The cross product of a vector and itself. Vector product of vectors given by coordinates

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Definition. The vector product of a vector a (multiplier) by a vector (multiplier) that is not collinear to it is the third vector c (product), which is constructed as follows:

1) its modulus is numerically equal to the area of ​​the parallelogram in fig. 155), built on vectors, i.e., it is equal to the direction perpendicular to the plane of the mentioned parallelogram;

3) in this case, the direction of the vector c is chosen (out of two possible ones) so that the vectors c form a right-handed system (§ 110).

Designation: or

Addendum to the definition. If the vectors are collinear, then considering the figure as a (conditionally) parallelogram, it is natural to assign zero area. That's why vector product collinear vectors is considered equal to the null vector.

Since the null vector can be assigned any direction, this convention does not contradict items 2 and 3 of the definition.

Remark 1. In the term "vector product", the first word indicates that the result of an action is a vector (as opposed to a scalar product; cf. § 104, remark 1).

Example 1. Find the vector product where the main vectors of the right coordinate system (Fig. 156).

1. Since the lengths of the main vectors are equal to the scale unit, the area of ​​the parallelogram (square) is numerically equal to one. Hence, the modulus of the vector product is equal to one.

2. Since the perpendicular to the plane is the axis, the desired vector product is a vector collinear to the vector k; and since both of them have modulus 1, the required cross product is either k or -k.

3. Of these two possible vectors, the first must be chosen, since the vectors k form a right system (and the vectors form a left one).

Example 2. Find the cross product

Solution. As in example 1, we conclude that the vector is either k or -k. But now we need to choose -k, since the vectors form the right system (and the vectors form the left). So,

Example 3 The vectors have lengths of 80 and 50 cm, respectively, and form an angle of 30°. Taking a meter as a unit of length, find the length of the vector product a

Solution. The area of ​​a parallelogram built on vectors is equal to The length of the desired vector product is equal to

Example 4. Find the length of the cross product of the same vectors, taking a centimeter as a unit of length.

Solution. Since the area of ​​the parallelogram built on vectors is equal to the length of the vector product is 2000 cm, i.e.

Comparison of examples 3 and 4 shows that the length of the vector depends not only on the lengths of the factors, but also on the choice of the length unit.

The physical meaning of the vector product. Of the many physical quantities represented by the vector product, we will consider only the moment of force.

Let A be the point of application of the force. The moment of force relative to the point O is called the vector product. Since the module of this vector product is numerically equal to the area of ​​the parallelogram (Fig. 157), the module of the moment is equal to the product of the base by the height, i.e., the force multiplied by the distance from the point O to the straight line along which the force acts.

In mechanics, it is proved that for the equilibrium of a rigid body it is necessary that not only the sum of the vectors representing the forces applied to the body, but also the sum of the moments of forces should be equal to zero. In the case when all forces are parallel to the same plane, the addition of the vectors representing the moments can be replaced by the addition and subtraction of their moduli. But for arbitrary directions of forces, such a replacement is impossible. In accordance with this, the cross product is defined precisely as a vector, and not as a number.

The online calculator calculates the cross product of vectors. A detailed solution is given. To calculate the cross product of vectors, enter the coordinates of the vectors in the cells and click on the "Calculate."

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Data entry instruction. Numbers are entered as whole numbers (examples: 487, 5, -7623, etc.), decimal numbers (eg. 67., 102.54, etc.) or fractions. The fraction must be typed in the form a/b, where a and b (b>0) are integer or decimal numbers. Examples 45/5, 6.6/76.4, -7/6.7, etc.

Cross product of vectors

Before proceeding to the definition of the vector product of vectors, consider the concepts ordered triple of vectors, left triple of vectors, right triple of vectors.

Definition 1. Three vectors are called ordered triple(or triple) if it is indicated which of these vectors is the first, which is the second and which is the third.

Recording cba- means - the first is a vector c, the second is the vector b and the third is the vector a.

