Before giving the concept of a vector product, let us turn to the question of the orientation of an ordered triple of vectors a →, b →, c → in three-dimensional space.

To begin with, let’s set aside the vectors a → , b → , c → from one point. The orientation of the triple a → , b → , c → can be right or left, depending on the direction of the vector c → itself. The type of triple a → , b → , c → will be determined from the direction in which the shortest turn is made from vector a → to b → from the end of vector c → .

If the shortest turn is carried out counterclockwise, then the triple of vectors a → , b → , c → is called right, if clockwise – left.

Next, take two non-collinear vectors a → and b →. Let us then plot the vectors A B → = a → and A C → = b → from point A. Let's construct a vector A D → = c →, which is simultaneously perpendicular to both A B → and A C →. Thus, when constructing the vector itself A D → = c →, we can do two things, giving it either one direction or the opposite (see illustration).

An ordered triple of vectors a → , b → , c → can be, as we found out, right or left depending on the direction of the vector.

From the above we can introduce the definition of a vector product. This definition is given for two vectors defined in a rectangular coordinate system of three-dimensional space.

Definition 1

The vector product of two vectors a → and b → we will call such a vector defined in a rectangular coordinate system of three-dimensional space such that:

• if the vectors a → and b → are collinear, it will be zero;
• it will be perpendicular to both vector a → ​​​​ and vector b → i.e. ∠ a → c → = ∠ b → c → = π 2 ;
• its length is determined by the formula: c → = a → · b → · sin ∠ a → , b → ;
• the triple of vectors a → , b → , c → has the same orientation as the given coordinate system.

The vector product of vectors a → and b → has the following notation: a → × b →.

Coordinates of the vector product

Since any vector has certain coordinates in the coordinate system, we can introduce a second definition of a vector product, which will allow us to find its coordinates using the given coordinates of the vectors.

Definition 2

In a rectangular coordinate system of three-dimensional space vector product of two vectors a → = (a x ; a y ; a z) and b → = (b x ; b y ; b z) is called a vector c → = 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 → , where i → , j → , k → are coordinate vectors.

The vector product can be represented as the determinant of a third-order square matrix, where the first row contains the vector vectors i → , j → , k → , the second row contains the coordinates of the vector a → , and the third row contains the coordinates of the vector b → in a given rectangular coordinate system, this is the determinant of the matrix looks like this: c → = a → × b → = i → j → k → a x a y a z b x b y b z

Expanding this determinant into the elements of the first row, we obtain the equality: c → = a → × b → = i → j → k → a x a y a z b x b y b z = 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 → = = 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 →

Properties of a cross product

It is known that the vector product in coordinates is represented as the determinant of the matrix c → = a → × b → = i → j → k → a x a y a z b x b y b z , then on the basis properties of the matrix determinant the following are displayed properties of a vector product:

1. anticommutativity a → × b → = - b → × a → ;
2. distributivity a (1) → + a (2) → × b = a (1) → × b → + a (2) → × b → or a → × b (1) → + b (2) → = a → × b (1) → + a → × b (2) → ;
3. associativity λ a → × b → = λ a → × b → or a → × (λ b →) = λ a → × b →, where λ is an arbitrary real number.

These properties have simple proofs.

As an example, we can prove the anticommutative property of a vector product.

Proof of anticommutativity

By definition, a → × b → = i → j → k → a x a y a z b x b y b z and b → × a → = i → j → k → b x b y b z a x a y a z . And if two rows of the matrix are swapped, then the value of the determinant of the matrix should change to the opposite, therefore, a → × b → = i → j → k → a x a y a z b x b y b z = - i → j → k → b x b y b z a x a y a z = - b → × a → , which and proves that the vector product is anticommutative.

Vector product - examples and solutions

In most cases, there are three types of problems.

In problems of the first type, the lengths of two vectors and the angle between them are usually given, and you need to find the length of the vector product. In this case, use the following formula c → = a → · b → · sin ∠ a → , b → .

