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 
Decision: 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 
Decision: 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 
Decision: 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) 
Decision: The test is based on one of the statements in this lesson: if the vectors are collinear, then their cross product is zero (zero vector): 
.
a) Find the vector product: 
So the vectors are not collinear.
b) Find the vector product: 
Answer: a) not collinear, b)
Here, perhaps, is all the basic information about the vector product of vectors.
This section will not be very large, since there are few problems where the mixed product of vectors is used. In fact, everything will rest on the definition, geometric meaning and a couple of working formulas.
The mixed product of vectors is the product of three vectors:
This is how they lined up like a train and wait, they can’t wait until they are calculated.
First again the definition and picture:
Definition: Mixed product non-coplanar vectors , taken in this order, is called volume of the parallelepiped, built on these vectors, equipped with a "+" sign if the basis is right, and a "-" sign if the basis is left.
Let's do the drawing. Lines invisible to us are drawn by a dotted line: 
Let's dive into the definition:
2) Vectors taken in a certain order, that is, the permutation of vectors in the product, as you might guess, does not go without consequences.
3) Before commenting on the geometric meaning, I will note the obvious fact: the mixed product of vectors is a NUMBER: . In educational literature, the design may be somewhat different, I used to designate a mixed product through, and the result of calculations with the letter "pe".
A-priory the mixed product is the volume of the parallelepiped, built on vectors (the figure is drawn with red vectors and black lines). That is, the number is equal to the volume of the given parallelepiped.
Note : The drawing is schematic.
4) Let's not bother again with the concept of the orientation of the basis and space. The meaning of the final part is that a minus sign can be added to the volume. In simple terms, the mixed product can be negative: .
The formula for calculating the volume of a parallelepiped built on vectors follows directly from the definition.
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 arm
3) forms a right triple with vectors OA and A B .
Therefore, M \u003d OA x F.
Finding the linear speed of rotation
Speed v point M of a rigid body rotating at an angular velocity w around a fixed axis, is determined by the Euler formula v \u003d w x r, where r \u003d OM, where O is some fixed point of the axis (see Fig. 21).
Before giving the concept of a vector product, let us turn to the question of the orientation of the ordered triple of vectors a → , b → , c → in three-dimensional space.
To begin with, let's set aside the vectors a → , b → , c → from one point. The orientation of the triple a → , b → , c → is right or left, depending on the direction of the vector c → . From the direction in which the shortest turn is made from the vector a → to b → from the end of the vector c → , the form of the triple a → , b → , c → will be determined.
If the shortest rotation is counterclockwise, then the triple of vectors a → , b → , c → is called right if clockwise - left.
Next, take two non-collinear vectors a → and b → . Let us then postpone the vectors A B → = a → and A C → = b → from the point A. Let us construct a vector A D → = c → , which is simultaneously perpendicular to both A B → and A C → . Thus, when constructing the vector A D → = c →, we can do two things, giving it either one direction or the opposite (see illustration).
The ordered trio of vectors a → , b → , c → can be, as we found out, right or left depending on the direction of the vector.
From the above, we can introduce the definition of a vector product. This definition is given for two vectors defined in a rectangular coordinate system of three-dimensional space.
Definition 1
The vector product of two vectors a → and b → we will call such a vector given in a rectangular coordinate system of three-dimensional space such that:
- if the vectors a → and b → are collinear, it will be zero;
- it will be perpendicular to both vector a → and vector b → i.e. ∠ a → c → = ∠ b → c → = π 2 ;
- its length is determined by the formula: c → = a → b → sin ∠ a → , b → ;
- the triplet of vectors a → , b → , c → has the same orientation as the given coordinate system.
The cross product of vectors a → and b → has the following notation: a → × b → .
Cross product coordinates
Since any vector has certain coordinates in the coordinate system, it is possible to introduce a second definition of the vector product, which will allow you to find its coordinates from the given coordinates of the vectors.
Definition 2
In a rectangular coordinate system of three-dimensional space vector product of two vectors a → = (a x ; a y ; a z) and b → = (b x ; b y ; b z) call the vector c → = a → × b → = (a y b z - a z b y) i → + (a z b x - a x b z) j → + (a x b y - a y b x) k → , where i → , j → , k → are coordinate vectors.
The vector product can be represented as a determinant of a square matrix of the third order, where the first row is the orta vectors i → , j → , k → , the second row contains the coordinates of the vector a → , and the third is the coordinates of the vector b → in a given rectangular coordinate system, this matrix determinant looks like this: c → = a → × b → = i → j → k → a x a y a z b x b y b z
Expanding this determinant over the elements of the first row, we obtain the equality: c → = a → × b → = i → j → k → a x a y a z b x b y b z = a y a z b y b z i → - a x a z b x b z j → + a x a y b x b y k → = = a → × b → = (a y b z - a z b y) i → + (a z b x - a x b z) j → + (a x b y - a y b x) k →
Cross product properties
It is known that the vector product in coordinates is represented as the determinant of the matrix c → = a → × b → = i → j → k → a x a y a z b x b y b z , then on the base matrix determinant properties the following vector product properties:
- anticommutativity a → × b → = - b → × a → ;
- distributivity a (1) → + a (2) → × b = a (1) → × b → + a (2) → × b → or a → × b (1) → + b (2) → = a → × b (1) → + a → × b (2) → ;
- associativity λ a → × b → = λ a → × b → or a → × (λ b →) = λ a → × b → , where λ is an arbitrary real number.
