In this article, we will dwell on the concept of the cross product of two vectors. We will give the necessary definitions, write down a formula for finding the coordinates of a vector product, list and substantiate its properties. After that, we will dwell on the geometric meaning of the vector product of two vectors and consider solutions to various typical examples.
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Definition of a vector product.
Before defining a vector product, let's figure out the orientation of an ordered triple of vectors in three-dimensional space.
Set aside vectors from one point. Depending on the direction of the vector, the triplet can be right or left. Let's look from the end of the vector at how the shortest rotation from vector to occurs. If the shortest rotation occurs counterclockwise, then the triplet of vectors is called right, otherwise - left.
Now we take two non-collinear vectors and. Let us set aside vectors and from point A. Let us construct some vector perpendicular to both and and. Obviously, when constructing a vector, we can do two things, giving it either one direction or the opposite (see illustration).
Depending on the direction of the vector, the ordered triplet of vectors can be right or left.
So we come close to the definition of a vector product. It is given for two vectors given in a rectangular coordinate system of three-dimensional space.
Definition.
The vector product of two vectors and, given in a rectangular coordinate system of three-dimensional space, is called a vector such that
The vector product of vectors and is denoted as.
Vector product coordinates.
Now let's give the second definition of a vector product, which allows you to find its coordinates by the coordinates of the given vectors and.
Definition.
In a rectangular coordinate system of three-dimensional space cross product of two vectors 
and 
is a vector, where are coordinate vectors.
This definition gives us the cross product in coordinate form.
It is convenient to represent the vector product in the form of a determinant of a square matrix of the third order, the first row of which is the unit vectors, the second row contains the coordinates of the vector, and the third contains the coordinates of the vector in a given rectangular coordinate system: 
If we expand this determinant by the elements of the first line, then we get equality from the definition of a vector product in coordinates (if necessary, refer to the article): 
It should be noted that the coordinate form of the cross product is fully consistent with the definition given in the first paragraph of this article. Moreover, these two definitions of a cross product are equivalent. You can see the proof of this fact in the book listed at the end of the article.
Vector product properties.
Since the cross product in coordinates can be represented in the form of a matrix determinant, the following can be easily justified on the basis of vector product properties:
As an example, let us prove the property of anti-commutativity of a vector product.
By definition 
and 
... We know that the value of the determinant of the matrix is \u200b\u200breversed if two rows are swapped, therefore, 
, which proves the property of anti-commutativity of the vector product.
Vector product - examples and solutions.
There are basically three types of tasks.
In problems of the first type, the lengths of two vectors and the angle between them are given, and it is required to find the length of the vector product. In this case, the formula is used 
.
Example.
Find the length of the vector product of vectors and, if known 
.
Decision.
We know from the definition that the length of the vector product of vectors and is equal to the product of the lengths of the vectors and the sine of the angle between them, therefore, 
.
Answer:
.
Problems of the second type are related to the coordinates of vectors, in them the cross product, its length, or something else is sought through the coordinates of given vectors and .
There are many different options possible. For example, not the coordinates of the vectors and can be given, but their expansions in coordinate vectors of the form 
and, or vectors and can be specified by the coordinates of their start and end points.
Let's consider typical examples.
Example.
Two vectors are given in a rectangular coordinate system 
... Find their cross product.
Decision.
According to the second definition, the cross product of two vectors in coordinates is written as: 
We would arrive at the same result if the cross product were written through the determinant 
Answer:
.
Example.
Find the length of the vector product of vectors and, where are the unit vectors of a rectangular Cartesian coordinate system.
Decision.
First, we find the coordinates of the vector product 
in a given rectangular coordinate system.
Since vectors and have coordinates and, accordingly (if necessary, see the article coordinates of a vector in a rectangular coordinate system), then by the second definition of a cross product we have 
That is, the cross product has coordinates in a given coordinate system.
