Dot product properties

Dot product of vectors, definition, properties

Linear operations on vectors.

Vectors, basic concepts, definitions, linear operations on them

A vector on a plane is an ordered pair of its points, while the first point is called the beginning, and the second the end - of the vector

Two vectors are called equal if they are equal and codirectional.

Vectors that lie on the same line are called codirectional if they are codirectional with some of the same vector that does not lie on this line.

Vectors that lie on the same line or on parallel lines are called collinear, and collinear but not codirectional are called oppositely directed.

Vectors lying on perpendicular lines are called orthogonal.

Definition 5.4. sum a+b vectors a and b is called the vector coming from the beginning of the vector a to the end of the vector b , if the beginning of the vector b coincides with the end of the vector a .

Definition 5.5. difference a - b vectors a and b such a vector is called With , which together with the vector b gives a vector a .

Definition 5.6. workk a vector a per number k called vector b , collinear vector a , which has module equal to | k||a |, and a direction that is the same as the direction a at k>0 and opposite a at k<0.

Properties of multiplication of a vector by a number:

Property 1. k(a+b ) = k a+ k b.

Property 2. (k+m)a = k a+ m a.

Property 3. k(m a) = (km)a .

Consequence. If non-zero vectors a and b are collinear, then there is a number k, what b= k a.

The scalar product of two nonzero vectors a and b called a number (scalar) equal to the product of the lengths of these vectors and the cosine of the angle φ between them. The scalar product can be expressed in various ways, for example, as ab, a · b, (a , b), (a · b). So the dot product is:

a · b = |a| · | b| cos φ

If at least one of the vectors is equal to zero, then the scalar product is equal to zero.

Permutation property: a · b = b · a(the scalar product does not change from permutation of factors);

distribution property: a · ( b · c) = (a · b) · c(the result does not depend on the order of multiplication);

Combination property (in relation to the scalar factor): (λ a) · b = λ ( a · b).

Property of orthogonality (perpendicularity): if the vector a and b non-zero, then their dot product is zero only when these vectors are orthogonal (perpendicular to each other) a ┴ b;

Square property: a · a = a 2 = |a| 2 (the scalar product of a vector with itself is equal to the square of its modulus);

If the coordinates of the vectors a=(x 1 , y 1 , z 1 ) and b=(x 2 , y 2 , z 2 ), then the scalar product is a · b= x 1 x 2 + y 1 y 2 + z 1 z 2 .

Vector holding vectors. Definition: The vector product of two vectors and is understood as a vector for which:

The module is equal to the area of ​​the parallelogram built on these vectors, i.e. , where is the angle between the vectors and

This vector is perpendicular to the multiplied vectors, i.e.

If the vectors are non-collinear, then they form a right triple of vectors.

Cross product properties:

1. When the order of the factors is changed, the vector product changes its sign to the opposite, preserving the module, i.e.

2 .Vector square is equal to zero-vector, i.e.

3 .The scalar factor can be taken out of the sign of the vector product, i.e.

4 .For any three vectors, the equality

5 .Necessary and sufficient condition for the collinearity of two vectors and :

MIXED PRODUCT OF THREE VECTORS AND ITS PROPERTIES

mixed product three vectors is called a number equal to . Denoted . Here the first two vectors are multiplied vectorially and then the resulting vector is multiplied scalarly by the third vector . Obviously, such a product is some number.

Consider the properties of the mixed product.

1. geometric sense mixed product. The mixed product of 3 vectors, up to a sign, is equal to the volume of the parallelepiped built on these vectors, as on edges, i.e. .

   Thus, and .

   

   Proof. Let's postpone the vectors from the common origin and build a parallelepiped on them. Let us denote and note that . By definition of the scalar product

   Assuming that and denoting through h the height of the parallelepiped, we find .

   Thus, at

   If , then and . Consequently, .

   Combining both these cases, we get or .

   From the proof of this property, in particular, it follows that if the triple of vectors is right, then the mixed product , and if it is left, then .

2. For any vectors , , the equality

   The proof of this property follows from property 1. Indeed, it is easy to show that and . Moreover, the signs "+" and "-" are taken simultaneously, because the angles between the vectors and and and are both acute or obtuse.

3. When any two factors are interchanged, the mixed product changes sign.

   Indeed, if we consider the mixed product , then, for example, or

4. A mixed product if and only if one of the factors is equal to zero or the vectors are coplanar.

   Proof.

   Thus, a necessary and sufficient condition for the complanarity of 3 vectors is the equality to zero of their mixed product. In addition, it follows from this that three vectors form a basis in space if .

   If the vectors are given in coordinate form, then it can be shown that their mixed product is found by the formula:

   .

   Thus, the mixed product is equal to a third-order determinant whose first line contains the coordinates of the first vector, the second line contains the coordinates of the second vector, and the third line contains the coordinates of the third vector.

   Examples.

ANALYTICAL GEOMETRY IN SPACE

The equation F(x, y, z)= 0 defines in space Oxyz some surface, i.e. locus of points whose coordinates x, y, z satisfy this equation. This equation is called the surface equation, and x, y, z– current coordinates.

However, often the surface is not defined by an equation, but as a set of points in space that have one property or another. In this case, it is required to find the equation of the surface, based on its geometric properties.

PLANE.

NORMAL PLANE VECTOR.

EQUATION OF A PLANE PASSING THROUGH A GIVEN POINT

Consider an arbitrary plane σ in space. Its position is determined by setting a vector perpendicular to this plane, and some fixed point M0(x0, y 0, z0) lying in the plane σ.

The vector perpendicular to the plane σ is called normal vector of this plane. Let the vector have coordinates .

We derive the equation for the plane σ passing through the given point M0 and having a normal vector . To do this, take an arbitrary point on the plane σ M(x, y, z) and consider the vector .

For any point MÎ σ vector. Therefore, their scalar product is equal to zero. This equality is the condition that the point MО σ. It is valid for all points of this plane and is violated as soon as the point M will be outside the plane σ.

If we denote by the radius vector the points M, is the radius vector of the point M0, then the equation can be written as

This equation is called vector plane equation. Let's write it in coordinate form. Since then

So, we have obtained the equation of the plane passing through the given point. Thus, in order to compose the equation of the plane, you need to know the coordinates of the normal vector and the coordinates of some point lying on the plane.

Note that the equation of the plane is an equation of the 1st degree with respect to the current coordinates x, y and z.

Examples.

GENERAL EQUATION OF THE PLANE

It can be shown that any equation of the first degree with respect to Cartesian coordinates x, y, z is an equation of some plane. This equation is written as:

Ax+By+Cz+D=0

and called general equation plane, and the coordinates A, B, C here are the coordinates of the normal vector of the plane.

Let us consider particular cases of the general equation. Let's find out how the plane is located relative to the coordinate system if one or more coefficients of the equation vanish.

A is the length of the segment cut off by the plane on the axis Ox. Similarly, one can show that b and c are the lengths of the segments cut off by the considered plane on the axes Oy and Oz.

It is convenient to use the equation of a plane in segments for constructing planes.

Definition. The vector product of a vector a and vector b is a vector denoted by the symbol [«, b] (or l x b), such that 1) the length of the vector [a, b] is equal to (p, where y is the angle between the vectors a and b ( 31); 2) the vector [a, b) is perpendicular to the vectors a and b, i.e. perpendicular to the plane of these vectors; 3) the vector [a, b] is directed in such a way that from the end of this vector the shortest turn from a to b is seen to occur counterclockwise (Fig. 32). Rice. 32 Fig.31 In other words, the vectors a, b and [а, b) form the right triple of vectors, i.e. located like the thumb, index and middle fingers of the right hand. If the vectors a and b are collinear, we will assume that [a, b] = 0. By definition, the length of the vector product is numerically equal to the area Sa of the parallelogram (Fig. 33) built on the multiplied vectors a and b as on the sides: 6.1 . Properties of a vector product 1. A vector product is equal to a zero vector if and only if at least one of the multiplied vectors is zero or when these vectors are collinear (if the vectors a and b are collinear, then the angle between them is either 0 or 7r). This is easy to obtain from the fact that If we consider the zero vector collinsar to any vector, then the condition for the collinarity of the vectors a and b can be expressed as follows 2. The vector product is anticommutative, i.e. always. Indeed, the vectors (a, b) and have the same length and are collinear. The directions of these vectors are opposite, since from the end of the vector [a, b] the shortest turn from a to b will be seen occurring counterclockwise, and from the end of the vector [b, a] - clockwise (Fig. 34). 3. The vector product has a distributive property with respect to addition 4. The numerical factor A can be taken out of the sign of the vector product 6.2. Vector product of vectors given by coordinates Let the vectors a and b be given by their coordinates in the basis. Using the distribution property of the vector product, we find the vector product of the vectors given by the coordinates. Mixed work. Let us write out the vector products of coordinate orts (Fig. 35): Therefore, for the vector product of vectors a and b, we obtain from formula (3) the following expression determinant over the elements of the 1st row, we obtain (4). Examples. 1. Find the area of ​​a parallelogram built on vectors Find the area of ​​the triangle (Fig. 36). It is clear that the area b "d of the triangle JSC is equal to half the area S of the parallelogram O AC B. Calculating the vector product (a, b | of the vectors a \u003d OA and b \u003d b \u003d ob), we obtain (a, b), c) = [a, |b, c)) is not true in the general case. For example, for a = ss j we have § 7. Mixed product of vectors Let we have three vectors a, b and c. Multiply the vectors a and 1> vectorially. As a result, we obtain the vector [a, 1>]. We multiply it scalarly by the vector c: (k b), c. The number ([a, b], e) is called the mixed product of the vectors a, b. c and is denoted by the symbol (a, 1), e). 7.1. The geometric meaning of the mixed product Let us set aside the vectors a, b and from the general point O (Fig. 37). If all four points O, A, B, C lie in the same plane ( vectors a, b and c are called in this case coplanar), then the mixed product ([a, b], c) = 0. This follows from the fact that the vector [a, b| is perpendicular to the plane in which the vectors a and 1 lie ", and hence the vector c. / If t points O, A, B, C do not lie in the same plane (vectors a, b and c are non-coplanar), we will build a parallelepiped on the edges OA, OB and OS (Fig. 38 a). By the definition of a cross product, we have (a,b) = So c, where So is the area of ​​the OADB parallelogram, and c is a unit vector perpendicular to the vectors a and b and such that the triple a, b, c is right, i.e. vectors a, b and c are located respectively as the thumb, index and middle fingers of the right hand (Fig. 38 b). Multiplying both parts of the last equality on the right scalar by the vector c, we get that the vector product of the vectors given by the coordinates. Mixed work. The number rc c is equal to the height h of the constructed parallelepiped, taken with the “+” sign if the angle between the vectors c and c is acute (the triple a, b, c is right), and with the sign “-” if the angle is obtuse (the triple a, b, c - left), so that Thus, the mixed product of the vectors a, b and c is equal to the volume V of the parallelepiped built on these vectors as on edges if the triple a, b, c is right, and -V if the triple a , b, c - left. Based on the geometric meaning of the mixed product, we can conclude that by multiplying the same vectors a, b and c in any other order, we will always get either +7 or -K. The sign of pro- Fig. 38 reference will depend only on which triplet the multiplied vectors form - right or left. If the vectors a, b, c form a right triple, then the triples b, c, a and c, a, b will also be right. At the same time, all three triplets b, a, c; a, c, b and c, b, a - left. Thus, (a, b, c) = (b, c, a) = (c, a, b) = - (b, a, c) = - (a, c, b) = - (c, b ,a). We emphasize once again that the mixed product of vectors is equal to zero if and only if the multiplied vectors a, b, c are coplanar: (a, b, c are coplanar) 7.2. Mixed Product in Coordinates Let the vectors a, b, c be given by their coordinates in the basis i, j, k: a = (x\,y\,z]), b= (x2,y2>z2), c = (x3, uz, 23). Let us find an expression for their mixed product (a, b, c). We have a mixed product of vectors given by their coordinates in the basis i, J, k, equal to the third-order determinant, the lines of which are composed, respectively, of the coordinates of the first, second and third of the multiplied vectors. The necessary and sufficient condition for the complanarity of the vectors a y\, Z|), b = (xx, y2.22), c = (x3, uz, 23) can be written in the following form z, ar2 y2 -2 =0. Uz Example. Check whether the vectors v = (7,4,6), b = (2, 1,1), c = (19, II, 17) are coplanar. The vectors under consideration will be coplanar or non-coplanar, depending on whether the determinant is equal to zero or not. Expanding it in terms of the elements of the first row, we obtain 7.3. Double cross product The double cross product [a, [b, c]] is a vector perpendicular to the vectors a and [b, c]. Therefore, it lies in the plane of the vectors b and c and can be expanded in these vectors. It can be shown that the formula [a, [!>, c]] = b(a, e) - c(a, b) is valid. Exercises 1. Three vectors AB = c, W? = o and CA = b serve as sides of the triangle. Express in terms of a, b and c the vectors coinciding with the medians AM, DN, CP of the triangle. 2. What condition must be connected between the vectors p and q so that the vector p + q divides the angle between them in half? It is assumed that all three vectors are related to a common origin. 3. Calculate the length of the diagonals of the parallelogram built on the vectors a = 5p + 2q and b = p - 3q, if it is known that |p| = 2v/2, |q| = 3 H-(p7ci) = f. 4. Denoting by a and b the sides of the rhombus emerging from a common vertex, prove that the diagonals of the rhombus are mutually perpendicular. 5. Calculate the dot product of the vectors a = 4i + 7j + 3k and b = 31 - 5j + k. 6. Find the unit vector a0 parallel to the vector a = (6, 7, -6). 7. Find the projection of the vector a = l+ j- kHa vector b = 21 - j - 3k. 8. Find the cosine of the angle between the vectors IS "w, if A (-4.0.4), B (-1.6.7), C (1.10.9). 9. Find a unit vector p° that is simultaneously perpendicular to the vector a = (3, 6, 8) and the x-axis. 10. Calculate the sine of the angle between the diagonals of the parallelopham built on the vectors a = 2i+J-k, b=i-3j + k as on the sides. Calculate the height h of the parallelepiped built on the vectors a = 31 + 2j - 5k, b = i-j + 4knc = i-3j + k, if the parallelogram built on the vectors a and I is taken as the base). Answers

vector product is a pseudovector perpendicular to the plane constructed by two factors, which is the result of the binary operation "vector multiplication" on vectors in three-dimensional Euclidean space. The vector product does not have the properties of commutativity and associativity (it is anticommutative) and, unlike the scalar product of vectors, is a vector. Widely used in many technical and physical applications. For example, the angular momentum and the Lorentz force are mathematically written as a cross product. The cross product is useful for "measuring" the perpendicularity of vectors - the modulus of the cross product of two vectors is equal to the product of their moduli if they are perpendicular, and decreases to zero if the vectors are parallel or anti-parallel.

You can define a vector product in different ways, and theoretically, in a space of any dimension n, you can calculate the product of n-1 vectors, while obtaining a single vector perpendicular to them all. But if the product is limited to non-trivial binary products with vector results, then the traditional vector product is defined only in three-dimensional and seven-dimensional spaces. The result of the vector product, like the scalar product, depends on the metric of the Euclidean space.

Unlike the formula for calculating the scalar product from the coordinates of the vectors in a three-dimensional rectangular coordinate system, the formula for the vector product depends on the orientation of the rectangular coordinate system, or, in other words, its “chirality”.

Definition:
The vector product of a vector a and vector b in the space R 3 is called a vector c that satisfies the following requirements:
the length of the vector c is equal to the product of the lengths of the vectors a and b and the sine of the angle φ between them:
|c|=|a||b|sin φ;
the vector c is orthogonal to each of the vectors a and b;
the vector c is directed so that the triple of vectors abc is right;
in the case of the space R7, the associativity of the triple of vectors a,b,c is required.
Designation:
c===a×b

Rice. 1. The area of ​​a parallelogram is equal to the modulus of the cross product

Geometric properties of the cross product:
A necessary and sufficient condition for the collinearity of two non-zero vectors is the equality of their vector product to zero.

Cross product module equals area S parallelogram built on vectors reduced to a common origin a and b(see fig. 1).

If a e- unit vector orthogonal to the vectors a and b and chosen so that the triple a,b,e- right, and S- the area of ​​the parallelogram built on them (reduced to a common origin), then the following formula is true for the vector product:
=S e

Fig.2. The volume of the parallelepiped when using the vector and scalar product of vectors; the dotted lines show the projections of the vector c on a × b and the vector a on b × c, the first step is to find the inner products

If a c- any vector π - any plane containing this vector, e- unit vector lying in the plane π and orthogonal to c,g- unit vector orthogonal to the plane π and directed so that the triple of vectors ecg is right, then for any lying in the plane π vector a the correct formula is:
=Pr e a |c|g
where Pr e a is the projection of the vector e onto a
|c|-modulus of vector c

When using vector and scalar products, you can calculate the volume of a parallelepiped built on vectors reduced to a common origin a, b and c. Such a product of three vectors is called mixed.
V=|a (b×c)|
The figure shows that this volume can be found in two ways: the geometric result is preserved even when the "scalar" and "vector" products are interchanged:
V=a×b c=a b×c

The value of the cross product depends on the sine of the angle between the original vectors, so the cross product can be thought of as the degree of "perpendicularity" of the vectors, just as the dot product can be thought of as the degree of "parallelism". The cross product of two unit vectors is equal to 1 (a unit vector) if the initial vectors are perpendicular, and equal to 0 (zero vector) if the vectors are parallel or antiparallel.

Cross product expression in Cartesian coordinates
If two vectors a and b are defined by their rectangular Cartesian coordinates, or more precisely, they are represented in an orthonormal basis
a=(a x ,a y ,a z)
b=(b x ,b y ,b z)
and the coordinate system is right, then their vector product has the form
=(a y b z -a z b y ,a z b x -a x b z ,a x b y -a y b x)
To remember this formula:
i =∑ε ijk a j b k
where ε ijk- the symbol of Levi-Civita.

In this lesson, we will look at two more operations with vectors: cross product of vectors and mixed product of vectors (immediate link for those who need it). It's okay, it sometimes happens that for complete happiness, in addition to dot product of vectors, more and more is needed. Such is vector addiction. One may get the impression that we are getting into the jungle of analytic geometry. This is not true. In this section of higher mathematics, there is generally little firewood, except perhaps enough for Pinocchio. In fact, the material is very common and simple - hardly more difficult than the same scalar product, even there will be fewer typical tasks. The main thing in analytic geometry, as many will see or have already seen, is NOT TO MISTAKE CALCULATIONS. Repeat like a spell, and you will be happy =)

If the vectors sparkle somewhere far away, like lightning on the horizon, it doesn't matter, start with the lesson Vectors for dummies to restore or reacquire basic knowledge about vectors. More prepared readers can get acquainted with the information selectively, I tried to collect the most complete collection of examples that are often found in practical work

What will make you happy? When I was little, I could juggle two and even three balls. It worked out well. Now there is no need to juggle at all, since we will consider only space vectors, and flat vectors with two coordinates will be left out. Why? This is how these actions were born - the vector and mixed product of vectors are defined and work in three-dimensional space. Already easier!

In this operation, in the same way as in the scalar product, two vectors. Let it be imperishable letters.

The action itself denoted in the following way: . There are other options, but I'm used to designating the cross product of vectors in this way, in square brackets with a cross.

And immediately question: if in dot product of vectors two vectors are involved, and here two vectors are also multiplied, then what is the difference? A clear difference, first of all, in the RESULT:

The result of the scalar product of vectors is a NUMBER:

The result of the cross product of vectors is a VECTOR: , that is, we multiply the vectors and get a vector again. Closed club. Actually, hence the name of the operation. In various educational literature, the designations may also vary, I will use the letter .

Definition of cross product

First there will be a definition with a picture, then comments.

Definition: cross product non-collinear vectors , taken in this order, is called VECTOR, length which is numerically equal to the area of ​​the parallelogram, built on these vectors; vector orthogonal to vectors, and is directed so that the basis has a right orientation:

We analyze the definition by bones, there is a lot of interesting things!

So, we can highlight the following significant points:

1) Source vectors , indicated by red arrows, by definition not collinear. It will be appropriate to consider the case of collinear vectors a little later.

2) Vectors taken in a strict order: – "a" is multiplied by "be", not "be" to "a". The result of vector multiplication is VECTOR , which is denoted in blue. If the vectors are multiplied in reverse order, then we get a vector equal in length and opposite in direction (crimson color). That is, the equality .

3) Now let's get acquainted with the geometric meaning of the vector product. This is a very important point! The LENGTH of the blue vector (and, therefore, the crimson vector ) is numerically equal to the AREA of the parallelogram built on the vectors . In the figure, this parallelogram is shaded in black.

Note : the drawing is schematic, and, of course, the nominal length of the cross product is not equal to the area of ​​the parallelogram.

