Def.20 A set of fixed point O and orthonormal basis( ; ) is called Cartesian(or rectangular) coordinate system on a plane. Point O is called origin of coordinates. The straight lines Ox and Oy passing through the origin of coordinates in the direction of the basis vectors and (Fig. 13) are called coordinate axes: Ox - axis abscissa, Oy - axis ordinate. We will denote the coordinate system as O or xOy, and the plane with the corresponding coordinate system will be called the Oxy plane.
It is easy to see that the Cartesian coordinate system on a plane is specified by two mutually perpendicular straight lines - axes, on each of which a positive direction is chosen and a segment of unit length is specified. The coordinate axes divide the plane into four regions - quarters or quadrants.
The quarters are numbered counterclockwise, as in Fig. 13.
Let's consider an arbitrary point M of the Oxy plane (Fig. 13). Radius vector point M in relation to point O is called a vector connecting the origin of coordinates with a given point.
Coordinates points M in the coordinate system O are called the coordinates of the radius vector in the basis ( ; ). If =( X; at), then the coordinates of point M are written as follows: M( X; at), number X called abscissa points M, at - ordinate points M.
The coordinates of a point can be found as projections of the radius vector onto each of the axes, x = a x = Pr oh and y = a y = Pr ou , = (a x; a y).
Conversely: if M( X; at), then =( X; at).
Def.21 The set of a fixed point O and an orthonormal basis (;;) is called Cartesian(or rectangular) coordinate system in space of dimension n=3.
As on the plane, point O is called origin. The straight lines Ox, Oy and Oz, passing through the origin of coordinates in the direction of the basis vectors , , (Fig. 14), are called coordinate axes:
Ox – axis abscissa, Oy – axis ordinate, Oz – axis applicate.
Planes passing through the coordinate axes are called coordinate planes. They divide the space into eight areas - octants. Point coordinates M are the coordinates of the radius vector in the basis ( ; ; ), and if =( X; y; z), then they write M( X; y; z), where x is the abscissa, y is the ordinate, z is the applicate of point M.
Conversely: if M( X; y; z), then =( X; y; z).
Rectangular system coordinates in space makes it possible to establish a one-to-one correspondence between points in space and ordered triplets of numbers (their coordinates), and on the plane - between points of the plane and ordered pairs of numbers.
In the Cartesian coordinate system, an ordered pair of numbers simultaneously specifies both a point on a given plane, the radius vector of this point, and a whole set of vectors equal to it. The same is true in the three-dimensional case. In the future, we will define vectors not by two points (start and end), but only by the end point, indicating its coordinates. We count starting point of all vectors (unless otherwise stated separately), point O is the origin of coordinates.
Similar to the considered cases n=2 and n=3, we can introduce the concept of a Cartesian coordinate system of n-dimensional space, which can be denoted R n. Points of such a space, like vectors, are specified by specifying an ordered set of n numbers 
- its Cartesian coordinates.
Actions on vectors in coordinate form
Let the vectors be given in the Cartesian coordinate system Oxyz
=x 1 × +y 1 × +z 1 × and =x 2 × +y 2 × +z 2 ×,
those. =(x 1; y 1; z 1) And =(x 2; y 2; z 2) .
Cartesian, unless specifically stated otherwise.
With such vectors you can do the following: actions:
comparison, sum (difference), multiplication by number, scalar product, find the magnitude of the vector, the angle between the vectors and check the vectors for collinearity. Let's look at them in more detail:
Comparison
Two vectors are equal if and only if they are equal All their coordinates of the same name , = (x 1 = x 2, y 1 =y 2, z 1 =z 2).
Violation of at least one equality indicates inequality of vectors.
Vectors of different dimensions are incomparable.
