Find the volume built on vectors. Vector product of vectors. Mixed product of vectors. Mixed product properties

For vectors , and , given by coordinates , , the mixed product is calculated by the formula: .

Mixed product is used: 1) to calculate the volumes of a tetrahedron and a parallelepiped built on vectors , and , as on edges, according to the formula: ; 2) as a condition for the complanarity of the vectors , and : and are coplanar.

Topic 5. Lines on the plane.

Normal line vector , any non-zero vector perpendicular to the given line is called. Direction vector straight , any non-zero vector parallel to the given line is called.

Straight on surface in the coordinate system can be given by an equation of one of the following types:

1) - general equation straight line, where is the normal vector of the straight line;

2) - the equation of a straight line passing through a point perpendicular to a given vector ;

3) - equation of a straight line passing through a point parallel to a given vector ( canonical equation );

4) - equation of a straight line passing through two given points , ;

5) - line equations with slope , where is the point through which the line passes; () - the angle that the line makes with the axis; - the length of the segment (with the sign ) cut off by a straight line on the axis (sign “ ” if the segment is cut off on the positive part of the axis and “ ” if on the negative part).

6) - straight line equation in cuts, where and are the lengths of the segments (with the sign ) cut off by a straight line on the coordinate axes and (the sign “ ” if the segment is cut off on the positive part of the axis and “ ” if on the negative one).

Distance from point to line , given by the general equation on the plane, is found by the formula:

Corner , ( )between straight lines and , given by general equations or equations with a slope, is found by one of the following formulas:

If or .

If or

Coordinates of the point of intersection of lines and are found as a solution to a system of linear equations: or .

Topic 10. Sets. Numeric sets. Functions.

Under many understand a certain set of objects of any nature, distinguishable from each other and conceivable as a single whole. The objects that make up a set call it elements . A set can be infinite (consists of an infinite number of elements), finite (consists of a finite number of elements), empty (does not contain a single element). Sets are denoted by , and their elements by . The empty set is denoted by .

Set call subset set if all elements of the set belong to the set and write .

Sets and called equal , if they consist of the same elements and write . Two sets and will be equal if and only if and .

Set call universal (within the framework of this mathematical theory) , if its elements are all objects considered in this theory.

Many can be set: 1) enumeration of all its elements, for example: (only for finite sets); 2) by setting a rule for determining whether an element of a universal set belongs to a given set : .

Association

crossing sets and is called a set

difference sets and is called a set

Supplement sets (up to a universal set) is called a set.

The two sets and are called equivalent and write ~ if a one-to-one correspondence can be established between the elements of these sets. The set is called countable , if it is equivalent to the set of natural numbers : ~ . The empty set is, by definition, countable.

Valid (real) number is called an infinite decimal fraction, taken with the sign "+" or "". Real numbers are identified with points on the number line.

module (absolute value) of a real number is a non-negative number:

The set is called numerical if its elements are real numbers. Numeric at intervals are called sets

numbers: , , , , , , , , .

The set of all points on the number line that satisfy the condition , where is an arbitrarily small number, is called -neighborhood (or just a neighborhood) of a point and is denoted by . The set of all points by the condition , where is an arbitrarily large number, is called - neighborhood (or just a neighborhood) of infinity and is denoted by .

A quantity that retains the same numerical value is called constant. A quantity that takes on different numerical values is called variable. Function the rule is called, according to which each number is assigned one well-defined number, and they write. The set is called domain of definition functions, - many ( or region ) values functions, - argument , - function value . The most common way to specify a function is the analytical method, in which the function is given by a formula. natural domain function is the set of values of the argument for which this formula makes sense. Function Graph , in a rectangular coordinate system , is the set of all points of the plane with coordinates , .

The function is called even on the set , symmetric with respect to the point , if the following condition is satisfied for all: and odd if the condition is met. Otherwise, a generic function or neither even nor odd .

The function is called periodical on the set if there exists a number ( function period ) such that the following condition is satisfied for all: . The smallest number is called the main period.

The function is called monotonically increasing (waning ) on the set if the larger value of the argument corresponds to the larger (smaller) value of the function .

The function is called limited on the set , if there exists a number such that the following condition is satisfied for all : . Otherwise, the function is unlimited .

Reverse to function , , is a function that is defined on a set and assigns to each such that . To find the function inverse to the function , you need to solve the equation relatively . If the function , is strictly monotonic on , then it always has an inverse, and if the function increases (decreases), then the inverse function also increases (decreases).

A function represented as , where , are some functions such that the domain of the function definition contains the entire set of values of the function , is called complex function independent argument. The variable is called an intermediate argument. A complex function is also called a composition of functions and , and is written: .

