In order to consider such a topic in detail, it is necessary to cover several more sections. The topic is directly related to terms such as dot product and vector product. In this article we have tried to give precise definition, indicate a formula that will help determine the product using the coordinates of the vectors. In addition, the article includes sections listing the properties of the work and presents detailed analysis typical equalities and problems.

Term

In order to determine what this term is, you need to take three vectors.

Definition 1

Mixed work a → , b → and d → is the value that is equal to the scalar product of a → × b → and d → , where a → × b → is the multiplication of a → and b → . The multiplication operation a →, b → and d → is often denoted a → · b → · d →. You can transform the formula like this: a → · b → · d → = (a → × b → , d →) .

Multiplication in a coordinate system

We can multiply vectors if they are specified on the coordinate plane.

Let's take i → , j → , k →

The product of vectors in this particular case will have the following form: a → × b → = (a y · b z - a z · b y) · i → + (a z · b x + a x · b z) · j → + (a x · b y + a y · b x) k → = a y a z b y b z i → - a x a z b x b z j → + a x a y b x b y k →

Definition 2

To do the dot product in the coordinate system it is necessary to add the results obtained during the multiplication of coordinates.

Therefore:

a → × b → = (a y b z - a z b y) i → + (a z b x + a x b z) j → + (a x b y + a y b x) k → = a y a z b y b z i → - a x a z b x b z · j → + a x a y b x b y · k →

We can also define a mixed product of vectors if a given coordinate system specifies the coordinates of the vectors that are being multiplied.

a → × b → = (a y a z b y b z · i → - a x a z b x b z · j → + a x a y b x b y · k → , d x · i → + d y · j → + d z · k →) = = a y a z b y b z · d x - a x a z b x b z · d y + x a y b x b y d z = a x a y a z b x b y b z d x d y d z

Thus, we can conclude that:

a → · b → · d = a → × b → , d → = a x a y a z b x b y b z d x d y d z

Definition 3

A mixed product can be equated to the determinant of a matrix whose rows are vector coordinates. Visually it looks like this: a → · b → · d = a → × b → , d → = a x a y a z b x b y b z d x d y d z .

Properties of operations on vectors From the features that stand out in a scalar or vector product, we can derive the features that characterize the mixed product. Below we present the main properties.

1. (λ a →) b → d → = a → (λ b →) d → = a → b → (λ d →) = λ a → b → d → λ ∈ R ;
2. a → · b → · d → = d → · a → · b → = b → · d → · a → ; a → · d → · b → = b → · a → · d → = d → · b → · a → ;
3. (a (1) → + a (2) →) · b → · d → = a (1) → · b → · d → + a (2) → · b → · d → a → · (b (1 ) → + b (2) →) d → = a → b (1) → d → + a → b (2) → d → a → b → (d (1) → + d (2) →) = a → b → d (2) → + a → b → d (2) →

In addition to the above properties, it should be clarified that if the multiplier is zero, then the result of the multiplication will also be zero.

The result of multiplication will also be zero if two or more factors are equal.

Indeed, if a → = b →, then, following the definition of the vector product [ a → × b → ] = a → · b → · sin 0 = 0 , therefore, the mixed product is equal to zero, since ([ a → × b → ] , d →) = (0 → , d →) = 0 .

If a → = b → or b → = d →, then the angle between the vectors [a → × b →] and d → is equal to π 2. By definition of the scalar product of vectors ([ a → × b → ], d →) = [ a → × b → ] · d → · cos π 2 = 0 .

The properties of the multiplication operation are most often required when solving problems.
In order to examine in detail this topic, let's take a few examples and describe them in detail.

Example 1

Prove the equality ([ a → × b → ], d → + λ a → + b →) = ([ a → × b → ], d →), where λ is some real number.