Operations “< “ и “ >” on vectors are not specified;
7.2 Sum and difference of vectors:
the coordinates of the sum (difference) of two vectors are equal to the sum (difference)
coordinates of these vectors of the same name,
± =(x 1 × +y 1 × +z 1 × ) ± (x 2 × +y 2 × +z 2 × )= (x 1 ±x 2)× +(y 1 ±y 2)× +(z 1 ±z 2)× ,
(x 1 ; y 1 ; z 1) ± (x 2 ; y 2 ; z 2) = (x 1 ±x 2 ; y 1 ±y 2 ; z 1 ±z 2);
7.3 At multiplying a vector by a number all of them are multiplied by this number
coordinates , l× =l×(x 1 +y 1 +z 1 )=(lх 1)× +(ly 1)× +(lz 1)× ,
l×(x 1 ; y 1 ; z 1)= (l×x 1 ; l×y 1 ; l×z 1);
7.4 Dot product two vectors in coordinate form:
(8) × =x 1 ×x 2 +y 1 ×y 2 + z 1 ×z 2,
i.e. the scalar product of two vectors is equal to the sum of pairwise
products of coordinates of the same name.
Let's derive this formula:
× =(x 1 +y 1 +z×)×(x 2 +y 2 +z×)=
= x 1 x 2 × 2 + x 1 y 2 × × + x 1 z 2 × +
+ y 1 x 2 × × + y 1 y 2 × 2 + y 1 z 2 × +
+ z 1 x 2 × + z 1 y 2 × + z 1 z 2 × 2 .
Vectors , , are orthonormal,
those. for them × = × = = = = = 0, 2 = 2 = 2 =1 ,
That's why × =x 1 x 2 × 1+y 1 y 2 × 1+ z 1 z 2 × 1;
Vector module
When = formula (8) takes the form × = 2 = x 1 2 +y 1 2 +z 1 2 , –
Definition 2.8. Three mutually perpendicular axes in space with a common origin ABOUT and with the same scale unit they form Cartesian coordinate system in space . The axes are numbered in a certain order and are called: first - axis Oh or abscissa axis , the second is the axis OU or ordinate axis , third – axis Oz or axis applicate .
Let M is an arbitrary point in space, and , and are projections of this point on the axis Oh, OU And Oz, accordingly (see Fig. 2.2).
Definition 2.9. Cartesian coordinates of point M are called algebraic quantities of vectors , and . This is indicated as follows: M(X, at, z), where , , (Fig. 2.2). Cartesian coordinates X, at And z points M they also call her abscissa, ordinate and applicate , respectively.
Definition 2.10. Let a vector be given in space A = . Cartesian vector coordinates A are called projections , and this vector onto the coordinate axes (see Fig. 2.3). Designation: A = .
Rice. 2.2 Fig. 2.3
If the coordinates of the points are known 
And 
, then the coordinates of the vector are calculated using the formulas:
Calculation formulas vector length A, and distances between points And :
. (2.2a)
The Cartesian coordinates of a vector on a plane are determined similarly, with the difference that there is no applicate axis and, accordingly, a third coordinate. Thus, if A = and , then obviously
(2.2b)
Definition 2.11. Let us denote a, b and g – the angles of inclination of the vector A to coordinate axes Oh, OU And Oz, respectively. The three numbers cosa, cosb and cosg are called direction cosines of the vector A .
The following equalities are valid:
Formulas for calculating direction cosines:
If equalities (2.4) are squared and added, we obtain:
Thus, the sum of the squares of the direction cosines of any vector is equal to one.
Since any vector is uniquely determined by specifying its three coordinates, we now see that any vector is also uniquely determined by specifying its length and direction cosines.
Linear operations It is customary to call the operation of adding vectors and the operation of multiplying a vector by a number.
Definition 2.12. Summoya +b two vectors A And b is called a vector that comes from the beginning of the vector A to the end of the vector b , provided that the vector b attached to the end of the vector A .
There are two ways to add vectors: according to the triangle rule (Fig. 2.4) and according to the parallelogram rule (Fig. 2.5).
Rice. 2.4 Fig. 2.5
Vector addition has the following properties:
1) commutative property: a + b = b + a ;
2) associative property: ( a+b ) + c = a + (b+c ).
