Let a vector ( X , at , z ).

Let us denote the angles of inclination of this vector to the axes Ooh, ooh and Oz respectively letters ,and. three numbers cos, cos and cos called direction cosines of the vector. Assuming = (1; 0; 0 ) we obtain from (9)

Similarly

From formulas (11) - (13) it follows:

1) cos 2 + cos 2 + cos 2 = 1 ,

those. the sum of the squared direction cosines of any non-zero vector is equal to one;

those.the direction cosines of this vector are proportional to its corresponding projections.

Note. From formulas (11)-(13) it can be seen that the projections of any unit vector on the coordinate axes, respectively, coincide with its direction cosines and, therefore,

Example. Find direction cosines of a vector (1; 2; 2). According to formulas (11)-(13) we have

4. Vector product of two vectors and its main properties.

Definition. Vector product of two vectorsand a new vector is called, the modulus of which is equal to the area of ​​a parallelogram built on vectors and reduced to a common origin, and which is perpendicular to the multiplied vectors (in other words, perpendicular to the plane of the parallelogram built on them) and directed in such a direction that the shortest turn from around the resulting vector appears to be occurring counterclockwise when viewed from the end of the vector (Fig. 40).

If the vectors are collinear, then their vector product is considered equal to the zero vector. From this definition it follows that

|| = || || sin,

where is the angle between the vectors and ( 0 ). The cross product of vectors and is denoted by the symbol

x or or [,].

Let us find out the physical meaning of the vector product. If the vector represents the applied at some point M s silo, and the vector goes from some point O exactly M, then the vector = represents the moment of force about the point O.

Cross product properties

1 . When the factors are rearranged, the vector product changes sign, i.e.

x = -(x).

()x=x()=(x), where is a scalar.

3. The vector product obeys the distribution law, i.e.

4. If the vector product of two vectors is equal to the zero vector, then either at least one of the multiplied vectors is equal to the zero vector (trivial case), or the sine of the angle between them is equal to zero, i.e. the vectors are collinear.

Back, if two non-zero vectors are collinear, then their vector product is equal to the zero vector.

In this way , for two non-zero vectors u to be collinear, it is necessary and sufficient that their vector product equals the zero vector.

From this, in particular, it follows that the vector product of a vector and itself is equal to the zero vector:

x =0

(X also called vector square vector .

5. Mixed product of three vectors and its main properties.

Let three vectors , and be given. Imagine that the vector is multiplied vectorially and the resulting vector x is multiplied scalarly by the vector, thereby determining the number (x). It is called or mixed product three vectors, and.

For brevity, the mixed product (x) will be denoted by or ().

Let us find out the geometric meaning of the mixed product. Let the considered vectors be non-complanar. Let's construct a parallelepiped on vectors, and as on edges.

The cross product x is a vector (=) numerically equal to the area of ​​the parallelogram OADB (the base of the constructed parallelepiped), built on vectors and directed perpendicular to the plane of the parallelogram (Fig. 41).

The scalar product (x)=is the product of the modulus of the vector and the projection of the vector (see item 1, (2)).

The height of the constructed parallelepiped is the absolute value of this projection.

Therefore, the product | | in absolute value is equal to the product of the area of ​​the base of the parallelepiped and its height, ie the volume of a parallelepiped built on vectors, and.

It is important to note that the scalar product gives the volume of the parallelepiped, sometimes with a positive and sometimes with a negative sign. A positive sign is obtained if the angle between the vectors is sharp; negative - if stupid. With an acute angle between the and the vector is located on the same side of the plane OADB , which is the vector and, therefore, from the end of the vector, the rotation from k will be seen in the same way as from the end of the vector, i.e. in the positive direction (counterclockwise).

At an obtuse angle between the vector located on the other side of the plane OADB than the vector, and hence from the end of the vector, the rotation from k will be seen in the negative (clockwise) direction. In other words, the product is positive if the vectors and form a system of the same name with the main Oxyz (mutually located in the same way as the axes Ox, Oy, Oz), and it is negative if the vectors form a system that is different from the main one.

In this way, mixed product is a number,the absolute value of which expresses the volume of the parallelepiped,built on vectors,like on the ribs.

The sign of the product is positive if the vectors ,, form a system with the same name as the main one, and negative otherwise.

It follows from this that the absolute value of the product = (x) will remain the same, in whatever order we take the factors,,. As for the sign, it will be positive in some cases, negative in others; it depends on whether our three vectors, taken in a certain order, form a system with the same name as the main one, or not. Note that our coordinate axes are located in such a way that they follow one after another counterclockwise, if you look into the inner part (Fig. 42). The order of succession is not violated if we start the tour from the second axis or from the third one, as long as it is made in the same direction, i.e. counterclock-wise. In this case, the multipliers are rearranged in a circular order (cyclically). Thus, we get the following property:

The mixed product does not change with a circular (cyclic) permutation of its factors. Permuting two neighboring factors changes the sign of the product

= ==-()=-()=-().

Finally, from geometric sense mixed product, the following assertion immediately follows.

A necessary and sufficient condition for the complanarity of vectors,,is the equality to zero of their mixed product:

Def. 1.5.6. Direction cosines vector a let's call the cosines of those angles that this vector forms with the basis vectors, respectively, i , j , k .

Vector direction cosines a = (X, at, z) are found by the formulas:

The sum of the squares of the direction cosines is equal to one:

Vector direction cosines a are the coordinates of its orth: .