Basic elementary functions are: power function , demonstration function ( , ), logarithmic function ( , ), trigonometric functions , , , , inverse trigonometric functions , , , . Elementary is called a function obtained from basic elementary functions by a finite number of their arithmetic operations and compositions.

The graph of the function is a parabola with vertex at , whose branches are directed upwards if or downwards if .

In some cases, when constructing a graph of a function, it is advisable to divide its domain of definition into several non-intersecting intervals and sequentially build a graph on each of them.

Any ordered set of real numbers is called dot-dimensional arithmetic (coordinate) space and denoted or , while the numbers are called its coordinates .

Let and be some sets of points and . If each point is assigned, according to some rule, one well-defined real number , then they say that a numerical function of variables is given on the set and write or briefly and , while called domain of definition , - set of values , - arguments (independent variables) functions.

A function of two variables is often denoted, a function of three variables -. The domain of definition of a function is a certain set of points in the plane, functions are a certain set of points in space.

Topic 7. Numerical sequences and series. Sequence limit. Limit of a function and continuity.

If, according to a certain rule, each natural number is associated with one well-defined real number, then they say that numerical sequence . Briefly denote . The number is called common member of the sequence . A sequence is also called a function of a natural argument. A sequence always contains an infinite number of elements, some of which may be equal.

The number is called sequence limit , and write if for any number there is a number such that the inequality is satisfied for all .

A sequence that has a finite limit is called converging , otherwise - divergent .

: 1) waning , if ; 2) increasing , if ; 3) non-decreasing , if ; 4) non-increasing , if . All of the above sequences are called monotonous .

The sequence is called limited , if there is a number such that the following condition is satisfied for all: . Otherwise, the sequence is unlimited .

Every monotone bounded sequence has a limit ( Weierstrass theorem).

The sequence is called infinitesimal , if . The sequence is called infinitely large (converging to infinity) if .

number is called the limit of the sequence, where

The constant is called the nonpeer number. The base logarithm of a number is called the natural logarithm of a number and is denoted .

An expression of the form , where is a sequence of numbers, is called numerical series and are marked. The sum of the first terms of the series is called th partial sum row.

The row is called converging if there is a finite limit and divergent if the limit does not exist. The number is called the sum of a convergent series , while writing.

If the series converges, then (a necessary criterion for the convergence of the series ) . The converse is not true.

If , then the series diverges ( a sufficient criterion for the divergence of the series ).

Generalized harmonic series is called a series that converges at and diverges at .

Geometric series call a series that converges at , while its sum is equal to and diverges at . find a number or symbol. (left semi-neighbourhood, right semi-neighborhood) and

For vectors , and , given by their coordinates , , the mixed product is calculated by the formula: .

Topic 5. Straight lines and planes.

Normal line vector , any non-zero vector perpendicular to the given line is called. Direction vector straight , any non-zero vector parallel to the given line is called.

Straight on surface

1) - general equation straight line, where is the normal vector of the straight line;

2) - the equation of a straight line passing through a point perpendicular to a given vector ;

3) canonical equation );

Distance from point to line , given by the general equation on the plane, is found by the formula:

Corner , ( )between straight lines and , given by general equations or equations with a slope, is found by one of the following formulas:

If or .

If or

Coordinates of the point of intersection of lines and are found as a solution to a system of linear equations: or .

The normal vector of the plane , any non-zero vector perpendicular to the given plane is called.

Plane in the coordinate system can be given by an equation of one of the following types:

1) - general equation plane, where is the normal vector of the plane;

2) - the equation of the plane passing through the point perpendicular to the given vector ;

3) - equation of the plane passing through three points , and ;

4) - plane equation in cuts, where , and are the lengths of the segments (with the sign ) cut off by the plane on the coordinate axes , and (sign “ ” if the segment is cut off on the positive part of the axis and “ ” if on the negative one).

Distance from point to plane , given by the general equation , is found by the formula:

Corner ,( )between planes and , given by general equations, is found by the formula:

Straight in space in the coordinate system can be given by an equation of one of the following types:

1) - general equation a straight line, as the lines of intersection of two planes, where and are the normal vectors of the planes and;

2) - equation of a straight line passing through a point parallel to a given vector ( canonical equation );

3) - equation of a straight line passing through two given points , ;

4) - equation of a straight line passing through a point parallel to a given vector, ( parametric equation );

Corner , ( ) between straight lines and in space , given by canonical equations, is found by the formula:

The coordinates of the point of intersection of the line , given by the parametric equation and plane , given by the general equation, are found as a solution to the system of linear equations: .

Corner , ( ) between the line , given by the canonical equation and plane , given by the general equation is found by the formula: .

Topic 6. Curves of the second order.

