Given 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 decimal numbers. 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 is called right (left) if, when reduced to a common origin, 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).
Troika 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) |
Vector artwork 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, you can calculate the product of n-1 vectors, thereby obtaining 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; the 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.
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 dot 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 and 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 as follows: . 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's 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 various 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 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's remember one of 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 triangle. 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 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 index finger 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 sides. 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.
Special case– vector product of a vector with itself:
Using the cross product, you can check the collinearity of three-dimensional vectors, and this task among others, we will also analyze.
To solve practical examples may be required 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 independent decision:
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 an independent solution:
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 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”.
By definition 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, the mixed product can be negative: .
Directly from the definition follows the formula for calculating the volume of a parallelepiped built on vectors.
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). Therefore 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 and let 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).