Definition 2.13. Vector – A called reverse vector A , if it is collinear A , has length equal to | a |, and is directed in the opposite direction.
Obviously, a + (-A ) = 0.
Definition 2.14. By difference two vectors a And b called a vector a – b = a + (–b ).
In Fig. Figure 2.6 shows how to construct the vector difference in two different ways.
Definition 2.15. Product of a number a to vector A called vector a A , collinear to the vector A , having length |a|×| a | and directed the same way A , if a > 0, and in the opposite direction, if a< 0. Если a = 0, то aA = 0.
Let us list the properties that the operation of multiplying a vector by a number has.
1. Distribution property relative to the sum of vectors: a( a + b ) = a a + a b.
2. Distributive property regarding the sum of numbers: (a + b) A =a A +b A .
3. Combination property: a(b A ) = (ab) A .
4. If the vector b collinear to the vector A , then there exists a number l such that b = l a .
Vector in Cartesian coordinate system
Definition. A vector is an ordered pair of points (the beginning of the vector and its end). If , then the vector has coordinates .
A vector in the Oxyz coordinate space can be represented as
Where the triple is called the coordinates of the vector. Vectors are unit vectors (orts) directed to positive side coordinate axes Ox, Oy and Oz, respectively. The length (modulus) of a vector is the number.
Linear operations with vectors
The addition of vectors is determined by the parallelogram rule: the vector is the diagonal of a parallelogram built on the vectors and (Fig. 1a).
The difference between two vectors is determined by the formula , where is a vector of the same length as the vector , but in the opposite direction. To find the difference vector, you need to set aside the vectors and from a common point, connect the ends of the vectors with a vector directed from the “subtracted” to the “reduced” (that is, from to) (Fig. 1b). The constructed vector will be the required difference.
When adding several vectors, each coordinate of the sum is the sum of the corresponding coordinates of the component vectors; when multiplying a vector by a given number, the coordinates of the vector are also multiplied by the same number:
b) , where is a scalar factor.
Several vectors are called collinear (coplanar) if they are parallel to the same line (plane). Vectors and are parallel (collinear), that is, the corresponding coordinates of these vectors are proportional with the same proportionality coefficient: .
Basis on the plane and in space
Definition. A basis on a plane (in space) is an ordered pair (triple) of non-collinear (non-coplanar) vectors. Any vector can be uniquely decomposed into a basis. The expansion coefficients are called the coordinates of this vector relative to a given basis. The vectors form a basis in the Cartesian coordinate space Oxyz.
Vectors are given. Show that the vectors and form a basis on the plane and find the coordinates of the vector in this basis.
Solution. If two vectors are non-collinear (), then they form a basis in the plane. Since , the vectors and are non-collinear and, therefore, form a basis. Let the vector have coordinates in this basis, then the expansion of the vector into vectors and has the form , or in coordinate form
Having solved the resulting system of equations in some way, we obtain that.
Means . Thus, in the basis the vector has coordinates .
Scalar, vector, mixed product of vectors.
Definition. The scalar product of two vectors is the number defined by the equality:
where is the angle between the vectors and . If, then.
Knowing the scalar product, you can determine the angle between two vectors using the formula: .
The condition for perpendicularity of non-zero vectors (the angle between them is 90°) has the form: , or , and the condition for their collinearity: , or .
Properties of the dot product:
1) ; 2) ; 3) ; 4) , and .
Example 2. Find the angle between the vectors and if , , , .
Solution. Let's use the formula. Let's determine the coordinates of the vectors and , taking into account that when adding vectors we add coordinates of the same name, and when multiplying a vector by a number, we multiply each coordinate of this vector by this number, and: , .
Let us find the scalar product of the vectors and and their lengths. . . . Substituting into the formula, we get . From here.
Definition. The cross product of a vector and a vector is a vector (another notation) that:
a) has length , where is the angle between the vectors and ;
b) perpendicular to the vectors and () (that is, perpendicular to the plane in which the vectors and lie);
c) is directed so that the vectors , , form a right-hand triple of vectors, that is, from the end of the third vector the shortest turn from the first to the second is visible counterclockwise (Fig. 2).