Definition 2. A triple of non-coplanar vectors abc called right (left) if, when reduced to a common beginning, these vectors are arranged as they are respectively large, unbent index and middle fingers right (left) hand.

Definition 2 can be formulated in another way.

Definition 2. A triple of non-coplanar vectors abc is called right (left) if, when reduced to a common origin, the vector c located on the other side of the plane defined by the vectors a And b, whence the shortest turn from a To b performed counterclockwise (clockwise).

Vector trio abc shown in fig. 1 is right and triple abc shown in fig. 2 is left.

If two triples of vectors are right or left, then they are said to have the same orientation. Otherwise, they are said to be of opposite orientation.

Definition 3. A Cartesian or affine coordinate system is called right (left) if the three basis vectors form a right (left) triple.

For definiteness, in what follows we will consider only right-handed coordinate systems.

Definition 4. vector art vector a per vector b called vector With, denoted by the symbol c=[ab] (or c=[a,b], or c=a×b) and satisfying the following three requirements:

• vector length With is equal to the product of the lengths of the vectors a And b to the sine of the angle φ between them:

|c|=|[ab]|=|a||b|sinφ;

(1)

• vector With orthogonal to each of the vectors a And b;
• vector c directed so that the three abc is right.

The cross product of vectors has the following properties:

• [ab]=−[ba] (antipermutability factors);
• [(λa)b]=λ [ab] (compatibility relative to the numerical factor);
• [(a+b)c]=[ac]+[bc] (distribution relative to the sum of vectors);
• [aa]=0 for any vector a.

Geometric properties of the cross product of vectors

Theorem 1. For two vectors to be collinear, it is necessary and sufficient that their vector product be equal to zero.

Proof. Necessity. Let the vectors a And b collinear. Then the angle between them is 0 or 180° and sinφ=sin180=sin 0=0. Therefore, taking into account expression (1), the length of the vector c equals zero. Then c null vector.

Adequacy. Let the cross product of vectors a And b nav to zero: [ ab]=0. Let us prove that the vectors a And b collinear. If at least one of the vectors a And b zero, then these vectors are collinear (because the zero vector has an indefinite direction and can be considered collinear to any vector).

If both vectors a And b nonzero, then | a|>0, |b|>0. Then from [ ab]=0 and from (1) it follows that sinφ=0. Hence the vectors a And b collinear.

The theorem has been proven.

Theorem 2. The length (modulus) of the vector product [ ab] equals the area S parallelogram built on vectors reduced to a common origin a And b.

Proof. As you know, the area of ​​a parallelogram is equal to the product of the adjacent sides of this parallelogram and the sine of the angle between them. Hence:

Then the cross product of these vectors has the form:

Expanding the determinant over the elements of the first row, we get the decomposition of the vector a×b basis i, j, k, which is equivalent to formula (3).

Proof of Theorem 3. Compose all possible pairs of basis vectors i, j, k and calculate their vector product. It should be taken into account that the basis vectors are mutually orthogonal, form a right triple, and have unit length (in other words, we can assume that i={1, 0, 0}, j={0, 1, 0}, k=(0, 0, 1)). Then we have:

From the last equality and relations (4), we obtain:

Compose a 3×3 matrix, the first row of which are the basis vectors i, j, k, and the remaining rows are filled with elements of vectors a And b.

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:

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 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. searched through known coordinates 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 set 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 cross product in terms of the matrix determinant, then the solution 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 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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Dot product properties

Dot product of vectors, definition, properties

Linear operations on vectors.

Vectors, basic concepts, definitions, linear operations on them

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

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

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

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

Vectors lying on perpendicular lines are called orthogonal.

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

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

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

Properties of multiplication of a vector by a number:

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

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

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

Consequence. If non-zero vectors A And b are collinear, then there is a number k, What b= k a.

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

a · b = |a| · | b| cos φ

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

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

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

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

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

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

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

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

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

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

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

Cross product properties:

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

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

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

4 .For any three vectors, the equality

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