Example 1

Find the length of the vector product of vectors a → and b → if you know a → = 3, b → = 5, ∠ a →, b → = π 4.

Solution

By determining the length of the vector product of vectors a → and b →, we solve this problem: a → × b → = a → · b → · sin ∠ a → , b → = 3 · 5 · sin π 4 = 15 2 2 .

Answer: 15 2 2 .

Problems of the second type have a connection with the coordinates of vectors, in them the vector product, its length, etc. are searched through the known coordinates of given vectors a → = (a x; a y; a z) And b → = (b x ; b y ; b z) .

For this type of problem, you can solve a lot of task options. For example, not the coordinates of the vectors a → and b → can be specified, but their expansions into coordinate vectors of the form b → = b x · i → + b y · j → + b z · k → and c → = 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 →, or vectors a → and b → can be specified by the coordinates of their start and end points.

Consider the following examples.

Example 2

In a rectangular coordinate system, two vectors are given: a → = (2; 1; - 3), b → = (0; - 1; 1). Find their cross product.

Solution

By the second definition, we find the vector product of two vectors in given coordinates: 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 → = = (1 · 1 - (- 3) · (- 1)) · i → + ((- 3) · 0 - 2 · 1) · j → + (2 · (- 1) - 1 · 0) · k → = = - 2 i → - 2 j → - 2 k → .

If we write the vector product through the determinant of the matrix, then the solution to this example looks like this: a → × b → = i → j → k → a x a y a z b x b y b z = i → j → k → 2 1 - 3 0 - 1 1 = - 2 i → - 2 j → - 2 k → .

Answer: a → × b → = - 2 i → - 2 j → - 2 k → .

Example 3

Find the length of the vector product of vectors i → - j → and i → + j → + k →, where i →, j →, k → are the unit vectors of the rectangular Cartesian coordinate system.

Solution

First, let's find the coordinates of a given vector product i → - j → × i → + j → + k → in a given rectangular coordinate system.

It is known that the vectors i → - j → and i → + j → + k → have coordinates (1; - 1; 0) and (1; 1; 1), respectively. Let's find the length of the vector product using the determinant of the matrix, then we have i → - j → × i → + j → + k → = i → j → k → 1 - 1 0 1 1 1 = - i → - j → + 2 k → .

Therefore, the vector product i → - j → × i → + j → + k → has coordinates (- 1 ; - 1 ; 2) in the given coordinate system.

We find the length of the vector product using the formula (see the section on finding the length of a vector): i → - j → × i → + j → + k → = - 1 2 + - 1 2 + 2 2 = 6.

Answer: i → - j → × i → + j → + k → = 6 . .

Example 4

In a rectangular Cartesian coordinate system, the coordinates of three points A (1, 0, 1), B (0, 2, 3), C (1, 4, 2) are given. Find some vector perpendicular to A B → and A C → at the same time.

Solution

Vectors A B → and A C → have the following coordinates (- 1 ; 2 ; 2) and (0 ; 4 ; 1) respectively. Having found the vector product of the vectors A B → and A C →, it is obvious that it is a perpendicular vector by definition to both A B → and A C →, that is, it is a solution to our problem. Let's find it A B → × A C → = i → j → k → - 1 2 2 0 4 1 = - 6 i → + j → - 4 k → .

Answer: - 6 i → + j → - 4 k → . - one of the perpendicular vectors.

Problems of the third type are focused on using the properties of the vector product of vectors. After applying which, we will obtain a solution to the given problem.

Example 5

Vectors a → and b → are perpendicular and their lengths are 3 and 4, respectively. Find the length of the vector product 3 a → - b → × a → - 2 b → = 3 a → × a → - 2 b → + - b → × a → - 2 b → = = 3 a → × a → + 3 · a → × - 2 · b → + - b → × a → + - b → × - 2 · b → .

