In this article, we will dwell on the concept of the cross product of two vectors. We will give the necessary definitions, write down a formula for finding the coordinates of a vector product, list and justify its properties. After that, we will dwell on the geometric meaning of the cross product of two vectors and consider the solutions of various typical examples.
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Definition of a vector product.
Before giving a definition of a cross product, let's deal with the orientation of an ordered triple of vectors in three-dimensional space.
Let's postpone vectors from one point. Depending on the direction of the vector, the triple can be right or left. Let's look from the end of the vector to see how the shortest turn from the vector to . If the shortest rotation is counterclockwise, then the triple of vectors is called right, otherwise - left.
Now let's take two non-collinear vectors and . Set aside vectors and from point A. Let's construct some vector that is perpendicular to and and at the same time. 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 triple of vectors can be right or left.
So we came close to the definition of a vector product. It is given for two vectors given in a rectangular coordinate system of three-dimensional space.
Definition.
Vector product of two vectors and , given in a rectangular coordinate system of three-dimensional space, is called a vector such that
The cross product of vectors and is denoted as .
Vector product coordinates.
Now we give the second definition of a vector product, which allows us to find its coordinates from the coordinates of the given vectors and.
Definition.
In a rectangular coordinate system of three-dimensional space cross product of two vectors 
And 
is a vector , where are coordinate vectors.
This definition gives us the cross product in coordinate form.
It is convenient to represent the vector product as a determinant of a square matrix of the third order, the first row of which is the orts, the second row contains the coordinates of the vector, and the third row contains the coordinates of the vector in a given rectangular coordinate system: 
If we expand this determinant by the elements of the first row, then we get 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 cross 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. The proof of this fact can be found in the book indicated at the end of the article.
Vector product properties.
Since the vector product in coordinates can be represented as the determinant of the matrix , the following can be easily substantiated on the basis vector product properties:
As an example, let us prove the anticommutativity property of a vector product.
A-priory 
And 
. We know that the value of the determinant of a matrix is reversed when two rows are swapped, so, 
, which proves the anticommutativity property of the vector product.
Vector product - examples and solutions.
Basically there are three types of tasks.
In problems of the first type, the lengths of two vectors and the angle between them are given, and it is required to find the length of the cross product. In this case, the formula is used 
.
Example.
Find the length of the cross product of vectors and if known 
.
Solution.
We know from the definition that the length of the cross product of vectors and is equal to the product of the lengths of vectors and times the sine of the angle between them, therefore, 
.
Answer:
.
Tasks of the second type are associated with the coordinates of vectors, in which the vector product, its length, or something else is searched through the coordinates of the given vectors And .
There are many different options available here. For example, not the coordinates of the vectors and , but their expansions in coordinate vectors of the form 
and , or vectors and can be specified by the coordinates of their start and end points.
Let's consider typical examples.
Example.
Two vectors are given in a rectangular coordinate system 
. Find their vector product.
Solution.
According to the second definition, the cross product of two vectors in coordinates is written as: 
We would have come to the same result if we had written the vector product through the determinant 
Answer:
.
Example.
Find the length of the cross product of vectors and , where are the orts of the rectangular Cartesian coordinate system.
Solution.
First, find the coordinates of the vector product 
in a given rectangular coordinate system.
Since the vectors and have coordinates and respectively (if necessary, see the article coordinates of a vector in a rectangular coordinate system), then according to the second definition of a cross product, we have 
That is, the vector product has coordinates in the 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.
The coordinates of three points are given in a rectangular Cartesian coordinate system. Find some vector that is perpendicular to 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 cross product of vectors and , then by definition it is a vector perpendicular to both to and to, that is, it is the solution to our problem. Let's find him 
Answer:
is one of the perpendicular vectors.
In tasks of the third type, the skill of using the properties of the vector product of vectors is checked. After properties are applied, the corresponding formulas are applied.
Example.
The vectors and are perpendicular and their lengths are 3 and 4 respectively. Find the length of the vector product 
.
Solution.
By the distributivity property of the vector product, we can write 
By virtue of the associative property, we take out the numerical coefficients for the sign of vector products in the last expression: 
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 have come to 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:
.
The geometric meaning of the vector product.
By definition, the length of the cross product of vectors is 
. And from the 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. Therefore, the length of the cross product is equal to twice the area of a triangle with sides of the vectors and , if they are postponed from one point. In other words, the length of the cross product of vectors and is equal to the area of a parallelogram with sides and and an angle between them equal to . This is the geometric meaning of the vector product.
Before giving the concept of a vector product, let us turn to the question of the orientation of the 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 → is right or left, depending on the direction of the vector c → . From the direction in which the shortest turn is made from the vector a → to b → from the end of the vector c → , the form of the triple a → , b → , c → will be determined.
If the shortest rotation is 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 postpone the vectors A B → = a → and A C → = b → from the point A. Let us construct a vector A D → = c → , which is simultaneously perpendicular to both A B → and A C → . Thus, when constructing the vector A D → = c →, we can do two things, giving it either one direction or the opposite (see illustration).
The ordered trio 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 given 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 triplet of vectors a → , b → , c → has the same orientation as the given coordinate system.
The cross product of vectors a → and b → has the following notation: a → × b → .
Cross product coordinates
Since any vector has certain coordinates in the coordinate system, it is possible to introduce a second definition of the vector product, which will allow you to find its coordinates from 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) call the 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 a determinant of a square matrix of the third order, where the first row is the orta vectors i → , j → , k → , the second row contains the coordinates of the vector a → , and the third is the coordinates of the vector b → in a given rectangular coordinate system, this matrix determinant looks like this: c → = a → × b → = i → j → k → a x a y a z b x b y b z
Expanding this determinant over 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 →
Cross product properties
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 base matrix determinant properties the following vector product properties:
- anticommutativity a → × b → = - b → × a → ;
- distributivity a (1) → + a (2) → × b = a (1) → × b → + a (2) → × b → or a → × b (1) → + b (2) → = a → × b (1) → + a → × b (2) → ;
- associativity λ a → × b → = λ a → × b → or a → × (λ b →) = λ a → × b → , where λ is an arbitrary real number.
These properties have not complicated proofs.
For example, we can prove the anticommutativity 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 interchanged, 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 the anticommutativity of the vector product.
Vector Product - Examples and Solutions
In most cases, there are three types of tasks.
In problems of the first type, the lengths of two vectors and the angle between them are usually given, but you need to find the length of the cross product. In this case, use the following formula c → = a → b → sin ∠ a → , b → .
Example 1
Find the length of the cross product of vectors a → and b → if a → = 3 , b → = 5 , ∠ a → , b → = π 4 is known.
Solution
Using the definition of 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 .
Tasks of the second type have a connection with the coordinates of vectors, they contain a vector product, its length, etc. are searched through the known coordinates of the given vectors a → = (a x ; a y ; a z) And b → = (b x ; b y ; b z) .
For this type of task, you can solve a lot of options for tasks. For example, not the coordinates of the vectors a → and b → , but their expansions in 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 the vectors a → and b → can be given by the coordinates of their start and end points.
Consider the following examples.
Example 2
Two vectors are given in a rectangular coordinate system a → = (2 ; 1 ; - 3) , b → = (0 ; - 1 ; 1) . Find their vector product.
Solution
According to 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 matrix determinant, then the solution of this example is as follows: 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 cross product of vectors i → - j → and i → + j → + k → , where i → , j → , k → - orts of a rectangular Cartesian coordinate system.
Solution
First, let's find the coordinates of the given vector product i → - j → × i → + j → + k → in the 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. Find the length of the vector product using the matrix determinant, 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 by the formula (see the section on finding the length of the vector): i → - j → × i → + j → + k → = - 1 2 + - 1 2 + 2 2 = 6 .
Answer: i → - j → × i → + j → + k → = 6 . .
Example 4
The coordinates of three points A (1 , 0 , 1) , B (0 , 2 , 3) , C (1 , 4 , 2) are given in a rectangular Cartesian coordinate system. 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 the solution to our problem. 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 → . is 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
The vectors a → and b → are perpendicular and their lengths are 3 and 4 respectively. Find the length of the cross 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 distributivity property of the 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 out the numerical coefficients beyond the sign of 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 it remains only to substitute the found values into the corresponding 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 cross product of vectors by definition is 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. Therefore, the length of the vector product is equal to the area of a parallelogram - a doubled triangle, namely, the product of the sides in the form of vectors a → and b → , laid off from one point, by the sine of the angle between them sin ∠ a → , b → .
This is the geometric meaning of the vector product.
The 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 force relative to a point in space.
Definition 3
Under the moment of force F → , applied to point B , relative to point A we will understand the following vector product A B → × F → .
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7.1. Definition of cross product
Three non-coplanar vectors a , b and c , taken in the indicated order, form a right triple 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 one if clockwise (see Fig. . 16).
The vector product of a 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. It has a length numerically equal to the area of the parallelogram built on the vectors a andb as on the sides (see fig. 17), i.e.
3. The vectors a , b and c form a right triple.
The vector product is denoted a x b or [a,b]. From the definition of a vector product, the following relations between the orts i follow directly, j And k(see fig. 18):
i x j \u003d k, j x k \u003d i, k x i \u003d j.
Let us prove, for example, that i xj \u003d 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. Cross product properties
1. When the factors are rearranged, the vector product changes sign, i.e. and xb \u003d (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 = -(bxa).
2. The vector product has a combination property with respect to a scalar factor, i.e. l (a xb) \u003d (l a) x b \u003d a x (l b).
Let l >0. The vector l (a xb) is perpendicular to the vectors a and b. Vector ( l a) x b is also perpendicular to the vectors a and b(vectors a, l but lie in the same plane). So 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 similarly 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., and ||b<=>and xb \u003d 0.
In particular, i *i =j *j =k *k =0 .
4. The vector product has a distribution property:
(a+b) xs = a xs + b xs .
Accept without proof.
7.3. Cross product expression in terms of coordinates
We will use the vector cross product table 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 match, the third vector is taken with a minus sign.
Let two vectors a =a x i +a y j+az k and b=bx i+by j+bz 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 shorter:
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 the cross product
Establishing collinearity of vectors
Finding the area of a parallelogram and a triangle
According to the definition of the cross product of vectors A and b |a xb | =| a | * |b |sin g , i.e. S par = |a x b |. And, therefore, D S \u003d 1/2 | a x b |.
Determining 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 torque F relative to the point ABOUT called 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 the force and the arm
3) forms a right triple with vectors OA and A B .
Therefore, M \u003d OA x F.
Finding the linear speed of rotation
Speed v point M of a rigid body rotating at an angular velocity w around a fixed axis, is determined by the Euler formula v \u003d w x r, where r \u003d OM, where O is some fixed point of the axis (see Fig. 21).
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.
- 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 . Hence, .
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 .
- 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.
- When any two factors are interchanged, the mixed product changes sign.
Indeed, if we consider the mixed product , then, for example, or
- 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.
The equation of a plane in segments is convenient to use for constructing planes.
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 wrong. 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 be returned 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 understood 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. 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 required 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 in 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".
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 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.