Algebraic curve of the second order in the coordinate system is called a curve, general equation which looks like:

where numbers - are not equal to zero at the same time. There is the following classification of second-order curves: 1) if , then the general equation defines the curve elliptical type (circle (for ), ellipse (for ), empty set, point); 2) if , then - curve hyperbolic type (hyperbola, a pair of intersecting lines); 3) if , then - curve parabolic type(parabola, empty set, line, pair of parallel lines). Circle, ellipse, hyperbola and parabola are called non-degenerate curves of the second order.

The general equation , where , which defines a non-degenerate curve (circle, ellipse, hyperbola, parabola), can always (using the full squares selection method) be reduced to an equation of one of the following types:

1a) - circle equation centered at a point and radius (Fig. 5).

1b)- the equation of an ellipse centered at a point and axes of symmetry parallel to the coordinate axes. The numbers and - are called semi-axes of an ellipse the main rectangle of the ellipse; the vertices of the ellipse .

To build an ellipse in the coordinate system: 1) mark the center of the ellipse; 2) we draw through the center with a dotted line the axis of symmetry of the ellipse; 3) we build the main rectangle of an ellipse with a dotted line with a center and sides parallel to the axes of symmetry; 4) we draw an ellipse with a solid line, inscribing it in the main rectangle so that the ellipse touches its sides only at the vertices of the ellipse (Fig. 6).

Similarly, a circle is constructed, the main rectangle of which has sides (Fig. 5).

Fig.5 Fig.6

2) - equations of hyperbolas (called conjugate) centered at a point and symmetry axes parallel to the coordinate axes. The numbers and - are called semiaxes of hyperbolas ; a rectangle with sides parallel to the axes of symmetry and centered at a point - the main rectangle of hyperbolas; points of intersection of the main rectangle with the axes of symmetry - vertices of hyperbolas; straight linespassing through opposite vertices of the main rectangle - asymptotes of hyperbolas .

To build a hyperbola in the coordinate system: 1) mark the center of the hyperbola; 2) we draw through the center with a dotted line the axis of symmetry of the hyperbola; 3) we build the main rectangle of a hyperbola with a dotted line with a center and sides and parallel to the axes of symmetry; 4) we draw straight lines through the opposite vertices of the main rectangle with a dashed line, which are asymptotes of the hyperbola, to which the branches of the hyperbola approach indefinitely close, at an infinite distance from the origin of coordinates, without crossing them; 5) we depict the branches of a hyperbola (Fig. 7) or hyperbola (Fig. 8) with a solid line.

Fig.7 Fig.8

3a)- the equation of a parabola with a vertex at a point and an axis of symmetry parallel to the coordinate axis (Fig. 9).

3b)- the equation of a parabola with a vertex at a point and an axis of symmetry parallel to the coordinate axis (Fig. 10).

To build a parabola in the coordinate system: 1) mark the top of the parabola; 2) we draw through the vertex with a dotted line the axis of symmetry of the parabola; 3) we depict a parabola with a solid line, directing its branch, taking into account the sign of the parabola parameter: at - in the positive direction of the coordinate axis parallel to the axis of symmetry of the parabola (Fig. 9a and 10a); at - in the negative side of the coordinate axis (Fig. 9b and 10b) .

Rice. 9a Fig. 9b

Rice. 10a Fig. 10b

Topic 7. Sets. Numeric sets. Function.

Set call subset set if all elements of the set belong to the set and write . Sets and called equal , if they consist of the same elements and write . Two sets and will be equal if and only if and .

Set call universal (within the framework of this mathematical theory) , if its elements are all objects considered in this theory.

Many can be set: 1) enumeration of all its elements, for example: (only for finite sets); 2) by setting a rule for determining whether an element of a universal set belongs to a given set : .

Association

crossing sets and is called a set

difference sets and is called a set

Supplement sets (up to a universal set) is called a set.

The concept of cardinality of a set arises when sets are compared by the number of elements they contain. The cardinality of the set is denoted by . The cardinality of a finite set is the number of its elements.

Equivalent sets have the same cardinality. The set is called uncountable if its cardinality is greater than the cardinality of the set .

Valid (real) number is called an infinite decimal fraction, taken with the sign "+" or "". Real numbers are identified with points on the number line. module (absolute value) of a real number is a non-negative number:

The set is called numerical if its elements are real numbers. Numeric at intervals sets of numbers are called: , , , , , , , , .

The function is called monotonically increasing (waning ) on the set if the larger value of the argument corresponds to the larger (smaller) value of the function .

The function is called limited on the set , if there exists a number such that the following condition is satisfied for all : . Otherwise, the function is unlimited .