We find the length of the vector product as the square root of the sum of the squares of its coordinates (we obtained this formula for the length of a vector in the section on finding the length of a vector):
Answer:
.
Example.
The coordinates of three points are given in a rectangular Cartesian coordinate system. Find some vector perpendicular and at the same time.
Decision.
Vectors and have coordinates and, respectively (see the article on finding vector coordinates through point coordinates). If we find the vector product of vectors and, then by definition it is a vector perpendicular to both k and k, that is, it is a solution to our problem. Let's find him 
Answer:
- one of the perpendicular vectors.
In problems of the third type, the skill of using the properties of the vector product of vectors is tested. After applying the properties, the corresponding formulas are applied.
Example.
The vectors and are perpendicular and their lengths are 3 and 4, respectively. Find the length of the cross product 
.
Decision.
By the property of distributivity of a vector product, we can write 
Due to the combination property, we take out the numerical coefficients outside the sign of the vector products in the last expression: 
The vector products and are equal to zero, since 
and 
, then.
Since the cross product is anticommutative, then.
So, using the properties of the cross product, we came to the equality 
.
By condition the vectors and are perpendicular, that is, the angle between them is equal. That is, we have all the data to find the required length 
Answer:
.
The geometric meaning of the vector product.
By definition, the length of the vector product of vectors is 
... And from a high school geometry course, we know that the area of \u200b\u200ba triangle is half the product of the lengths of the two sides of the triangle by the sine of the angle between them. Consequently, the length of the vector product is equal to twice the area of \u200b\u200ba triangle with vectors and sides, if they are set aside from one point. In other words, the length of the vector product of vectors and is equal to the area of \u200b\u200ba parallelogram with sides and and the angle between them equal to. This is the geometric meaning of a vector product.
Before giving the concept of a vector product, let us turn to the question of the orientation of an ordered triple of vectors a →, b →, c → in three-dimensional space.
Let's put aside vectors a →, b →, c → from one point. The orientation of the triple a →, b →, c → can be right or left, depending on the direction of the vector c → itself. From the direction in which the shortest rotation 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 triplet of vectors a →, b →, c → is called rightif clockwise - left.
Next, we take two non-collinear vectors a → and b →. Let us then postpone vectors A B → \u003d a → and A C → \u003d b → from point A. Let's construct a vector A D → \u003d c →, which is simultaneously perpendicular to both A B → and A C →. Thus, when constructing the vector itself A D → \u003d c → we can do two things, giving it either one direction or the opposite (see illustration).
The ordered triple 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 cross 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 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 → \u003d ∠ b → c → \u003d π 2;
- its length is determined by the formula: c → \u003d a → b → sin ∠ a →, b →;
- the triplet of vectors a →, b →, c → has the same orientation as the given coordinate system.
The vector product of vectors a → and b → has the following notation: a → × b →.
Vector product coordinates
Since any vector has certain coordinates in the coordinate system, you can enter a second definition of the cross product, which will allow you to find its coordinates by the given coordinates of the vectors.
Definition 2
In a rectangular coordinate system of three-dimensional space vector product of two vectors a → \u003d (a x; a y; a z) and b → \u003d (b x; b y; b z) called the vector c → \u003d a → × b → \u003d (ay bz - az by) i → + (az bx - ax bz) j → + (ax by - ay bx) 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 vectors of the unit vectors i →, j →, k →, the second row contains the coordinates of the vector a →, and the third contains the coordinates of the vector b → in a given rectangular coordinate system, this determinant of the matrix looks like this: c → \u003d a → × b → \u003d i → j → k → axayazbxbybz
Expanding this determinant over the elements of the first row, we obtain the equality: c → \u003d a → × b → \u003d i → j → k → axayazbxbybz \u003d ayazbybz i → - axazbxbz j → + axaybxby k → \u003d \u003d a → × b → \u003d (ay bz - az by) i → + (az bx - ax bz) j → + (ax by - ay bx) k →
Vector product properties
It is known that the vector product in coordinates is represented as the determinant of the matrix c → \u003d a → × b → \u003d i → j → k → a x a y a z b x b y b z, then on the basis properties of the determinant of the matrix the following vector product properties:
- anticommutativity a → × b → \u003d - b → × a →;
- distributivity a (1) → + a (2) → × b \u003d a (1) → × b → + a (2) → × b → or a → × b (1) → + b (2) → \u003d a → × b (1) → + a → × b (2) →;
- associativity λ a → × b → \u003d λ a → × b → or a → × (λ b →) \u003d λ a → × b →, where λ is an arbitrary real number.
