.
2. Dot product of any perpendicular vectors a and b are equal to zero.
3. The scalar product of two vectors is zero if and only if they are perpendicular or at least one of them is zero.
4. The scalar product of two vectors a and b is positive if and only if there is an acute angle between them.
5. The scalar product of two vectors a and b is negative if and only if there is an obtuse angle between them.
Alternative definition dot product, or calculating the scalar product of two vectors specified by their coordinates.
(Calculating the coordinates of a vector if the coordinates of its beginning and end are given is very simple:
Let there be a vector AB, A - the beginning of the vector, B - the end, and the coordinates of these points
A=(a1,a2,a3), B=(b1,b2,b3)
Then the coordinates of the vector AB are:
AB=(b1-a1, b2-a2, b3-a3) .
Similarly in two-dimensional space - there are simply no third coordinates)
So, let two vectors be given, defined by a set of their coordinates:
a) In two-dimensional space (on a plane)..gif" width="49" height="19 src=">
Then their scalar product can be calculated using the formula:
b) B three-dimensional space: ;
Similar to the two-dimensional case, their scalar product is calculated using the formula:
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So, let us have two vectors: https://pandia.ru/text/78/214/images/image104_4.gif" width="73" height="23 src=">
And we need to find the angle between them. Using their coordinates, we find their lengths, and then simply equate the two formulas for the scalar product. This way we get the cosine of the desired angle.
Vector length A calculated as the root of the scalar square of the vector A, which we calculate using the formula for the scalar product of vectors specified by the coordinates:
https://pandia.ru/text/78/214/images/image107_3.gif" width="365" height="23">
Means, 
, 
The required angle has been found.
13. Vector artwork.
http://www. dpva. info/Guide/GuideMathematics/linearAlgebra/vectorVectorsMultiplication/
Cross product of two vectors a and b is an operation on them, defined only in three-dimensional space, the result of which is vector with the following properties:
https://pandia.ru/text/78/214/images/image111_3.gif" width="83" height="27">, where a And b.
3) The vector is directed in such a way that if you bring the vector https://pandia.ru/text/78/214/images/image117_3.gif" width="13" height="24 src=">.gif" width=" 13" height="24"> to the vector will be COUNTER CLOCKWISE.
For greater clarity, let's give an example - in the figure on the right there is a vector - vector product vectors a and b. As stated in the definition, we have reduced all three vectors to general beginning, and then, if you look at vectors a and b from the end of the vector, the shortest turn from vector a to vector b will be counterclockwise.
https://pandia.ru/text/78/214/images/image119_3.gif" width="76" height="25">
Also, directly from the definition it follows that for any scalar factor k (number) the following is true: 
det A https://pandia.ru/text/78/214/images/image182_2.gif" width="56 height=32" height="32"> 
7.2 Finding the Determinant of a 3rd Order Matrix using the Triangle Rule
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Each element of a square Matrix (the order of which is greater than or equal to three) can be associated with two numbers called MINOR or ALGEBRAIC COMPLEMENT. The minor of an element Aij of a square Matrix A (of any order) is called the DETERMINANT OF THE MATRIX, obtained from Matrix A by deleting the row and column at the intersection of which the element Aij stands. The sign M is the designation of Minor.
https://pandia.ru/text/78/214/images/image034_15.gif" width="72" height="51 src=">.gif" width="35" height="19">
https://pandia.ru/text/78/214/images/image194_1.gif" width="96 height=82" height="82">
ELEMENTS | Minor | Algebraic Complement |
||||||||||||||||
Let A = some third-order Matrix, then the determinant of matrix A is equal to:
Note: The determinant can be calculated from the elements any strings or any column of this Matrix.
# Find the determinant of the Matrix by the elements of the first row and first column:
https://pandia.ru/text/78/214/images/image201_0.gif" width="58" height="56 src=">
https://pandia.ru/text/78/214/images/image203_0.gif" width="253" height="34 src=">
7.3 Nth order MATRIX DETERMINANT
Let A - square matrix nth order. Then, the Determinant of the nth order Matrix will look like this:
Having decomposed the elements of 1 row, find the elements of Matrix A
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2) a12=0*(2*0*1+1*0*0+1*2*0)-0*(0*0*0+1*1*1+2*0*2)=0
3) a13=2*(2*2*1+1*1*0+0*0*2)-2*(0*2*0+1*0*1+2*2*1)=0
4) a14=-1*(2*2*0+1*1*1+0*0*0)-1*(1*2*0+1*0*0+2*1*0)=- 1
6. BASIC PROPERTIES OF THE DETERMINANT
1. The determinant will not change if its rows are swapped with the corresponding columns (transpose)
2. When rearranging two rows or columns, the Definition will change its sign to the opposite.
3. Total multiplier all elements of a row (column) can be taken out of the determinant sign
4. A determinant with two identical rows or columns is always equal to zero.
5. If the elements of two rows (columns) of the determinant are proportional, then the determinant is equal to zero.
6. If in some row or column of the determinant we add, respectively, the elements of another row or column, multiplied by the same number, then the determinant will not change its value.
https://pandia.ru/text/78/214/images/image208_0.gif" width="48" height="12"> etc.