Solution

By the distributive property of a vector product, we can write 3 a → - b → × a → - 2 b → = 3 a → × a → - 2 b → + - b → × a → - 2 b → = = 3 a → × a → + 3 a → × - 2 b → + - b → × a → + - b → × - 2 b →

By the property of associativity, we take the numerical coefficients out of the sign of the vector products in the last expression: 3 · a → × a → + 3 · a → × - 2 · b → + - b → × a → + - b → × - 2 · b → = = 3 · a → × a → + 3 · (- 2) · a → × b → + (- 1) · b → × a → + (- 1) · (- 2) · b → × b → = = 3 a → × a → - 6 a → × b → - b → × a → + 2 b → × b →

The vector products a → × a → and b → × b → are equal to 0, since a → × a → = a → · a → · sin 0 = 0 and b → × b → = b → · b → · sin 0 = 0, then 3 · a → × a → - 6 · a → × b → - b → × a → + 2 · b → × b → = - 6 · a → × b → - b → × a → . .

From the anticommutativity of the vector product it follows - 6 · a → × b → - b → × a → = - 6 · a → × b → - (- 1) · a → × b → = - 5 · a → × b → . .

Using the properties of the vector product, we obtain the equality 3 · a → - b → × a → - 2 · b → = = - 5 · a → × b → .

By condition, the vectors a → and b → are perpendicular, that is, the angle between them is equal to π 2. Now all that remains is to substitute the found values ​​into the appropriate formulas: 3 a → - b → × a → - 2 b → = - 5 a → × b → = = 5 a → × b → = 5 a → b → · sin (a → , b →) = 5 · 3 · 4 · sin π 2 = 60 .

Answer: 3 a → - b → × a → - 2 b → = 60.

The length of the vector product of vectors by definition is equal to a → × b → = a → · b → · sin ∠ a → , b → . Since it is already known (from the school course) that the area of ​​a triangle is equal to half the product of the lengths of its two sides multiplied by the sine of the angle between these sides. Consequently, the length of the vector product is equal to the area of ​​the parallelogram - a doubled triangle, namely the product of the sides in the form of vectors a → and b →, laid down from one point, by the sine of the angle between them sin ∠ a →, b →.

This is the geometric meaning of a vector product.

Physical meaning of the vector product

In mechanics, one of the branches of physics, thanks to the vector product, you can determine the moment of a force relative to a point in space.

Definition 3

By the moment of force F → applied to point B, relative to point A, we will understand the following vector product A B → × F →.

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

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, 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 take 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.

In this article we will take a closer look at the concept of the cross product of two vectors. We will give the necessary definitions, write a formula for finding the coordinates of a vector product, list and justify its properties. After this, we will dwell on the geometric meaning of the vector product of two vectors and consider solutions to various typical examples.

Page navigation.

Definition of cross product.

Before defining a vector product, let's understand the orientation of an ordered triple of vectors in three-dimensional space.

Let's plot the vectors from one point. Depending on the direction of the vector, the three can be right or left. Let's look from the end of the vector at how the shortest turn from the vector to . If the shortest rotation occurs counterclockwise, then the triple of vectors is called right, otherwise - left.

Now let's take two non-collinear vectors and . Let us plot the vectors and from point A. Let's construct some vector perpendicular to both and and . Obviously, when constructing a vector, we can do two things, giving it either one direction or the opposite (see illustration).

Depending on the direction of the vector, the ordered triplet of vectors can be right-handed or left-handed.

This brings us close to the definition of a vector product. It is given for two vectors defined in a rectangular coordinate system of three-dimensional space.

Definition.

The cross product of two vectors and , specified in a rectangular coordinate system of three-dimensional space, is called a vector such that

The cross product of vectors and is denoted as .

Coordinates of the vector product.

Now we will give the second definition of a vector product, which allows you to find its coordinates from the coordinates of given vectors and.

Definition.

In a rectangular coordinate system of three-dimensional space vector product of two vectors And is a vector , where are the coordinate vectors.