We recall one of the geometric formulas: the area of ​​a parallelogram is equal to the product of adjacent sides and the sine of the angle between them. Therefore, based on the foregoing, the formula for calculating the LENGTH of a vector product is valid:

I emphasize that in the formula we are talking about the LENGTH of the vector, and not about the vector itself. What is the practical meaning? And the meaning is such that in problems of analytic geometry, the area of ​​a parallelogram is often found through the concept of a vector product:

We get the second important formula. The diagonal of the parallelogram (red dotted line) divides it into two equal triangles. Therefore, the area of ​​a triangle built on vectors (red shading) can be found by the formula:

4) An equally important fact is that the vector is orthogonal to the vectors , that is . Of course, the oppositely directed vector (crimson arrow) is also orthogonal to the original vectors .

5) The vector is directed so that basis It has right orientation. In a lesson about transition to a new basis I have spoken in detail about plane orientation, and now we will figure out what the orientation of space is. I will explain on your fingers right hand. Mentally combine forefinger with vector and middle finger with vector . Ring finger and little finger press into your palm. As a result thumb- the vector product will look up. This is the right-oriented basis (it is in the figure). Now swap the vectors ( index and middle fingers) in some places, as a result, the thumb will turn around, and the vector product will already look down. This is also a right-oriented basis. Perhaps you have a question: what basis has a left orientation? "Assign" the same fingers left hand vectors , and get the left basis and left space orientation (in this case, the thumb will be located in the direction of the lower vector). Figuratively speaking, these bases “twist” or orient space in different directions. And this concept should not be considered something far-fetched or abstract - for example, the most ordinary mirror changes the orientation of space, and if you “pull the reflected object out of the mirror”, then in general it will not be possible to combine it with the “original”. By the way, bring three fingers to the mirror and analyze the reflection ;-)

... how good it is that you now know about right and left oriented bases, because the statements of some lecturers about the change of orientation are terrible =)

Vector product of collinear vectors

The definition has been worked out in detail, it remains to find out what happens when the vectors are collinear. If the vectors are collinear, then they can be placed on one straight line and our parallelogram also “folds” into one straight line. The area of ​​such, as mathematicians say, degenerate parallelogram is zero. The same follows from the formula - the sine of zero or 180 degrees is equal to zero, which means that the area is zero

Thus, if , then and . Please note that the cross product itself is equal to the zero vector, but in practice this is often neglected and written that it is also equal to zero.

A special case is the vector product of a vector and itself:

Using the cross product, you can check the collinearity of three-dimensional vectors, and we will also analyze this problem, among others.

To solve practical examples, it may be necessary trigonometric table to find the values ​​of the sines from it.

Well, let's start a fire:

Example 1

a) Find the length of the vector product of vectors if

b) Find the area of ​​a parallelogram built on vectors if

Solution: No, this is not a typo, I intentionally made the initial data in the condition items the same. Because the design of the solutions will be different!

a) According to the condition, it is required to find length vector (vector product). According to the corresponding formula:

Answer:

Since it was asked about the length, then in the answer we indicate the dimension - units.

b) According to the condition, it is required to find square parallelogram built on vectors . The area of ​​this parallelogram is numerically equal to the length of the cross product:

Answer:

Please note that in the answer about the vector product there is no talk at all, we were asked about figure area, respectively, the dimension is square units.

We always look at WHAT is required to be found by the condition, and, based on this, we formulate clear answer. It may seem like literalism, but there are enough literalists among the teachers, and the task with good chances will return for revision. Although this is not a particularly strained nitpick - if the answer is incorrect, then one gets the impression that the person does not understand simple things and / or has not delved into the essence of the task. This moment should always be kept under control, solving any problem in higher mathematics, and in other subjects too.

Where did the big letter "en" go? In principle, it could be additionally stuck to the solution, but in order to shorten the record, I did not do this. I hope everyone understands that and is the designation of the same thing.

A popular example for a do-it-yourself solution:

Example 2

Find the area of ​​a triangle built on vectors if

The formula for finding the area of ​​a triangle through the vector product is given in the comments to the definition. Solution and answer at the end of the lesson.

In practice, the task is really very common, triangles can generally be tortured.

To solve other problems, we need:

Properties of the cross product of vectors

We have already considered some properties of the vector product, however, I will include them in this list.