Triangular determinant- this is the determinant for which all elements lying above (or below) the main diagonal are zeros, equal to the product elements of the main diagonal.
https://pandia.ru/text/78/214/images/image210_0.gif" width="37" height="28 src=">DIV_ADBLOCK263">
If an inverse Matrix A exists, then the Matrix is called INVERTIBLE. Finding a Square Matrix has great value when solving system linear equations.
17. Inverse matrix.
http://www. mathelp. *****/book1/omatrix. htm
1. Find the Determinant of Matiritsa A
2. Find the algebraic complement of all elements of Matrix A (Aij) and write a new Matrix
3. Transpose the new Matrix
4. Multiply the transposed Matrix by the inverse of the determinant. (For example: to the number 6 the inverse determinant will be the number)
Let us denote ∆ =det A. In order for a square Matrix A to have an inverse, it is necessary and sufficient that the Matrix be non-degenerate (non-zero). Matrix, matrix inverse A, denoted by A-1, so B = A-1..gif" width="12" height="19 src=">.gif" width="82" height="34 src="> - normalizing plane multiplier, the sign of which is chosen opposite sign D, if arbitrary, if D=0.
21. Curves of the 2nd (equation of a circle).
Definition 11.1.Second order curves on a plane are called lines of intersection circular cone with planes not passing through its vertex.
If such a plane intersects all the generatrices of one cavity of the cone, then in the section it turns out ellipse, at the intersection of the generatrices of both cavities – hyperbola, and if the cutting plane is parallel to any generatrix, then the section of the cone is parabola.
Comment. All second-order curves are specified by second-degree equations in two variables.
Classification of second order curves
Non-degenerate curves
non-degenerate, if the following options may occur:
Non-degenerate curve second order is called central if
· ellipse - provided D> 0 and Δ I < 0;
a special case of an ellipse - a circle - provided I 2 = 4D or a 11 = a 22,a 12 = 0;
imaginary ellipse (not a single real point) - subject to Δ I > 0;
· hyperbole - provided D < 0;
A non-degenerate second-order curve is called non-central if Δ I = 0
· parabola - provided D = 0.
Degenerate curves: The second order curve is called degenerate, if Δ = 0. The following options may occur:
· real point at the intersection of two imaginary lines (degenerate ellipse) - provided D > 0;
· a pair of real intersecting lines (degenerate hyperbola) - provided D < 0;
· degenerate parabola - provided D = 0:
· a pair of real parallel lines - provided B < 0;
· one real line (two merged parallel lines) - provided B = 0;
· a pair of imaginary parallel lines (not a single real point) - provided B > 0.
22. Ellipse and its equation.
Definition 11.2.Ellipse is the set of points in the plane for which the sum of the distances to two fixed points is F 1 and F 2 of this plane, called tricks, is a constant value.
Comment. When the points coincide F 1 and F 2 the ellipse turns into a circle.
Headmistress Di ellipse corresponding to the focus Fi, is called a straight line located in the same half-plane with Fi relative to the axis Oh perpendicular to the axis Oh at a distance a/e from the origin.Comment. With a different choice of coordinate system, the ellipse may not be specified canonical equation(11.1), but a second degree equation of a different type.
Ellipse properties:
1) An ellipse has two mutually perpendicular axes of symmetry (the main axes of the ellipse) and a center of symmetry (the center of the ellipse). If an ellipse is given by a canonical equation, then its main axes are the coordinate axes, and its center is the origin. Since the lengths of the segments formed by the intersection of the ellipse with the main axes are equal to 2 A and 2 b (2a>2b), That main axis, passing through the foci is called the major axis of the ellipse, and the second major axis is called the minor axis.
Then https://pandia.ru/text/78/214/images/image264.gif" width="141" height="122 src=">
Let us derive the canonical equation of a hyperbola by analogy with the derivation of the equation of an ellipse, using the same notation.
|r1 - r2 | = 2a, where. If we designate b² = c² - a², from here you can get https://pandia.ru/text/78/214/images/image267.gif" width="38" height="30 src=">.gif" width="87" height="44 src="> , (11.3`)
for which the real and imaginary axis are swapped while maintaining the same asymptotes.