In order to find a solution to this equality, its left side must be transformed. To do this, you need to use the third property mixed product which reads:

([ a → × b → ], d → + λ a → + b →) = ([ a → × b → ], d →) + ([ a → × b → ], λ a →) + ( [ a → × b → ] , b →)
We have seen that (([ a → × b → ] , b →) = 0 . It follows from this that
([ a → × b → ], d → + λ a → + b →) = ([ a → × b → ], d →) + ([ a → × b → ], λ a →) + ( [ a → × b → ] , b →) = = ([ a → × b → ] , d →) + ([ a → × b → ] , λ a →) + 0 = ([ a → × b → ] , d →) + ([ a → × b → ] , λ a →)

According to the first property, ([ a ⇀ × b ⇀ ], λ · a →) = λ · ([ a ⇀ × b ⇀ ], a →), and ([ a ⇀ × b ⇀ ], a →) = 0. Thus, ([ a ⇀ × b ⇀ ], λ · a →) . That's why,
([ a ⇀ × b ⇀ ], d → + λ a → + b →) = ([ a ⇀ × b ⇀ ], d →) + ([ a ⇀ × b ⇀ ], λ a →) = = ([ a ⇀ × b ⇀ ], d →) + 0 = ([ a ⇀ × b ⇀ ], d →)

Equality has been proven.

Example 2

It is necessary to prove that the modulus of the mixed product of three vectors is not greater than the product of their lengths.

Solution

Based on the condition, we can present the example in the form of an inequality a → × b → , d → ≤ a → · b → · d → .

By definition, we transform the inequality a → × b → , d → = a → × b → · d → · cos (a → × b → ^ , d →) = = a → · b → · sin (a → , b → ^) · d → · cos ([ a → × b → ^ ] , d)

Using elementary functions, we can conclude that 0 ≤ sin (a → , b → ^) ≤ 1, 0 ≤ cos ([ a → × b → ^ ], d →) ≤ 1.

From this we can conclude that
(a → × b → , d →) = a → · b → · sin (a → , b →) ^ · d → · cos (a → × b → ^ , d →) ≤ ≤ a → · b → · 1 d → 1 = a → b → d →

The inequality has been proven.

Analysis of typical tasks

In order to determine what the product of vectors is, you need to know the coordinates of the vectors being multiplied. For the operation, you can use the following formula a → · b → · d → = (a → × b → , d →) = a x a y a z b x b y b z d x d y d z .

Example 3

In a rectangular coordinate system, there are 3 vectors with the following coordinates: a → = (1, - 2, 3), b → (- 2, 2, 1), d → = (3, - 2, 5). It is necessary to determine what the product of the indicated vectors a → · b → · d → is equal to.

Based on the theory presented above, we can use the rule that the mixed product can be calculated through the determinant of the matrix. It will look like this: a → b → d → = (a → × b → , d →) = a x a y a z b x b y b z d x d y d z = 1 - 2 3 - 2 2 1 3 - 2 5 = = 1 2 5 + (- 1 ) 1 3 + 3 (- 2) (- 2) - 3 2 3 - (- 1) (- 2) 5 - 1 1 (- 2) = - 7

Example 4

It is necessary to find the product of vectors i → + j → , i → + j → - k → , i → + j → + 2 · k → , where i → , j → , k → are the unit vectors of the rectangular Cartesian coordinate system.

Based on the condition that states that the vectors are located in a given coordinate system, their coordinates can be derived: i → + j → = (1, 1, 0) i → + j → - k → = (1, 1, - 1) i → + j → + 2 k → = (1, 1, 2)

We use the formula that was used above
i → + j → × (i → + j → - k → , (i → + j → + 2 k →) = 1 1 0 1 1 - 1 1 1 2 = 0 i → + j → × (i → + j → - k → , (i → + j → + 2 k →) = 0

It is also possible to determine the mixed product using the length of the vector, which is already known, and the angle between them. Let's look at this thesis with an example.

Example 5

In a rectangular coordinate system there are three vectors a →, b → and d →, which are perpendicular to each other. They are a right-handed triple and their lengths are 4, 2 and 3. It is necessary to multiply the vectors.

Let us denote c → = a → × b → .

According to the rule, the result of multiplication scalar vectors is a number that is equal to the result of multiplying the lengths of the vectors used by the cosine of the angle between them. We conclude that a → · b → · d → = ([ a → × b → ], d →) = c → , d → = c → · d → · cos (c → , d → ^) .

We use the length of the vector d → specified in the example condition: a → b → d → = c → d → cos (c → , d → ^) = 3 c → cos (c → , d → ^) . It is necessary to determine c → and c → , d → ^ . By condition a →, b → ^ = π 2, a → = 4, b → = 2. Vector c → is found using the formula: c → = [ a → × b → ] = a → · b → · sin a → , b → ^ = 4 · 2 · sin π 2 = 8
We can conclude that c → is perpendicular to a → and b → . Vectors a → , b → , c → will be a right-hand triple, so the Cartesian coordinate system is used. Vectors c → and d → will be unidirectional, that is, c → , d → ^ = 0 . Using the derived results, we solve the example a → · b → · d → = 3 · c → · cos (c → , d → ^) = 3 · 8 · cos 0 = 24 .

a → · b → · d → = 24 .

We use the factors a → , b → and d → .

Vectors a → , b → and d → originate from the same point. We use them as sides to build a figure.

Let us denote that c → = [ a → × b → ] . For this case, we can define the product of vectors as a → · b → · d → = c → · d → · cos (c → , d → ^) = c → · n p c → d → , where n p c → d → is the numerical projection of the vector d → to the direction of the vector c → = [ a → × b → ] .

The absolute value n p c → d → is equal to the number, which is also equal to the height of the figure for which the vectors a → , b → and d → are used as sides. Based on this, it should be clarified that c → = [ a → × b → ] is perpendicular to a → both vector and vector according to the definition of vector multiplication. The value c → = a → x b → is equal to the area of ​​the parallelepiped built on the vectors a → and b →.

We conclude that the modulus of the product a → · b → · d → = c → · n p c → d → is equal to the result of multiplying the area of ​​the base by the height of the figure, which is built on the vectors a → , b → and d → .

Definition 4

The absolute value of the cross product is the volume of the parallelepiped: V par l l e l e p i p i d a = a → · b → · d → .

This formula is the geometric meaning.

Definition 5

Volume of a tetrahedron, which is built on a →, b → and d →, equals 1/6 of the volume of the parallelepiped. We get, V t e t r a e d a = 1 6 · V par l l e l e p i d a = 1 6 · a → · b → · d → .

In order to consolidate knowledge, let's look at a few typical examples.

Example 6

It is necessary to find the volume of a parallelepiped, the sides of which are A B → = (3, 6, 3), A C → = (1, 3, - 2), A A 1 → = (2, 2, 2), specified in a rectangular coordinate system . The volume of a parallelepiped can be found using the absolute value formula. It follows from this: A B → · A C → · A A 1 → = 3 6 3 1 3 - 2 2 2 2 = 3 · 3 · 2 + 6 · (- 2) · 2 + 3 · 1 · 2 - 3 · 3 · 2 - 6 1 2 - 3 (- 2) 2 = - 18

Then, V par l l e l e p e d a = - 18 = 18 .

V par l l e l e p i p i d a = 18

Example 7

The coordinate system contains points A (0, 1, 0), B (3, - 1, 5), C (1, 0, 3), D (- 2, 3, 1). It is necessary to determine the volume of the tetrahedron that is located at these points.

Let's use the formula V t e t r a e d r a = 1 6 · A B → · A C → · A D → . We can determine the coordinates of vectors from the coordinates of points: A B → = (3 - 0, - 1 - 1, 5 - 0) = (3, - 2, 5) A C → = (1 - 0, 0 - 1, 3 - 0 ) = (1 , - 1 , 3) ​​A D → = (- 2 - 0 , 3 - 1 , 1 - 0) = (- 2 , 2 , 1)

Next, we determine the mixed product A B → A C → A D → by vector coordinates: A B → A C → A D → = 3 - 2 5 1 - 1 3 - 2 2 1 = 3 (- 1) 1 + (- 2 ) · 3 · (- 2) + 5 · 1 · 2 - 5 · (- 1) · (- 2) - (- 2) · 1 · 1 - 3 · 3 · 2 = - 7 Volume V t et r a e d r a = 1 6 · - 7 = 7 6 .

V t e t r a e d r a = 7 6 .

If you notice an error in the text, please highlight it and press Ctrl+Enter

Mixed (or vector-scalar) product three vectors a, b, c (taken in the indicated order) is called the scalar product of vector a and the vector product b x c, i.e. the number a(b x c), or, what is the same, (b x c)a.
Designation: abc.

Purpose. The online calculator is designed to calculate the mixed product of vectors. The resulting solution is saved in a Word file. Additionally, a solution template is created in Excel.

Signs of coplanarity of vectors

Three vectors (or larger number) are called coplanar if they, being reduced to a common origin, lie in the same plane.
If at least one of the three vectors is zero, then the three vectors are also considered coplanar.

Sign of coplanarity. If the system a, b, c is right-handed, then abc>0 ; if left, then abc Geometric meaning of mixed product. The mixed product abc of three non-coplanar vectors a, b, c is equal to the volume of the parallelepiped built on the vectors a, b, c, taken with a plus sign if the system a, b, c is right-handed, and with a minus sign if this system is left-handed.

Properties of a mixed product

1. When the factors are rearranged circularly, the mixed product does not change; when two factors are rearranged, the sign is reversed: abc=bca=cab=-(bac)=-(cba)=-(acb)
   It follows from the geometric meaning.
2. (a+b)cd=acd+bcd ( distributive property). Extends to any number of terms.
   Follows from the definition of a mixed product.
3. (ma)bc=m(abc) ( associative property relative to the scalar factor).
   Follows from the definition of a mixed product. These properties make it possible to apply transformations to mixed products that differ from ordinary algebraic ones only in that the order of the factors can be changed only taking into account the sign of the product.
4. A mixed product that has at least two equal factors is equal to zero: aab=0.

Example No. 1. Find a mixed product.

ab(3a+2b-5c)=3aba+2abb-5abc=-5abc .

Example No. 2. (a+b)(b+c)(c+a)= (axb+axc+bxb+bxc)(c+a)= (axb+axc +bxc)(c+a)=abc+acc+aca+ aba+bcc+bca. All terms except the two extreme ones are equal to zero. Also, bca=abc . Therefore (a+b)(b+c)(c+a)=2abc .
Solution Example No. 3. Calculate the mixed product of three vectors a=15i+20j+5k, b=2i-4j+14k, c=3i-6j+21k.

. To calculate the mixed product of vectors, it is necessary to find the determinant of a system composed of vector coordinates. Let's write the system in the form. 8.1. Definitions of a mixed product, its

geometric meaning Consider the product of vectors a, b

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

Let's find out the geometric meaning of the expression (a xb)*c. Let's build a parallelepiped whose edges are the vectors a, b, c and the vector d = a x (see Fig. 22). We have: (a x b) c = d c = |d | (see Fig. 22). etc (see Fig. 22). d with , |d |=|a x b | =S, where S is the area of ​​a parallelogram built on vectors a and b, pr= Н For the right triple of vectors, etc. , |d |=|a x b | =S, where S is the area of ​​a parallelogram built on vectors a and b, pr= - H for the left, where H is the height of the parallelepiped. We get: ( Consider the product of vectors a, axb

Thus, 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.

8.2. Properties of a mixed product

1. The mixed product does not change when its factors are cyclically rearranged, i.e. (a x b) c =( Consider the product of vectors a, x c) a = (c x a) b.

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

2. The mixed product does not change when the signs of vector and scalar multiplication are swapped, i.e. (a xb) c =a *( b x With ).

Indeed, (a xb) c =±V and a (b xc)=(b xc) a =±V. We take the same sign on the right side of these equalities, since the triples of vectors a, b, c and b, c, a are of the same orientation.

Therefore, (a xb) c =a (b xc). This allows you to write the mixed product of vectors (a x b)c in the form abc without vector or scalar multiplication signs.

3. The mixed product changes its sign when changing the places of any two factor vectors, i.e. abc = -acb, abc = -bac, abc = -cba.

Indeed, such a rearrangement is equivalent to rearranging the factors in a vector product, changing the sign of the product.

4. The mixed product of non-zero vectors a, b and c is equal to zero whenever and only if they are coplanar.

If abc =0, then a, b and c are coplanar.

Let's assume that this is not the case. It would be possible to build a parallelepiped with volume V ¹ 0. But since abc =±V , we would get that abc ¹ 0 . This contradicts the condition: abc =0 .

Conversely, let vectors a, b, c be coplanar. Then vector d =a x Consider the product of vectors a, will be perpendicular to the plane in which the vectors a, b, c lie, and therefore d ^ c. Therefore d c =0, i.e. abc =0.

8.3. Expressing a mixed product in terms of coordinates

Let the vectors a =a x i +a y be given j+a z k, b = b x i+b y j+b z k, с =c x i+c y j+c z k. Let's find their mixed product using expressions in coordinates for the vector and scalar products:

The resulting formula can be written more briefly:

since the right-hand side of equality (8.1) represents the expansion of the third-order determinant into 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.

8.4.

Some mixed product applications

Determination of the relative orientation of vectors a, Consider the product of vectors a, and c is based on the following considerations. If abc > 0, then a, b, c are a right triple; if abc<0 , то а , b , с - левая тройка.