For arbitrary vectors and an arbitrary number, the following properties are true:

1) In other sources of information, this item is usually not distinguished in the properties, but it is very important in practical terms. So let it be.

2) - the property is also discussed above, sometimes it is called anticommutativity. In other words, the order of the vectors matters.

3) - combination or associative vector product laws. The constants are easily taken out of the limits of the vector product. Really, what are they doing there?

4) - distribution or distribution vector product laws. There are no problems with opening brackets either.

As a demonstration, consider a short example:

Example 3

Find if

Solution: By condition, it is again required to find the length of the vector product. Let's paint our miniature:

(1) According to the associative laws, we take out the constants beyond the limits of the vector product.

(2) We take the constant out of the module, while the module “eats” the minus sign. The length cannot be negative.

(3) What follows is clear.

Answer:

It's time to throw wood on the fire:

Example 4

Calculate the area of ​​a triangle built on vectors if

Solution: Find the area of ​​a triangle using the formula . The snag is that the vectors "ce" and "te" are themselves represented as sums of vectors. The algorithm here is standard and is somewhat reminiscent of examples No. 3 and 4 of the lesson. Dot product of vectors. Let's break it down into three steps for clarity:

1) At the first step, we express the vector product through the vector product, in fact, express the vector in terms of the vector. No word on length yet!

(1) We substitute expressions of vectors .

(2) Using distributive laws, open the brackets according to the rule of multiplication of polynomials.

(3) Using the associative laws, we take out all the constants beyond the vector products. With little experience, actions 2 and 3 can be performed simultaneously.

(4) The first and last terms are equal to zero (zero vector) due to the pleasant property . In the second term, we use the anticommutativity property of the vector product:

(5) We present similar terms.

As a result, the vector turned out to be expressed through a vector, which was what was required to be achieved:

2) At the second step, we find the length of the vector product we need. This action is similar to Example 3:

3) Find the area of ​​the desired triangle:

Steps 2-3 of the solution could be arranged in one line.

Answer:

The considered problem is quite common in tests, here is an example for an independent solution:

Example 5

Find if

Short solution and answer at the end of the lesson. Let's see how attentive you were when studying the previous examples ;-)

Cross product of vectors in coordinates

, given in the orthonormal basis , is expressed by the formula:

The formula is really simple: we write the coordinate vectors in the top line of the determinant, we “pack” the coordinates of the vectors into the second and third lines, and we put in strict order- first, the coordinates of the vector "ve", then the coordinates of the vector "double-ve". If the vectors need to be multiplied in a different order, then the lines should also be swapped:

Example 10

Check if the following space vectors are collinear:
a)
b)

Solution: The test is based on one of the statements in this lesson: if the vectors are collinear, then their cross product is zero (zero vector): .

a) Find the vector product:

So the vectors are not collinear.

b) Find the vector product:

Answer: a) not collinear, b)

Here, perhaps, is all the basic information about the vector product of vectors.

This section will not be very large, since there are few problems where the mixed product of vectors is used. In fact, everything will rest on the definition, geometric meaning and a couple of working formulas.

The mixed product of vectors is the product of three vectors:

This is how they lined up like a train and wait, they can’t wait until they are calculated.

First again the definition and picture:

Definition: Mixed product non-coplanar vectors , taken in this order, is called volume of the parallelepiped, built on these vectors, equipped with a "+" sign if the basis is right, and a "-" sign if the basis is left.

Let's do the drawing. Lines invisible to us are drawn by a dotted line:

Let's dive into the definition:

2) Vectors taken in a certain order, that is, the permutation of vectors in the product, as you might guess, does not go without consequences.

3) Before commenting on the geometric meaning, I will note the obvious fact: the mixed product of vectors is a NUMBER: . In educational literature, the design may be somewhat different, I used to designate a mixed product through, and the result of calculations with the letter "pe".

By definition the mixed product is the volume of the parallelepiped, built on vectors (the figure is drawn with red vectors and black lines). That is, the number is equal to the volume of the given parallelepiped.

Note : The drawing is schematic.

4) Let's not bother again with the concept of the orientation of the basis and space. The meaning of the final part is that a minus sign can be added to the volume. In simple terms, the mixed product can be negative: .

The formula for calculating the volume of a parallelepiped built on vectors follows directly from the definition.