These properties are easy to prove.
For example, we can prove the property of anti-commutativity of a vector product.
Proof of Anticommutativity
By definition, a → × b → \u003d i → j → k → a x a y a z b x b y b z and b → × a → \u003d i → j → k → b x b y b z a x a y a z. And if two rows of the matrix are rearranged, then the value of the determinant of the matrix should change to the opposite, therefore, a → × b → \u003d i → j → k → axayazbxbybz \u003d - i → j → k → bxbybzaxayaz \u003d - b → × a →, which and proves the anti-commutativity 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 → \u003d a → b → sin ∠ a →, b →.
Example 1
Find the length of the vector product of vectors a → and b → if you know a → \u003d 3, b → \u003d 5, ∠ a →, b → \u003d π 4.
Decision
By determining the length of the vector product of vectors a → and b → we solve this problem: a → × b → \u003d a → b → sin ∠ a →, b → \u003d 3 5 sin π 4 \u003d 15 2 2.
Answer: 15 2 2 .
Problems of the second type have a connection with the coordinates of vectors, in them the cross product, its length, etc. are searched through the known coordinates of the given vectors a → \u003d (a x; a y; a z) and b → \u003d (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 → can be specified, but their expansion in coordinate vectors of the form b → \u003d b x i → + b y j → + b z k → and c → \u003d a → × b → \u003d (ay bz - az by) i → + (az bx - ax bz) j → + (ax by - ay bx) k →, or vectors a → and b → can be specified by the coordinates of their start and end points.
Consider the following examples.
Example 2
Two vectors a → \u003d (2; 1; - 3), b → \u003d (0; - 1; 1) are given in a rectangular coordinate system. Find their cross product.
Decision
According to the second definition, we find the vector product of two vectors in given coordinates: a → × b → \u003d (ay bz - az by) i → + (az bx - ax bz) j → + (ax by - ay Bx) k → \u003d \u003d (1 1 - (- 3) (- 1)) i → + ((- 3) 0 - 2 1) j → + (2 (- 1) - 1 0) k → \u003d \u003d - 2 i → - 2 j → - 2 k →.
If we write the cross product through the determinant of the matrix, then the solution of this example looks like this: a → × b → \u003d i → j → k → axayazbxbybz \u003d i → j → k → 2 1 - 3 0 - 1 1 \u003d - 2 i → - 2 j → - 2 k →.
Answer: a → × b → \u003d - 2 i → - 2 j → - 2 k →.
Example 3
Find the length of the vector product of vectors i → - j → and i → + j → + k →, where i →, j →, k → are the unit vectors of a rectangular Cartesian coordinate system.
Decision
First, we find the coordinates of a given vector product i → - j → × i → + j → + k → in a 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. Let us find the length of the vector product using the determinant of the matrix, then we have i → - j → × i → + j → + k → \u003d i → j → k → 1 - 1 0 1 1 1 \u003d - 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 a vector): i → - j → × i → + j → + k → \u003d - 1 2 + - 1 2 + 2 2 \u003d 6.
Answer: i → - j → × i → + j → + k → \u003d 6. ...
Example 4
In a rectangular Cartesian coordinate system, the coordinates of three points A (1, 0, 1), B (0, 2, 3), C (1, 4, 2) are given. Find some vector perpendicular to A B → and A C → at the same time.