Reverse to function , , such a function is called , which is defined on the set and to each

Matches such that . To find the function inverse to the function , you need to solve the equation relatively . If the function , is strictly monotonic on , then it always has an inverse, and if the function increases (decreases), then the inverse function also increases (decreases).

If the graph of the function is given, then the construction of the graph of the function is reduced to a series of transformations (shift, compression or stretching, display) of the graph:

1) 2) the transformation displays the graph symmetrically about the axis ; 3) the transformation shifts the graph along the axis by units ( - to the right, - to the left); 4) the transformation shifts the chart along the axis by units ( - up, - down); 5) transformation graph along the axis stretches in times, if or compresses in times, if ; 6) transforming the graph along the axis compresses by a factor if or stretches by a factor if .

The sequence of transformations when plotting a function graph can be represented symbolically as:

Note. When performing a transformation, keep in mind that the amount of shift along the axis is determined by the constant that is added directly to the argument, and not to the argument.

The graph of the function is a parabola with vertex at , whose branches are directed upwards if or downwards if . The graph of a linear-fractional function is a hyperbola centered at the point , whose asymptotes pass through the center, parallel to the coordinate axes. , satisfying the condition. called.

Consider the product of vectors , and , composed as follows:
. Here the first two vectors are multiplied vectorially, and their result is scalarly multiplied by the third vector. Such a product is called a vector-scalar, or mixed, product of three vectors. The mixed product is some number.

Let us find out the geometric meaning of the expression
.

Theorem . The mixed product of three vectors is equal to the volume of the parallelepiped built on these vectors, taken with a plus sign if these vectors form a right triple, and with a minus sign if they form a left triple.

Proof.. We construct a parallelepiped whose edges are the vectors , , and vector
.

We have:
,
, where - area of the parallelogram built on vectors and ,
for the right triple of vectors and
for the left, where
is the height of the parallelepiped. We get:
, i.e.
, where - the volume of the parallelepiped formed by the vectors , and .

Mixed product properties

1. The mixed product does not change when cyclical permutation of its factors, i.e. .

Indeed, in this case, neither the volume of the parallelepiped nor the orientation of its edges change.

2. The mixed product does not change when the signs of vector and scalar multiplication are reversed, i.e.
.

Really,
and
. We take the same sign on the right side of these equalities, since the triples of vectors , , and , , - one orientation.

Consequently,
. This allows us to write the mixed product of vectors
as
without signs of vector, scalar multiplication.

3. The mixed product changes sign when any two factor vectors change places, i.e.
,
,
.

Indeed, such a permutation is equivalent to a permutation of the factors in the vector product, which changes the sign of the product.

4. Mixed Product of Nonzero Vectors , and is zero if and only if they are coplanar.

2.12. Computing the mixed product in coordinate form in an orthonormal basis

Let the vectors
,
,
. Let's find their mixed product using expressions in coordinates for vector and scalar products:

. (10)

The resulting formula can be written shorter:

since the right side of equality (10) is the expansion of the third order determinant in terms of the elements of the third row.

So, the mixed product of vectors is equal to the third order determinant, composed of the coordinates of the multiplied vectors.

2.13 Some applications of the mixed product

Determining the relative orientation of vectors in space

Determining the relative orientation of vectors , and based on the following considerations. If a
, then , , - right three if
, then , , - left three.

Complanarity condition for vectors

Vectors , and are coplanar if and only if their mixed product is zero (
,
,
):

vectors , , coplanar.

Determining the volumes of a parallelepiped and a triangular pyramid

It is easy to show that the volume of a parallelepiped built on vectors , and is calculated as
, and the volume of the triangular pyramid built on the same vectors is equal to
.

Example 1 Prove that the vectors
,
,
coplanar.

Solution. Let's find the mixed product of these vectors using the formula:

This means that the vectors
coplanar.

Example 2 Given the vertices of a tetrahedron: (0, -2, 5), (6, 6, 0), (3, -3, 6),
(2, -1, 3). Find the length of its height dropped from the vertex .

Solution. Let us first find the volume of the tetrahedron
. According to the formula we get:

Since the determinant is a negative number, in this case, you need to take a minus sign before the formula. Consequently,
.

The desired value h determine from the formula
, where S - base area. Let's determine the area S:

where

Because the

Substituting into the formula
values
and
, we get h= 3.

Example 3 Do vectors form
basis in space? Decompose Vector
on the basis of vectors .

Solution. If the vectors form a basis in space, then they do not lie in the same plane, i.e. are non-coplanar. Find the mixed product of vectors
:
,

Therefore, the vectors are not coplanar and form a basis in space. If vectors form a basis in space, then any vector can be represented as a linear combination of basis vectors, namely
,where
vector coordinates in vector basis
. Let's find these coordinates by compiling and solving the system of equations