These properties have not complicated proofs.
For example, we can prove the anticommutativity property of a vector product.
Proof of anticommutativity
By definition, a → × b → = i → j → k → a x a y a z b x b y b z and b → × a → = i → j → k → b x b y b z a x a y a z . And if two rows of the matrix are interchanged, then the value of the determinant of the matrix should change to the opposite, therefore, a → × b → = i → j → k → a x a y a z b x b y b z = - i → j → k → b x b y b z a x a y a z = - b → × a → , which and proves the anticommutativity of the vector product.
Vector Product - Examples and Solutions
In most cases, there are three types of tasks.
In problems of the first type, the lengths of two vectors and the angle between them are usually given, but you need to find the length of the cross product. In this case, use the following formula c → = a → b → sin ∠ a → , b → .
Example 1
Find the length of the cross product of vectors a → and b → if a → = 3 , b → = 5 , ∠ a → , b → = π 4 is known.
Decision
Using the definition of the length of the vector product of vectors a → and b →, we solve this problem: a → × b → = a → b → sin ∠ a → , b → = 3 5 sin π 4 = 15 2 2 .
Answer: 15 2 2 .
Tasks of the second type have a connection with the coordinates of vectors, they contain a vector product, its length, etc. are searched through the known coordinates of the given vectors a → = (a x ; a y ; a z) and b → = (b x ; b y ; b z) .
For this type of task, you can solve a lot of options for tasks. For example, not the coordinates of the vectors a → and b → , but their expansions in coordinate vectors of the form b → = b x i → + b y j → + b z k → and c → = a → × b → = (a y b z - a z b y) i → + (a z b x - a x b z) j → + (a x b y - a y b x) k → , or the vectors a → and b → can be given by the coordinates of their start and end points.
Consider the following examples.
Example 2
Two vectors are set in a rectangular coordinate system a → = (2 ; 1 ; - 3) , b → = (0 ; - 1 ; 1) . Find their vector product.
Decision
According to the second definition, we find the vector product of two vectors in given coordinates: a → × b → = (a y b z - a z b y) i → + (a z b x - a x b z) j → + (a x b y - a y b x) k → = = (1 1 - (- 3) (- 1)) i → + ((- 3) 0 - 2 1) j → + (2 (- 1) - 1 0) k → = = - 2 i → - 2 j → - 2 k → .
If we write the vector product through the matrix determinant, then the solution of this example is as follows: a → × b → = i → j → k → a x a y a z b x b y b z = i → j → k → 2 1 - 3 0 - 1 1 = - 2 i → - 2 j → - 2 k → .
Answer: a → × b → = - 2 i → - 2 j → - 2 k → .
Example 3
Find the length of the cross product of vectors i → - j → and i → + j → + k → , where i → , j → , k → - orts of a rectangular Cartesian coordinate system.
Decision
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.
Decision
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 → .
Decision
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 → .
If you notice a mistake in the text, please highlight it and press Ctrl+Enter
Angle between vectors
In order for us to introduce the concept of a cross product of two vectors, we must first deal with such a concept as the angle between these vectors.
Let us be given two vectors $\overline(α)$ and $\overline(β)$. Let us take some point $O$ in space and set aside the vectors $\overline(α)=\overline(OA)$ and $\overline(β)=\overline(OB)$ from it, then the angle $AOB$ will be called angle between these vectors (Fig. 1).
Notation: $∠(\overline(α),\overline(β))$
The concept of the cross product of vectors and the formula for finding
Definition 1
The vector product of two vectors is a vector perpendicular to both given vectors, and its length will be equal to the product of the lengths of these vectors with the sine of the angle between these vectors, and this vector with two initial ones has the same orientation as the Cartesian coordinate system.
Notation: $\overline(α)х\overline(β)$.
Mathematically it looks like this:
- $|\overline(α)x\overline(β)|=|\overline(α)||\overline(β)|sin∠(\overline(α),\overline(β))$
- $\overline(α)x\overline(β)⊥\overline(α)$, $\overline(α)x\overline(β)⊥\overline(β)$
- $(\overline(α)x\overline(β),\overline(α),\overline(β))$ and $(\overline(i),\overline(j),\overline(k))$ are the same oriented (Fig. 2)
Obviously, the outer product of vectors will equal the zero vector in two cases:
- If the length of one or both vectors is zero.
- If the angle between these vectors is equal to $180^\circ$ or $0^\circ$ (because in this case the sine is equal to zero).