The coordinates of the vector product of a vector and a vector are determined by the formula:
Geometric meaning vector product: the modulus of the vector is equal to the area of the parallelogram built on the vectors and.
Properties of a vector product:
3) ; 4) and collinear.
Example 3. The parallelogram is built on the vectors and , where , , . Calculate the length of the diagonals of this parallelogram, the angle between the diagonals and the area of the parallelogram.
Let us denote the angle between the diagonals by the letter , then
Hence, .
Using the properties of the vector product, we calculate the area of the parallelogram:
Definition. The mixed product of three vectors, , is the scalar product of a vector and a vector:
If then the mixed product can be calculated using the formula:
Properties of a mixed product:
1) When any two vectors are rearranged, the mixed product changes sign;
4) coplanar.
The geometric meaning of the mixed product: the volume of a parallelepiped built on the vectors , , (Fig. 4), and the volume of the triangular pyramid formed by them are found by the formulas.
Example 4. Are the vectors , , , coplanar?
Solution. If the vectors are coplanar, then by property 4) their mixed product is equal to zero. Let's check it out. Let's find the mixed product of these vectors by calculating the determinant:
vectors , , are non-coplanar.
Division of a segment in this respect.
Let a segment in Oxyz space be specified by points and . If it is divided by a point in the relation , then the coordinates of the point are as follows:
Example 5. Find the point dividing the segment in the relation if .
Solution. Let's determine the coordinates of the point:
Thus, .
Analytic geometry.
Equation of a plane. The general equation of a plane has the form: , , where is the normal vector of the plane (i.e. perpendicular to the plane), and the coefficient is proportional to the distance from the origin to the plane.
The equation of a plane passing through a point perpendicular to the vector has the form
The equation of a plane passing through three given points has the form:
The angle between two planes having normal vectors and is defined as the angle between the vectors and by the formula:
The distance from a point to a plane is calculated by the formula.
Example 6. Write the equation of the plane passing through the points , , .
Solution. Let's use the equation of a plane passing through three given points. Let's calculate the determinant
Or - the desired equation of the plane.
Equation of a straight line on a plane. The general equation of a straight line on a plane has the form: , where is the normal vector of the straight line (perpendicular to the straight line), and the coefficient is proportional to the distance from the origin to the straight line.
The equation of a line passing through a given point has the form
In another form, where is the tangent of the angle formed by the straight line and the positive direction of the Ox axis, called the angular coefficient, b is the ordinate of the point of intersection of the straight line with the Oy axis.
The equation of a straight line passing through two given points and has the form
The angle between two straight lines is determined by the formula
The distance from a point to a straight line is found by the formula
Example 7. Given are the equations of two sides of a rectangle and the equation of its diagonal. Write equations
the remaining sides and the second diagonal of this rectangle.
Solution. Let's make a schematic drawing (Fig. 6). Let's rewrite these equations in the form: , , . Since the angular coefficients of the lines defining the sides of the rectangle are the same, these equations define parallel lines, that is, the sides lying on them are opposite. Let's find the points of intersection of this diagonal with these sides. Let these be dots and . To do this, first equate 1 and 3, and then 2 and 3 equations:
; . Thus, .
The unknown sides are parallel to each other and perpendicular to the data (since it is a rectangle).
Comment. Angle coefficients perpendicular lines and are related by the relation .
Thus, the equations of the unknown sides of the rectangle are:
Substituting the coordinates of the point into the first equation and the points into the second, we obtain that and, therefore, , .
Let's find the coordinates of points and by equating the equations of the corresponding sides:
That is ;
That is .
We obtain the equation of the diagonal as the equation of a straight line passing through two given points and:
Equations of a straight line in space. A straight line in Oxyz space is defined as the line of intersection of two planes ( general equations straight line in space).
The canonical equations of a line in space have the form
where is the point through which the line passes, and the vector parallel to this line is called the direction vector of the line.
The equations of a straight line in space passing through two given points have the form
The angle between two straight lines with direction vectors is determined by the formula
Example 8. A pyramid is specified by the coordinates of its vertices , , . Need to find:
1) length of ribs and; 2) the angle between the ribs and ; 3) area of the face containing the vertices; 4) volume of the pyramid; 5) equations of lines and ;
6) equation of the height lowered from the vertex to the plane;
7) distance from the top to the plane; 8) the angle between the edge and the face containing the vertices.
Solution.1) The lengths of the edges and are determined as the module of the vectors and by the formulas;
2) Find the coordinates of the vectors and:
The lengths of these vectors, i.e. the lengths of the edges and are as follows:
The cosine of the angle between the edges and is calculated using the formula;
3) The area of the face (triangle) is equal to half the area of the parallelogram constructed on the vectors and , i.e. half the modulus of the vector product of these vectors, which is equal to
Then, (sq. units);
4) The volume of the pyramid is .
5) We will find the equations of lines and as equations of lines passing through two given points:
(): (abscissas of points and identical);
6) The direction vector of the height is the normal vector of the plane. We get the equation of the plane:
Equation of a plane. Then the normal vector of the plane has coordinates . The canonical equations of a line passing through a point parallel to a vector have the form: ;
7) To calculate the distance from the vertex to the plane, we use the formula. In our case, the equation of the plane and . So, ;
8) The angle between a straight line and a plane is found by the formula:
Where is the normal vector of the plane. and (see paragraph 7). Thus, ,
Second order curves
Definition. A parabola is a set of points on a plane (see Fig. 7a), for each of which the distance to a given point (the focus of the parabola) is equal to the distance to a certain given straight line (directrix). The distance from the focus of the parabola to the directrix is called the parameter of the parabola. Parabola is a symmetrical curve; the point of intersection of a parabola with its axis of symmetry is called the vertex of the parabola.
The canonical equation of a parabola in the Cartesian coordinate system: .
Definition. An ellipse is a set of points on a plane (see Fig. 7b), for each of which the sum of the distances to two given points and (foci) is constant and equal to .
The segment is called focal length and is denoted by . The midpoint is the center of the ellipse. The straight line on which the foci of the ellipse lie is called the first axis of the ellipse. The straight line passing through the center of the ellipse perpendicular to its first axis is called the second axis of the ellipse. The axes of an ellipse are its axes of symmetry. The points of intersection of the ellipse with the axes of symmetry are called its vertices. - major axis of the ellipse, - minor axis.
The directrix of an ellipse corresponding to a given focus is a straight line perpendicular to the first axis and spaced from the center of the ellipse at a distance where is the eccentricity of the ellipse.
The canonical equation of an ellipse in the Cartesian coordinate system: , where and are the major and minor semi-axes of the ellipse, respectively.
Definition. A hyperbola is a set of points on a plane (see Fig. 8), the modulus of the difference between the distances to two given points and (foci of the hyperbola) is constant and equal to . The focal length is denoted by . The line on which the foci lie is called the real (or focal axis) of the hyperbola. A straight line passing through the center of a hyperbola, perpendicular to the real axis, is called
imaginary axis.
The directrix of a hyperbola corresponding to a given focus is a straight line perpendicular to the real axis, spaced from the center by a distance and lying from the center on the same side as the focus, where is the eccentricity.
A hyperbola has two asymptotes given by the equations.
The canonical equation of a hyperbola in the Cartesian coordinate system: ,
where and are the halves of the sides of the main rectangle of the hyperbola.
Example 9. Determine the type of second-order line given by the equation
Solution. Selecting complete squares in x and y, we get:
those. we have a hyperbola whose center lies at point , .
Polar coordinates. For a point in the Oxy plane, its polar coordinates are determined by a pair of numbers, where is the length of the vector, and is the angle of inclination of the vector to the polar axis (the positive direction of the Ox axis), and is the length of the vector.
Cartesian and polar coordinates are related by the following relations.