This definition gives us the cross product in coordinate form.

It is convenient to represent the vector product as the determinant of a third-order square matrix, the first row of which is the vectors, the second row contains the coordinates of the vector, and the third contains the coordinates of the vector in a given rectangular coordinate system:

If we expand this determinant into the elements of the first row, we obtain the equality from the definition of the vector product in coordinates (if necessary, refer to the article):

It should be noted that the coordinate form of the vector product is fully consistent with the definition given in the first paragraph of this article. Moreover, these two definitions of a cross product are equivalent. You can see the proof of this fact in the book listed at the end of the article.

Properties of a vector product.

Since the vector product in coordinates can be represented as a determinant of the matrix, the following can easily be justified on the basis properties of the cross product:

As an example, let us prove the anticommutative property of a vector product.

A-priory And . We know that the value of the determinant of a matrix is ​​reversed if two rows are swapped, therefore, , which proves the anticommutative property of a vector product.

Vector product - examples and solutions.

There are mainly three types of problems.

In problems of the first type, the lengths of two vectors and the angle between them are given, and you need to find the length of the vector product. In this case, the formula is used .

Example.

Find the length of the vector product of the vectors and , if known .

Solution.

We know from the definition that the length of the vector product of vectors and is equal to the product of the lengths of vectors and by the sine of the angle between them, therefore, .

Answer:

.

Problems of the second type are related to the coordinates of vectors, in which the vector product, its length or anything else is searched through the coordinates of given vectors And .

There are a lot of different options possible here. For example, not the coordinates of the vectors and can be specified, but their expansions into coordinate vectors of the form and , or vectors and can be specified by the coordinates of their start and end points.

Let's look at typical examples.

Example.

Two vectors are given in a rectangular coordinate system . Find their cross product.

Solution.

According to the second definition, the vector product of two vectors in coordinates is written as:

We would have arrived at the same result if the vector product had been written in terms of the determinant

Answer:

.

Example.

Find the length of the vector product of the vectors and , where are the unit vectors of the rectangular Cartesian coordinate system.

Solution.

First we find the coordinates of the vector product in a given rectangular coordinate system.

Since vectors and have coordinates and respectively (if necessary, see the article coordinates of a vector in a rectangular coordinate system), then by the second definition of a vector product we have

That is, the vector product has coordinates in a given coordinate system.

We find the length of a vector product as the square root of the sum of the squares of its coordinates (we obtained this formula for the length of a vector in the section on finding the length of a vector):

Answer:

.

Example.

In a rectangular Cartesian coordinate system, the coordinates of three points are given. Find some vector that is perpendicular and at the same time.

Solution.

Vectors and have coordinates and respectively (see the article finding the coordinates of a vector through the coordinates of points). If we find the vector product of the vectors and , then by definition it is a vector perpendicular to both to and to , that is, it is a solution to our problem. Let's find him

Answer:

- one of the perpendicular vectors.

In problems of the third type, the skill of using the properties of the vector product of vectors is tested. After applying the properties, the corresponding formulas are applied.

Example.

The vectors and are perpendicular and their lengths are 3 and 4, respectively. Find the length of the cross product .

Solution.

By the distributive property of a vector product, we can write

Due to the combinational property, we take the numerical coefficients out of the sign of the vector products in the last expression:

The vector products and are equal to zero, since And , Then .

Since the vector product is anticommutative, then .

So, using the properties of the vector product, we arrived at the equality .

By condition, the vectors and are perpendicular, that is, the angle between them is equal to . That is, we have all the data to find the required length

Answer:

.

Geometric meaning of a vector product.

By definition, the length of the vector product of vectors is . And from a high school geometry course we know that the area of ​​a triangle is equal to half the product of the lengths of the two sides of the triangle and the sine of the angle between them. Consequently, the length of the vector product is equal to twice the area of ​​a triangle whose sides are the vectors and , if they are plotted from one point. In other words, the length of the vector product of vectors and is equal to the area of ​​a parallelogram with sides and and the angle between them equal to . This is the geometric meaning of the vector product.