Let the basis vectors i , j , k drawn from a common point O. We will assume that the orts set the positive directions of the axes Oh, OU, Oz. point collection O (origin) and an orthonormal basis i , j , k called Cartesian rectangular coordinate system in space. Let BUT is an arbitrary point in space. Vector a = OA= x i + y j + z k called radius vector points BUT, the coordinates of this vector ( x, y, z) are also called point coordinates BUT(symbol: BUT(x, y, z)). Coordinate axes Oh, OU, Oz also called, respectively, the axis abscissa, axis ordinate, axis applicate.

If the vector is given by the coordinates of its starting point AT 1 (x 1 , y 1 , z 1) and end point AT 2 (x 2 , y 2 , z 2), then the coordinates of the vector are equal to the difference between the coordinates of the end and the beginning: (since ).

Cartesian rectangular coordinate systems on the plane and on the line are determined in exactly the same way with corresponding quantitative (according to dimension) changes.

Solution of typical tasks.

Example 1 Find the length and direction cosines of a vector a = 6i – 2j -3k .

Solution. Vector length: . Direction cosines: .

Example 2 Find vector coordinates a , forming with coordinate axes equal sharp corners if the length of this vector is .

Solution. Since , then substituting into formula (1.6), we obtain . Vector a forms sharp angles with the coordinate axes, so the ortho . Therefore, we find the coordinates of the vector .

Example 3 Three non-coplanar vectors are given e 1 = 2i – k , e 2 = 3i + 3j , e 3 = 2i + 3k . Decompose Vector d = i + 5j - 2k basis e 1 , e 2 , e 3 .

Let a vector be given. Unit vector in the same direction as (vector vector ) is found by the formula:

.

Let the axis forms angles with the coordinate axes
.Direction cosines of the axis the cosines of these angles are called: If direction given by unit vector , then the direction cosines serve as its coordinates, i.e.:

.

The direction cosines are related by the relation:

If direction given by an arbitrary vector , then find the unit vector of this vector and, comparing it with the expression for the unit vector , get:

Scalar product

Dot product
two vectors and called a number equal to the product of their lengths by the cosine of the angle between them:
.

The scalar product has the following properties:

Consequently,
.

The geometric meaning of the scalar product: dot product of vector and unit vector equal to the projection of the vector in the direction determined , i.e.
.

From the definition of the scalar product follows the following table of multiplication of orts
:

.

If the vectors are given by their coordinates
and
, i.e.
,
, then, multiplying these vectors scalarly and using the multiplication table of orts, we obtain the expression for the scalar product
through the coordinates of the vectors:

.

vector product

Cross product of a vectorper vector called vector , the length and direction of which is determined by the conditions:

The vector product has the following properties:

It follows from the first three properties that the vector multiplication of a sum of vectors by a sum of vectors obeys the usual rules for polynomial multiplication. It is only necessary to ensure that the order of the multipliers does not change.

The basic unit vectors are multiplied as follows:

If a
and
, then taking into account the properties of the vector product of vectors, we can derive a rule for calculating the coordinates of the vector product from the coordinates of the factor vectors:

If we take into account the rules for multiplication of orts obtained above, then:

A more compact form of writing an expression for calculating the coordinates of the vector product of two vectors can be constructed if we introduce the concept of a matrix determinant.

Consider a special case when the vectors and belong to the plane
, i.e. they can be represented as
and
.

If the coordinates of the vectors are written in the form of a table as follows:
, then we can say that a square matrix of the second order is formed from them, i.e. size
, consisting of two rows and two columns. Each square matrix a number is assigned, which is calculated from the elements of the matrix according to certain rules and is called the determinant. The determinant of a second-order matrix is ​​equal to the difference between the products of the elements of the main diagonal and the secondary diagonal:

.

In this case:

The absolute value of the determinant is thus equal to the area of ​​the parallelogram built on the vectors and like on the sides.

If we compare this expression with the vector product formula (4.7), then:

This expression is a formula for calculating the determinant of a third-order matrix from the first row.

In this way:

Third order matrix determinant is calculated as follows:

and is the algebraic sum of six terms.

The formula for calculating the determinant of a third-order matrix is ​​easy to remember if you use ruleSarrus, which is formulated as follows:

Each term is the product of three elements located in different columns and different rows of the matrix;

The plus sign has the products of elements that form triangles with a side parallel to the main diagonal;

The minus sign is given to the products of the elements belonging to the secondary diagonal and to the two products of the elements that form triangles with a side parallel to the secondary diagonal.

DEFINITION

Vector is called an ordered pair of points and (that is, it is known exactly which of the points in this pair is the first).

The first point is called the beginning of the vector, and the second one is his the end.

The distance between the start and end of a vector is called long or vector module.

A vector whose beginning and end are the same is called zero and is denoted by ; its length is assumed to be zero. Otherwise, if the length of the vector is positive, then it is called non-zero.

Comment. If the length of a vector is equal to one, then it is called ortom or unit vector and is denoted.

EXAMPLE

Exercise	Check if vector is single.
Solution	Let's calculate the length of the given vector, it is equal to the square root of the sum of squared coordinates: Since the length of the vector is equal to one, then the vector is a vector.
Answer	The vector is single.

A non-zero vector can also be defined as a directed segment.

Comment. The direction of the null vector is not defined.

Vector direction cosines

DEFINITION

Direction cosines some vector are called the cosines of the angles that the vector forms with the positive directions of the coordinate axes.

Comment. The direction of a vector is uniquely determined by its direction cosines.

To find the direction cosines of a vector, it is necessary to normalize the vector (that is, divide the vector by its length):

Comment. The coordinates of the unit vector are equal to its direction cosines.

THEOREM

(Property of direction cosines). The sum of the squares of the direction cosines is equal to one: