Obviously, in the case of a vector product, the order in which the vectors are taken matters, moreover,
Also, directly from the definition it follows that for any scalar factor k (number) the following is true:
The cross product of collinear vectors is equal to the zero vector. Moreover, vector product of two vectors is equal to zero if and only if they are collinear. (In case one of them is a zero vector, it is necessary to remember that a zero vector is collinear to any vector by definition).
The vector product has distributive property, that is
Expressing the vector product through the coordinates of vectors.
Let two vectors be given
(how to find the coordinates of a vector from the coordinates of its beginning and end - see the article Dot product of vectors, item Alternative definition of the dot product, or calculating the dot product of two vectors specified by their coordinates.)
Why do you need a vector product?
There are many ways to use the cross product, for example, as written above, by calculating the cross product of two vectors you can find out whether they are collinear.
Or it can be used as a way to calculate the area of a parallelogram constructed from these vectors. Based on the definition, the length of the resulting vector is the area of the given parallelogram.
Also great amount applications exist in electricity and magnetism.Online vector product calculator.
To find the scalar product of two vectors using this calculator, you need to enter in the first line in order the coordinates of the first vector, in second - second. The coordinates of vectors can be calculated from the coordinates of their beginning and end (see article Dot product of vectors, item An alternative definition of the dot product, or calculating the dot product of two vectors given by their coordinates.)
The online calculator calculates the cross product of vectors. Given detailed solution. To calculate the cross product of vectors, enter the coordinates of the vectors in the cells and click on the "Calculate" button.
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Data entry instructions. Numbers are entered as integers (examples: 487, 5, -7623, etc.), decimals (ex. 67., 102.54, etc.) or fractions. The fraction must be entered in the form a/b, where a and b (b>0) are integers or decimals. Examples 45/5, 6.6/76.4, -7/6.7, etc.
Vector product of vectors
Before moving on to the definition of the vector product of vectors, let's consider the concepts ordered vector triplet, left vector triplet, right vector triplet.
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.
Record cba- means - the first is a vector c, the second is the vector b and the third is the vector a.
Definition 2. Triple of non-coplanar vectors abc called right (left) if, when reduced to general beginning, these vectors are located in the same way as the large, unbent index and middle fingers of the right (left) hand are located, respectively.
Definition 2 can be formulated differently.
Definition 2". Triple of non-coplanar vectors abc is called right (left) if, when reduced to a common origin, the vector c is located on the other side of the plane defined by the vectors a And b, where is the shortest turn from a To b performed counterclockwise (clockwise).
Three of vectors abc, shown in Fig. 1 is right, and three abc shown in Fig. 2 is the left one.
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If two triplets of vectors are right or left, then they are said to be of 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 three basis vectors form a right (left) triple.
For definiteness, in what follows we will consider only right-handed coordinate systems.
Definition 4. Vector artwork vector a to vector b called a 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 equal to the product of vector lengths a And b by the sine of the angle φ between them:
- vector With orthogonal to each of the vectors a And b;
- vector c directed so that the three abc is right.
| |c|=|[ab]|=|a||b|sinφ; | (1) |
The cross product of vectors has the following properties:
- [ab]=−[ba] (anti-permutability factors);
- [(λa)b]=λ [ab] (combination relative to the numerical factor);
- [(a+b)c]=[ac]+[bc] (distributiveness relative to the sum of vectors);
- [aa]=0 for any vector a.
Geometric properties of the vector 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 equal to zero. Then c zero vector.
Adequacy. Let the vector product of vectors a And b obviously 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 (since the zero vector has an indefinite direction and can be considered collinear to any vector).
If both vectors a And b non-zero, then | a|>0, |b|>0. Then from [ ab]=0 and from (1) it follows that sinφ=0. Therefore the vectors a And b collinear.
The theorem is proven.
Theorem 2. Length (modulus) of the vector product [ ab] equals area S parallelogram constructed 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 vector product of these vectors has the form:
Expanding the determinant over the elements of the first row, we obtain the decomposition of the vector a×b by basis i, j, k, which is equivalent to formula (3).
Proof of Theorem 3. Let's create 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-handed 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:
Let's create a 3x3 matrix, the first row of which is the basis vectors i, j, k, and the remaining lines are filled with vector elements a And b:
Thus, the result of the vector product of vectors a And b will be a vector:
 .
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Example 2. Find the vector product of vectors [ ab], where is the vector a represented by two points. Starting point of vector a: 
, end point of the vector a: 
, vector b looks like 
.
Solution: Move the first vector to the origin. To do this, subtract the coordinates of the starting point from the corresponding coordinates of the end point:
Let's calculate the determinant of this matrix by expanding it along the first row. The result of these calculations is the vector product of vectors a And b.
Vector artwork is a pseudovector perpendicular to a plane constructed from two factors, which is the result of the binary operation “vector multiplication” over vectors in three-dimensional Euclidean space. The vector product does not have the properties of commutativity and associativity (it is anticommutative) and, unlike the scalar product of vectors, is a vector. Widely used in many engineering and physics applications. For example, angular momentum and Lorentz force are written mathematically as a vector product. The cross product is useful for "measuring" the perpendicularity of vectors - the modulus of the cross product of two vectors is equal to the product of their moduli if they are perpendicular, and decreases to zero if the vectors are parallel or antiparallel.