Establishing coplanarity of vectors

Vectors a, Consider the product of vectors a, and c are coplanar if and only if their mixed product is equal to zero

Determination of the volumes of a parallelepiped and a triangular pyramid

It is easy to show that the volume of a parallelepiped built on vectors a, Consider the product of vectors a, and c is calculated as V =|abc |, and the volume of a triangular pyramid built on the same vectors is equal to V =1/6*|abc |.

Example 6.3.

The vertices of the pyramid are points A(1; 2; 3), B(0; -1; 1), C(2; 5; 2) and D (3; 0; -2). Find the volume of the pyramid.

Solution: We find vectors a, Consider the product of vectors a, is:

a=AB =(-1;-3;-2), b =AC=(1;3;-1), c=AD =(2; -2; -5).

We find Consider the product of vectors a, and with:

=-1 (-17)+3 (-3)-2 (-8)=17-9+16=24.

Therefore, V =1/6*24=4

In this lesson we will look at two more operations with vectors: vector product of vectors And mixed product of vectors (immediate link for those who need it). It’s okay, sometimes it happens that for complete happiness, in addition to scalar product of vectors, more and more are required. This is vector addiction. It may seem that we are getting into the jungle of analytical geometry. This is wrong. In this section of higher mathematics there is generally little wood, except perhaps enough for Pinocchio. In fact, the material is very common and simple - hardly more complicated than the same scalar product, there will even be fewer typical tasks. The main thing in analytical geometry, as many will be convinced or have already been convinced, is NOT TO MAKE MISTAKES IN CALCULATIONS. Repeat like a spell and you will be happy =)

If 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 right away? When I was little, I could juggle two or even three balls. It worked out well. Now you won't have to juggle at all, since we will consider only spatial 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. It's already easier!

This operation, just like the scalar product, involves two vectors. Let these be imperishable letters.

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

And right away question: if in scalar product of vectors two vectors are involved, and here two vectors are also multiplied, then what is the difference? The obvious difference is, first of all, in the RESULT:

The result of the scalar product of vectors is NUMBER:

The result of the cross product of vectors is VECTOR: , that is, we multiply the vectors and get a vector again. Closed club. Actually, this is where the name of the operation comes from. In different educational literature, 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: Vector product non-collinear vectors, taken in this order, 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:

Let’s break down the definition piece by piece, there’s a lot of interesting stuff here!

So, the following significant points can be highlighted:

1) The original 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 are taken in a strictly defined order: – "a" is multiplied by "be", not “be” with “a”. The result of vector multiplication is VECTOR, which is indicated in blue. If the vectors are multiplied in reverse order, we obtain a vector equal in length and opposite in direction (raspberry color). That is, the equality is true .

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 black.

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

Let us 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 above, the formula for calculating the LENGTH of a vector product is valid:

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

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

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

5) The vector is directed so that basis It has right orientation. In the lesson about transition to a new basis I spoke in sufficient detail about plane orientation, and now we will figure out what space orientation 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 it into your palm. As a result thumb– the vector product will look up. This is a right-oriented basis (it is this one in the figure). Now change 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. You may have a question: which basis has left orientation? “Assign” to the same fingers left hand vectors, and get the left basis and left orientation of space (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 orientation of space is changed by the most ordinary mirror, and if you “pull the reflected object out of the looking glass,” then in the general case it will not be possible to combine it with the “original.” By the way, hold three fingers up 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 a change in orientation are scary =)

Cross product of collinear vectors

The definition has been discussed 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 equal to zero. The same follows from the formula - the sine of zero or 180 degrees is equal to zero, which means the area is zero

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

A special case is the cross product of a vector with itself:

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

To solve practical examples you may need trigonometric table to find the values ​​of sines from it.

Well, let's light the 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 deliberately made the initial data in the clauses the same. Because the design of the solutions will be different!

a) According to the condition, you need to find length vector (cross product). According to the corresponding formula:

Answer:

If you were asked about length, then in the answer we indicate the dimension - units.

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

Answer:

Please note that the answer does not talk about the vector product at all; we were asked about area of ​​the figure, accordingly, the dimension is square units.