4) Eccentricity of the hyperbola e> 1.
5) Distance ratio ri from hyperbola point to focus Fi to the distance di from this point to the directrix corresponding to the focus is equal to the eccentricity of the hyperbola.
The proof can be carried out in the same way as for the ellipse.
23. Parabola.
Definition 11.8.Parabola is the set of points on the plane for which the distance to some fixed point is F this plane is equal to the distance to some fixed straight line. Dot F called focus parabolas, and the straight line is its headmistress.
To derive the equation of a parabola, we choose a Cartesian coordinate system so that its origin is the midpoint of the perpendicular FD, lowered from focus to the directrix, and the coordinate axes were located parallel and perpendicular to the directrix. Let the length of the segment FD
D O F x is equal to r. Then from the equality r = d it follows that https://pandia.ru/text/78/214/images/image271.gif" width="101 height=38" height="38">,
Using algebraic transformations, this equation can be reduced to the form:
y² = 2 px, (11.4) called canonical parabola equation.
Magnitude r called parameter parabolas.
Properties of a parabola :
1) A parabola has an axis of symmetry (parabola axis). The point where the parabola intersects the axis is called the vertex of the parabola. If a parabola is given by a canonical equation, then its axis is the axis Oh, and the vertex is the origin of coordinates.
2) The entire parabola is located in the right half-plane of the plane Ooh.
Comment. Using the properties of the directrixes of an ellipse and a hyperbola and the definition of a parabola, we can prove the following statement:
The set of points on the plane for which the relation e the distance to some fixed point to the distance to some straight line is a constant value, it is an ellipse (with e<1), гиперболу (при e>1) or parabola (with e=1).
Reducing a second order equation to canonical form.
Definition 11.9. Line defined general equation second order
https://pandia.ru/text/78/214/images/image274.gif" width="103 height=19" height="19"> you can set a matrix
https://pandia.ru/text/78/214/images/image276.gif" width="204" height="24 src="> (assuming that λ .
In the case where one of eigenvalues matrices A is equal to 0, equation (11.5) as a result of two coordinate transformations can be reduced to the form: , (11.8) which is the canonical equation of a parabola.
24. Rectangular coordinates in space.
Rectangular coordinate system in space formed by three mutually perpendicular coordinate axes OX, OY And OZ. The coordinate axes intersect at the point O, which is called the origin of coordinates, on each axis a positive direction is selected, indicated by arrows, and a unit of measurement for the segments on the axes. The units are usually the same for all axes (which is not mandatory). OX- abscissa axis, OY- ordinate axis, OZ- applicator axis.
If thumb right hand take for direction X, pointing for direction Y, and the average for the direction Z, then it is formed right coordinate system Similar fingers of the left hand form the left coordinate system. In other words, the positive direction of the axes is chosen so that when the axis rotates OX counterclockwise by 90° its positive direction coincides with the positive direction of the axis OY, if this rotation is observed from the positive direction of the axis OZ. It is impossible to combine the right and left coordinate systems so that the corresponding axes coincide (see Fig. 2).
Point position A in space is determined by three coordinates x, y And z. Coordinate x equal to the length of the segment O.B., coordinate y- length of the segment O.C., coordinate z- length of the segment O.D. in selected units of measurement. Segments O.B., O.C. And O.D. are determined by planes drawn from the point A parallel to the planes YOZ, XOZ And XOY respectively. Coordinate x called the abscissa of the point A, coordinate y- ordinate of the point A, coordinate z- applicate point A. Write it down like this: .
Let us now give the concept of the method of coordinates on a plane, that is, we will indicate a method that allows you to determine the position of points on a plane using numbers.
Let's take two mutually perpendicular straight lines and set a positive direction on each of them. These straight lines, relative to which we will determine the position of points on the plane, are called coordinate axes. Coordinate axes are usually positioned as shown in Fig. 6: one is horizontal and the positive direction on it is chosen from left to right, and the other is vertical and the positive direction on it is from bottom to top. One of the axes (usually horizontal) is called the abscissa axis (Ox axis), and the other is called
ordinate axis (Oy axis). The point of intersection of the coordinate axes is called the origin of coordinates (in Fig. 6, the origin of coordinates is indicated by the letter O). Finally, let's choose a scale unit (we will always assume that the same scale unit is selected on both coordinate axes).
