This 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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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 the large, unbent index and middle fingers of the right (left) hand are respectively located.
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.
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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:
- 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] (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. Need. 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. Consequently:
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.
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 O- some point in space (see Fig. 20).
It is known from physics that torque F relative to the point O called vector M , which passes through the point O and:
1) perpendicular to the plane passing through the points O, A, B;
2) numerically equal to the product of the force and the shoulder
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).
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. Therefore, the vector product of 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.
vector product is a pseudovector perpendicular to the plane constructed by two factors, which is the result of the binary operation "vector multiplication" on 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 technical and physical applications. For example, the angular momentum and the Lorentz force are mathematically written as a cross 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 anti-parallel.
You can define a vector product in different ways, and theoretically, in a space of any dimension n, you can calculate the product of n-1 vectors, while 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 the vector product, like the scalar product, depends on the metric of the Euclidean space.
Unlike the formula for calculating the scalar product from the coordinates of the vectors in a three-dimensional rectangular coordinate system, the formula for the vector product depends on the orientation of the rectangular coordinate system, or, in other words, its “chirality”.
Definition:
The vector product of a vector a and vector b in the space R 3 is called a vector c that satisfies the following requirements:
the length of the vector c is equal to the product of the lengths of the vectors a and b and the sine of the angle φ between them:
|c|=|a||b|sin φ;
the vector c is orthogonal to each of the vectors a and b;
the vector c is directed so that the triple of vectors abc is right;
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 cross product
Geometric properties of the cross product:
A necessary and sufficient condition for the collinearity of two non-zero vectors is the equality of their vector product to zero.
Cross product module equals area S parallelogram built on vectors reduced to a common origin a and b(see fig. 1).
If a e- unit vector orthogonal to the vectors a and b and chosen so that the triple a,b,e- right, and S- the area of the parallelogram built on them (reduced to a common origin), then the following formula is true for the vector product:
=S e
Fig.2. The volume of the parallelepiped when using the vector and scalar product of vectors; the dotted lines show the projections of the vector c on a × b and the vector a on b × c, the first step is to find the inner products
If a c- any 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 correct formula is:
=Pr e a |c|g
where Pr e a is the projection of the 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 interchanged:
V=a×b c=a b×c
The value of the cross product depends on the sine of the angle between the original vectors, so the cross product can be thought of as the degree of "perpendicularity" of the vectors, just as the dot product can be thought of as the degree of "parallelism". The cross product of two unit vectors is equal to 1 (a unit vector) if the initial vectors are perpendicular, and equal to 0 (zero vector) if the vectors are parallel or antiparallel.
Cross product expression in Cartesian coordinates
If two vectors a and b are defined by their rectangular Cartesian coordinates, or more precisely, they are 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, 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- the symbol of Levi-Civita.
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 not true. 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 desired 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".
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 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.