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 a certain 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.

7.1. Definition of cross product

Three non-coplanar vectors a, b and c, taken in the indicated order, form a right-handed triplet if, from the end of the third vector c, the shortest turn from the first vector a to the second vector b is seen to be counterclockwise, and a left-handed triplet if clockwise (see Fig. . 16).

The vector product of vector a and vector b is called vector c, which:

1. Perpendicular to vectors a and b, i.e. c ^ a and c ^ b ;

2. Has a length numerically equal to the area of ​​a parallelogram constructed on vectors a andb as on the sides (see Fig. 17), i.e.

3. Vectors a, b and c form a right-handed triple.

The cross product is denoted a x b or [a,b]. The following relations between the unit vectors i directly follow from the definition of the vector product, j And k(see Fig. 18):

i x j = k, j x k = i, k x i = j.
Let us prove, for example, that i xj =k.

1) k ^ i, k ^ j ;

2) |k |=1, but | i x j| = |i | |J | sin(90°)=1;

3) vectors i, j and k form a right triple (see Fig. 16).

7.2. Properties of a cross product

1. When rearranging the factors, the vector product changes sign, i.e. and xb =(b xa) (see Fig. 19).

Vectors a xb and b xa are collinear, have the same modules (the area of ​​the parallelogram remains unchanged), but are oppositely directed (triples a, b, a xb and a, b, b x a of opposite orientation). That is axb = -(b xa).

2. The vector product has a combining property with respect to the scalar factor, i.e. l ​​(a xb) = (l a) x b = a x (l b).

Let l >0. Vector l (a xb) is perpendicular to vectors a and b. Vector ( l a)x b is also perpendicular to vectors a and b(vectors a, l but lie in the same plane). This means that the vectors l(a xb) and ( l a)x b collinear. It is obvious that their directions coincide. They have the same length:

That's why l(a xb)= l a xb. It is proved in a similar way for l<0.

3. Two non-zero vectors a and b are collinear if and only if their vector product is equal to the zero vector, i.e. a ||b<=>and xb =0.

In particular, i *i =j *j =k *k =0 .

4. The vector product has the distribution property:

(a+b) xc = a xc + b xs.

We will accept without proof.

7.3. Expressing the cross product in terms of coordinates

We will use the cross product table of vectors i, j and k:

if the direction of the shortest path from the first vector to the second coincides with the direction of the arrow, then the product is equal to the third vector; if it does not coincide, the third vector is taken with a minus sign.

Let two vectors a =a x i +a y be given j+a z k and b =b x i+b y j+b z k. Let's find the vector product of these vectors by multiplying them as polynomials (according to the properties of the vector product):

The resulting formula can be written even more briefly:

since the right side of equality (7.1) corresponds to the expansion of the third-order determinant in terms of the elements of the first row. Equality (7.2) is easy to remember.

7.4. Some applications of cross product

Establishing collinearity of vectors

Finding the area of ​​a parallelogram and a triangle

According to the definition of the vector product of vectors A and b |a xb | =|a | * |b |sin g, i.e. S pairs = |a x b |. And, therefore, D S =1/2|a x b |.

Determination of the moment of force about a point

Let a force be applied at point A F =AB let it go ABOUT- some point in space (see Fig. 20).

It is known from physics that moment of force F relative to the point ABOUT called a vector M, which passes through the point ABOUT And:

1) perpendicular to the plane passing through the points O, A, B;

2) numerically equal to the product of force per arm

3) forms a right triple with vectors OA and A B.

Therefore, M = OA x F.

Finding linear rotation speed

Speed v point M of a rigid body rotating with angular velocity w around a fixed axis, is determined by Euler’s formula v =w xr, where r =OM, where O is some fixed point of the axis (see Fig. 21).