Now the position of any point on the plane can be determined by numbers - the coordinates of this point. Indeed, for every point M of the plane there correspond on the coordinate axes two points P and Q, which are its projections onto these axes (Fig. 6) and, conversely, knowing the points on the coordinate axes, we can construct a single point M on the plane for which P and Q are projections onto these axes. Thus, determining the position of point M on the plane is reduced to determining the positions of its projections P and Q onto the coordinate axes.
But we already know that the position of a point on the axis is completely determined by the coordinate. Let be the coordinate of point P on the abscissa axis and y the coordinate of point Q on the ordinate axis. The numbers x and y completely determine the position of the point M on the plane and are called the coordinates of the point; in this case it is called the abscissa of the point M, and y is its ordinate.
Thus, the abscissa of a point is the value of a directed segment of the Ox axis, the beginning of which is the origin of coordinates, and the end is the projection of the point onto this axis; The ordinate of a point is the value of a directed segment of the Oy axis, the beginning of which is the origin of coordinates, and the end is the projection of the point onto the ordinate axis.
So, the position of any point on the plane is completely determined by specifying a pair of numbers x and y, the first of which is the abscissa of the point, and the second is its ordinate.
We agree to write the coordinates of a point in parentheses, next to the letter denoting this point, putting the abscissa in the first place, and the ordinate in the second and separating them with a comma: When indicated in Fig. 6 the location of the coordinate axes for all points of the plane lying to the right of the Oy axis (ordinate axis), the abscissa is positive, and for points lying to the left of the Oy axis it is negative. The points on the Oy axis itself have an abscissa equal to zero. In exactly the same way, points of the plane lying above the Ox axis (abscissa axis) have a positive y ordinate, and points lying below the axis have a negative axis. The points of the Ox axis itself have an ordiate equal to zero. The origin has coordinates (0, 0).
Coordinate axes divide the plane into four parts called quarters or quadrants (sometimes also called coordinate axes
corners). The part of the plane enclosed between the positive semi-axes Ox and Oy is called the first quadrant. Next, the quadrants are numbered counterclockwise (Fig. 7). For all points of the 1st quadrant for points of the 2nd quadrant in the 3rd quadrant and in the 4th quadrant
The coordinates that are taken here to determine the position of a point on the plane are called rectangular coordinates, since the point M of the plane is obtained by the intersection of two straight lines PM and QM (Fig. 6), meeting at right angles, and also Cartesian, named after the mathematician and philosopher Descartes, who in 1637 published the first work on analytical geometry.
The Cartesian rectangular coordinate system is not the only one coordinate system, which allows us to determine the positions of points on the plane (see § 11 of this chapter), but it is the simplest and in the future we will mainly use it. The described coordinate method leads to the solution of two main problems.
Problem I. Given a given point M, find its coordinates.
From this point M we lower perpendiculars on the axis. The bases of these perpendiculars - points P and Q - will determine both required coordinates. The first coordinate of the point M, its abscissa, is equal to the value of the directed segment of the OR axis. The second coordinate of the point, its ordinate, is equal to the value of the directed segment of the OQ axis
Task I. Knowing the coordinates of point M, construct this point.
Let us plot along the Ox axis from point O a segment with a length of units to the right, if and to the left, if. The end of this segment - point P - will be the projection of the desired point M onto the Ox axis, plotting along the Oy axis from point O a segment with a length of units up, if and down, if we obtain point Q - the projection of the desired point onto the Oy axis. Knowing P and Q, it is easy to construct the desired point M from these points, as projections. To do this, you need to draw straight lines through P and Q, parallel to the coordinate axes; at the intersection of these lines the desired point will be obtained
Comment. If we agree to consider the directed segments RM and QM (Fig. 6) as segments of axes, the directions of which coincide with the directions of the coordinate axes parallel to them, then the abscissa of point M will be expressed not only by the value of the segment OP,
but also an equal value of the segment QM. The ordinate of the same point will be equally expressed both by the value of the segment OQ and by the equal value of the segment PM. We will call the directed segments OP, QM, OQ and PM coordinate segments of the point M. Then, when solving the two main problems considered, there is no need to determine both projections of the point M, it is enough to determine only one, for example, the projection on the abscissa axis. So, in problem 1 we lower a perpendicular from a given point M to the abscissa axis. Its base P determines the projection of point M onto this axis. The value of the directed segment OP will give the abscissa of a given point, and the value of the segment RM will give the ordinate y.
Example. Construct a point using coordinates. Lay a segment of 2 units to the right of O along the abscissa axis; through the end P of this segment we draw a straight line parallel to the ordinate axis, and on it we lay down a segment 3 units long from P; the end of this segment is the desired point M.