The vector product can be defined in different ways, and theoretically, in a space of any dimension n, one can calculate the product of n-1 vectors, thereby obtaining a single vector perpendicular to them all. But if the product is limited to non-trivial binary products with vector results, then the traditional vector product is defined only in three-dimensional and seven-dimensional spaces. The result of a vector product, like a scalar product, depends on the metric of Euclidean space.
Unlike the formula for calculating the scalar product vectors from coordinates in a three-dimensional rectangular coordinate system, the formula for the cross product depends on the orientation of the rectangular coordinate system, or, in other words, its “chirality”.
Definition:
The vector product of vector a and vector b in space R3 is a vector c that satisfies the following requirements:
the length of vector c is equal to the product of the lengths of vectors a and b and the sine of the angle φ between them:
|c|=|a||b|sin φ;
vector c is orthogonal to each of vectors a and b;
vector c is directed so that the triple of vectors abc is right-handed;
in the case of the space R7, the associativity of the triple of vectors a, b, c is required.
Designation:
c===a × b
Rice. 1. The area of a parallelogram is equal to the modulus of the vector product
Geometric properties of a cross product:
A necessary and sufficient condition for the collinearity of two nonzero vectors is that their vector product is equal to zero.
Cross Product Module equals area S parallelogram constructed on vectors reduced to a common origin a And b(see Fig. 1).
If e- unit vector orthogonal to the vectors a And b and chosen so that three a,b,e- right, and S is the area of the parallelogram constructed on them (reduced to a common origin), then the formula for the vector product is valid:
=S e
Fig.2. Volume of a parallelepiped using the vector and scalar product of vectors; dotted lines show the projections of vector c onto a × b and vector a onto b × c, the first step is to find the scalar products
If c- some vector, π
- any plane containing this vector, e- unit vector lying in the plane π
and orthogonal to c,g- unit vector orthogonal to the plane π
and directed so that the triple of vectors ecg is right, then for any lying in the plane π
vector a the formula is correct:
=Pr e a |c|g
where Pr e a is the projection of vector e onto a
|c|-modulus of vector c
When using vector and scalar products, you can calculate the volume of a parallelepiped built on vectors reduced to a common origin a, b And c. Such a product of three vectors is called mixed.
V=|a (b×c)|
The figure shows that this volume can be found in two ways: the geometric result is preserved even when the “scalar” and “vector” products are swapped:
V=a×b c=a b×c
The magnitude of the cross product depends on the sine of the angle between the original vectors, so the cross product can be perceived as the degree of “perpendicularity” of the vectors, just as the scalar product can be seen as the degree of “parallelism”. The vector product of two unit vectors is equal to 1 (unit vector) if the original vectors are perpendicular, and equal to 0 (zero vector) if the vectors are parallel or antiparallel.
Expression for the cross product in Cartesian coordinates
If two vectors a And b defined by their rectangular Cartesian coordinates, or more precisely, represented in an orthonormal basis
a=(a x ,a y ,a z)
b=(b x ,b y ,b z)
and the coordinate system is right-handed, then their vector product has the form
=(a y b z -a z b y ,a z b x -a x b z ,a x b y -a y b x)
To remember this formula:
i =∑ε ijk a j b k
Where ε ijk- symbol of Levi-Civita.
Properties of the dot product
Scalar product 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, with the first point being called the beginning and the second point being the end of the vector
Two vectors are said to be equal if they are equal and co-directional.
Vectors lying on the same line are called codirectional if they are codirectional with some of the same vector not lying on this line.
Vectors lying 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. Amount a+b vectors a And b is called a 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. By difference a – b vectors A And b such a vector is called With , which sums with the vector b gives a vector A .
Definition 5.6. The workk a vector A per number k called a vector b , collinear to the vector A , having a modulus equal to | k||a |, and the direction coinciding with the direction A at k>0 and the opposite A at k<0.
Properties of multiplying 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 such a number k, What b = k a.
The scalar product of two non-zero vectors a And b is a number (scalar) equal to the product of the lengths of these vectors and the cosine of the angle φ between them. The dot product can be denoted 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 rearranging the factors);
· Distribution property: a · ( b · c) = (a · b) · c(the result does not depend on the order of multiplication);
· Combination property (with respect to the scalar factor): (λ a) · b = λ ( a · b).
· Property of orthogonality (perpendicularity): if the vector a And b are non-zero, then their scalar product is equal to zero only when these vectors are orthogonal (perpendicular to each other) a ┴ b;
· Property of a square: 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 equal to a · b= x 1 x 2 + y 1 y 2 + z 1 z 2 .
Vector holding vectors. Definition: The vector product of two vectors is a vector for which:
The module is equal to the area of the parallelogram constructed on these vectors, i.e. , where is the angle between the vectors and
This vector is perpendicular to the vectors being multiplied, i.e.
If the vectors are non-collinear, then they form a right-hand triplet of vectors.
Properties of a cross product:
1. When changing the order of the factors, the vector product changes its sign to the opposite, preserving the modulus, i.e.
2 .The vector square is equal to the null 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 is true
5 .Necessary and sufficient condition for the collinearity of two vectors and :