We always look at WHAT we need to find according to the condition, and, based on this, we formulate clear answer. It may seem like literalism, but there are plenty of literalists among teachers, and the assignment has a good chance of being returned for revision. Although this is not a particularly far-fetched quibble - if the answer is incorrect, then one gets the impression that the person does not understand simple things and/or has not understood the essence of the task. This point must always be kept under control when solving any problem in higher mathematics, and in other subjects too.

Where did the big letter “en” go? In principle, it could have been additionally attached to the solution, but in order to shorten the entry, I did not do this. I hope everyone understands that and is a designation for the same thing.

A popular example for a DIY 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. The solution and answer are at the end of the lesson.

In practice, the task is really very common; triangles can generally torment you.

To solve other problems we will need:

Properties of the vector 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 highlighted 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) – associative or associative vector product laws. Constants can be easily moved outside the vector product. Really, what should they do there?

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

To demonstrate, let's look at a short example:

Example 3

Find if

Solution: The condition again requires finding the length of the vector product. Let's paint our miniature:

(1) According to associative laws, we take the constants outside the scope of the vector product.

(2) We move the constant outside the module, and the module “eats” the minus sign. The length cannot be negative.

(3) The rest is clear.

Answer:

It's time to add more wood to the fire:

Example 4

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

Solution: Find the area of ​​the triangle using the formula . The catch is that the vectors “tse” and “de” are themselves presented as sums of vectors. The algorithm here is standard and somewhat reminiscent of examples No. 3 and 4 of the lesson Dot product of vectors. For clarity, we will divide the solution into three stages:

1) At the first step, we express the vector product through the vector product, in fact, let's express a vector in terms of a vector. No word yet on lengths!

(1) Substitute the expressions of the vectors.

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

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

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

(5) We present similar terms.

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

2) In 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 required triangle:

Stages 2-3 of the solution could have been written in one line.

Answer:

The problem considered is quite common in tests, here is an example for solving it yourself:

Example 5

Find if

A 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

, specified in an orthonormal basis, expressed by the formula:

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

Example 10

Check whether the following space vectors are collinear:
A)
b)

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

a) Find the vector product:

Thus, 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 depend on the definition, geometric meaning and a couple of working formulas.

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

So they lined up like a train and can’t wait to be identified.

First, again, a definition and a picture:

Definition: Mixed work non-coplanar vectors, taken in this order, called parallelepiped volume, 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 with dotted lines:

Let's dive into the definition:

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

3) Before commenting on the geometric meaning, I will note an obvious fact: the mixed product of vectors is a NUMBER: . In educational literature, the design may be slightly different; I am used to denoting a mixed product by , and the result of calculations by the letter “pe”.

A-priory 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 a given parallelepiped.

Note : The drawing is schematic.

4) Let’s not worry again about the concept of 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 words, a mixed product can be negative: .

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

MIXED PRODUCT OF THREE VECTORS AND ITS PROPERTIES

Mixed work three vectors is called a number equal to . Designated . 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 a certain number.

Let's consider the properties of a mixed product.

1. Geometric meaning mixed work. 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 set aside the vectors from the common origin and construct a parallelepiped on them. Let us denote and note that . By definition of the scalar product

   Assuming that and denoting by h find the height of the parallelepiped.

   Thus, when

   If, then so. Hence, .

   Combining both of these cases, we get or .

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

2. For any vectors , , the equality is true

   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 and obtuse.

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

   Indeed, if we consider a 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 coplanarity of 3 vectors is that their mixed product is equal to zero. In addition, it follows 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 the third-order determinant, which has the coordinates of the first vector in the first line, the coordinates of the second vector in the second line, and the coordinates of the third vector in the third line.

   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 specified by an equation, but as a set of points in space that have one or another property. In this case, it is necessary 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

Let us consider an arbitrary plane σ in space. Its position is determined by specifying a vector perpendicular to this plane and some fixed point M0(x 0, y 0, z 0), lying in the σ plane.

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

Let us derive the equation of the plane σ passing through this 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 MО σ is a 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 σ plane.

If we denote the points by the radius vector M, – radius vector of the point M0, then the equation can be written in the form

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 this point. Thus, in order to create an equation of a 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 first degree equation with respect to Cartesian coordinates x, y, z represents the equation of some plane. This equation is written as:

Ax+By+Cz+D=0

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

Let us consider special 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 become zero.

A is the length of the segment cut off by the plane on the axis Ox. Similarly, it can be shown that b And c– lengths of segments cut off by the plane under consideration on the axes Oy And Oz.

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