Thus, in the chosen coordinate system, each point on the plane corresponds to a well-defined pair of coordinates x and y and, conversely, every pair of real numbers x, y defines a single point on the plane whose abscissa is x and whose ordinate is y. Therefore, to specify a point means to specify its coordinates; finding a point means finding its coordinates.
If we introduce a coordinate system on a plane or in three-dimensional space, we will be able to describe geometric shapes and their properties using equations and inequalities, that is, we will be able to use algebra methods. Therefore, the concept of a coordinate system is very important.
In this article we will show how a rectangular Cartesian coordinate system is defined on a plane and in three-dimensional space and find out how the coordinates of points are determined. For clarity, we provide graphic illustrations.
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Rectangular Cartesian coordinate system on a plane.
Let us introduce a rectangular coordinate system on the plane.
To do this, draw two mutually perpendicular lines on the plane and select on each of them positive direction, indicating it with an arrow, and select on each of them scale(unit of length). Let us denote the point of intersection of these lines by the letter O and consider it starting point. So we got rectangular coordinate system on a plane.
Each of the straight lines with a selected origin O, direction and scale is called coordinate line or coordinate axis.
A rectangular coordinate system on a plane is usually denoted by Oxy, where Ox and Oy are its coordinate axes. The Ox axis is called x-axis, and the Oy axis – y-axis.
Now let's agree on the image of a rectangular coordinate system on a plane.
Typically, the unit of measurement of length on the Ox and Oy axes is chosen to be the same and is laid off from the origin on each coordinate axis in the positive direction (marked with a dash on the coordinate axes and the unit is written next to it), the abscissa axis is directed to the right, and the ordinate axis is directed upward. All other options for the direction of the coordinate axes are reduced to the voiced one (Ox axis - to the right, Oy axis - up) by rotating the coordinate system at a certain angle relative to the origin and looking at it from the other side of the plane (if necessary).
The rectangular coordinate system is often called Cartesian, since it was first introduced on the plane by Rene Descartes. Even more commonly, a rectangular coordinate system is called a rectangular Cartesian coordinate system, putting it all together.
Rectangular coordinate system in three-dimensional space.
The rectangular coordinate system Oxyz is set in a similar way in three-dimensional Euclidean space, only not two, but three mutually perpendicular lines are taken. In other words, a coordinate axis Oz is added to the coordinate axes Ox and Oy, which is called axis applicate.
Depending on the direction of the coordinate axes, right and left rectangular coordinate systems in three-dimensional space are distinguished.
If viewed from the positive direction of the Oz axis and the shortest rotation from the positive direction of the Ox axis to the positive direction of the Oy axis occurs counterclockwise, then the coordinate system is called right.
If viewed from the positive direction of the Oz axis and the shortest rotation from the positive direction of the Ox axis to the positive direction of the Oy axis occurs clockwise, then the coordinate system is called left.
Coordinates of a point in a Cartesian coordinate system on a plane.
First, consider the coordinate line Ox and take some point M on it.
Each real number corresponds to a single point M on this coordinate line. For example, a point located on a coordinate line at a distance from the origin in the positive direction corresponds to the number , and the number -3 corresponds to a point located at a distance of 3 from the origin in negative direction. The number 0 corresponds to the starting point.
On the other hand, each point M on the coordinate line Ox corresponds to a real number. This real number is zero if point M coincides with the origin (point O). This real number is positive and equal to the length of the segment OM on a given scale if point M is removed from the origin in the positive direction. This real number is negative and equal to the length of the segment OM with a minus sign if point M is removed from the origin in the negative direction.
The number is called coordinate points M on the coordinate line.
Now consider a plane with the introduced rectangular Cartesian coordinate system. Let us mark on this plane arbitrary point M.
Let be the projection of point M onto the line Ox, and let be the projection of point M onto the coordinate line Oy (if necessary, see the article). That is, if through the point M we draw lines perpendicular to the coordinate axes Ox and Oy, then the points of intersection of these lines with the lines Ox and Oy are points and, respectively.
Let the number correspond to a point on the Ox coordinate axis, and the number to a point on the Oy axis.
Each point M of the plane in a given rectangular Cartesian system coordinates corresponds to a single ordered pair of real numbers called coordinates of point M on a plane. The coordinate is called abscissa of point M, A - ordinate of point M.
The converse statement is also true: each ordered pair of real numbers corresponds to a point M of the plane in given system coordinates
Coordinates of a point in a rectangular coordinate system in three-dimensional space.
Let us show how the coordinates of point M are determined in a rectangular coordinate system defined in three-dimensional space.
Let and be the projections of point M onto the coordinate axes Ox, Oy and Oz, respectively. Let these points on the coordinate axes Ox, Oy and Oz correspond to real numbers And .