To clearly see how the cross product of vectors is found, consider the following solution examples.
Example 1
Find the length of the vector $\overline(δ)$, which will be the result of the cross product of vectors, with coordinates $\overline(α)=(0,4,0)$ and $\overline(β)=(3,0,0 )$.
Decision.
Let's depict these vectors in the Cartesian coordinate space (Fig. 3):
Figure 3. Vectors in Cartesian coordinate space. Author24 - online exchange of student papers
We see that these vectors lie on the $Ox$ and $Oy$ axes, respectively. Therefore, the angle between them will be equal to $90^\circ$. Let's find the lengths of these vectors:
$|\overline(α)|=\sqrt(0+16+0)=4$
$|\overline(β)|=\sqrt(9+0+0)=3$
Then, by Definition 1, we obtain the module $|\overline(δ)|$
$|\overline(δ)|=|\overline(α)||\overline(β)|sin90^\circ=4\cdot 3\cdot 1=12$
Answer: $12$.
Calculation of the cross product by the coordinates of the vectors
Definition 1 immediately implies a way to find the cross product for two vectors. Since a vector, in addition to a value, also has a direction, it is impossible to find it only using a scalar value. But besides it, there is another way to find the vectors given to us using the coordinates.
Let us be given vectors $\overline(α)$ and $\overline(β)$, which will have coordinates $(α_1,α_2,α_3)$ and $(β_1,β_2,β_3)$, respectively. Then the vector of the cross product (namely, its coordinates) can be found by the following formula:
$\overline(α)x\overline(β)=\begin(vmatrix)\overline(i)&\overline(j)&\overline(k)\\α_1&α_2&α_3\\β_1&β_2&β_3\end(vmatrix)$
Otherwise, expanding the determinant, we obtain the following coordinates
$\overline(α)х\overline(β)=(α_2 β_3-α_3 β_2,α_3 β_1-α_1 β_3,α_1 β_2-α_2 β_1)$
Example 2
Find the vector of the cross product of collinear vectors $\overline(α)$ and $\overline(β)$ with coordinates $(0,3,3)$ and $(-1,2,6)$.
Decision.
Let's use the formula above. Get
$\overline(α)x\overline(β)=\begin(vmatrix)\overline(i)&\overline(j)&\overline(k)\\0&3&3\\-1&2&6\end(vmatrix)=(18 -6)\overline(i)-(0+3)\overline(j)+(0+3)\overline(k)=12\overline(i)-3\overline(j)+3\overline(k )=(12,-3,3)$
Answer: $(12,-3,3)$.
Properties of the cross product of vectors
For arbitrary mixed three vectors $\overline(α)$, $\overline(β)$ and $\overline(γ)$, as well as $r∈R$, the following properties hold:
Example 3
Find the area of a parallelogram whose vertices have coordinates $(3,0,0)$, $(0,0,0)$, $(0,8,0)$ and $(3,8,0)$.
Decision.
First, draw this parallelogram in coordinate space (Fig. 5):
Figure 5. Parallelogram in coordinate space. Author24 - online exchange of student papers
We see that the two sides of this parallelogram are constructed using collinear vectors with coordinates $\overline(α)=(3,0,0)$ and $\overline(β)=(0,8,0)$. Using the fourth property, we get:
$S=|\overline(α)x\overline(β)|$
Find the vector $\overline(α)х\overline(β)$:
$\overline(α)x\overline(β)=\begin(vmatrix)\overline(i)&\overline(j)&\overline(k)\\3&0&0\\0&8&0\end(vmatrix)=0\overline (i)-0\overline(j)+24\overline(k)=(0,0,24)$
Consequently
$S=|\overline(α)x\overline(β)|=\sqrt(0+0+24^2)=24$
Obviously, in the case of a cross 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, the cross product of two vectors is zero if and only if they are collinear. (In case one of them is a zero vector it is necessary to remember that the zero vector is collinear to any vector by definition).
Vector product has distributive property, that is
The expression of the cross product in terms of the coordinates of the vectors.
Let two vectors be given
(how to find the coordinates of a vector by the coordinates of its beginning and end - see the article Dot product of vectors, paragraph Alternative definition of the dot product, or calculating the dot product of two vectors given by their coordinates.)
Why do you need a vector product?
There are many ways to use the cross product, for example, as already written above, by calculating the cross product of two vectors, you can find out if they are collinear.
Or it can be used as a way to calculate the area of a parallelogram built from these vectors. Based on the definition, the length of the resulting vector is the area of this parallelogram.
Also, a huge number of applications exist in electricity and magnetism.Online calculator of vector product.
To find the scalar product of two vectors using this calculator, you need to enter the coordinates of the first vector in the first line in order, and the second vector in the second. The coordinates of vectors can be calculated from their start and end coordinates (see article Dot product of vectors , item An alternative definition of the dot product, or calculating the dot product of two vectors given their coordinates.)