Decision
The vectors A B → and A C → have the following coordinates (- 1; 2; 2) and (0; 4; 1), respectively. Having found the vector product of 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 a solution to our problem. Let's find it A B → × A C → \u003d i → j → k → - 1 2 2 0 4 1 \u003d - 6 i → + j → - 4 k →.
Answer: - 6 i → + j → - 4 k →. - 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 get a solution to the given problem.
Example 5
Vectors a → and b → are perpendicular and their lengths are 3 and 4, respectively. Find the length of the vector product 3 a → - b → × a → - 2 b → \u003d 3 a → × a → - 2 b → + - b → × a → - 2 b → \u003d \u003d 3 a → × a → + 3 a → × - 2 b → + - b → × a → + - b → × - 2 b →.
Decision
By the distributivity property of a vector product, we can write 3 a → - b → × a → - 2 b → \u003d 3 a → × a → - 2 b → + - b → × a → - 2 b → \u003d \u003d 3 a → × a → + 3 a → × - 2 b → + - b → × a → + - b → × - 2 b →
By the property of associativity, we move the numerical coefficients outside the sign of the vector products in the last expression: 3 a → × a → + 3 a → × - 2 b → + - b → × a → + - b → × - 2 b → \u003d \u003d 3 a → × a → + 3 (- 2) a → × b → + (- 1) b → × a → + (- 1) (- 2) b → × b → \u003d \u003d 3 a → × a → - 6 a → × b → - b → × a → + 2 b → × b →
The vector products a → × a → and b → × b → are 0 because a → × a → \u003d a → a → sin 0 \u003d 0 and b → × b → \u003d b → b → sin 0 \u003d 0, then 3 a → × a → - 6 a → × b → - b → × a → + 2 b → × b → \u003d - 6 a → × b → - b → × a →. ...
The anticommutativity of the vector product implies - 6 a → × b → - b → × a → \u003d - 6 a → × b → - (- 1) a → × b → \u003d - 5 a → × b →. ...
Using the properties of the vector product, we obtain the equality 3 a → - b → × a → - 2 b → \u003d \u003d - 5 a → × b →.
By condition, the vectors a → and b → are perpendicular, that is, the angle between them is π 2. Now it remains only to substitute the found values \u200b\u200binto the corresponding formulas: 3 a → - b → × a → - 2 b → \u003d - 5 a → × b → \u003d \u003d 5 a → × b → \u003d 5 a → b → · sin (a →, b →) \u003d 5 · 3 · 4 · sin π 2 \u003d 60.
Answer: 3 a → - b → × a → - 2 b → \u003d 60.
The length of the vector product of vectors by ordering is equal to a → × b → \u003d a → b → sin ∠ a →, b →. Since it is already known (from the school course) that the area of \u200b\u200ba triangle is half the product of the lengths of its two sides multiplied by the sine of the angle between these sides. Consequently, the length of the vector product is equal to the area of \u200b\u200ba parallelogram - a doubled triangle, namely the product of the sides in the form of vectors a → and b →, deferred 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 a 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
By the moment of force F → applied to point B, relative to point A, we mean the following vector product A B → × F →.
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In this lesson, we'll look at two more vector operations: vector product of vectors and mixed product of vectors (immediately link, who needs it exactly)... It's okay, it sometimes happens that for complete happiness, besides dot product of vectors, it takes more and more. Such is the vector addiction. One might 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 not enough firewood at all, except that there is enough for Buratino. In fact, the material is very common and simple - hardly more complicated than the same scalar product, there will even be fewer typical tasks. The main thing in analytic geometry, as many will be convinced or have already been convinced, is NOT TO BE MISTAKE IN THE CALCULATIONS. Repeat as a spell, and you will be happy \u003d)
If vectors sparkle somewhere far away, like lightning on the horizon, it doesn't matter, start with the lesson Vectors for dummiesto recover or regain basic knowledge of 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 works
How to please you right away? When I was little, I knew how to juggle with two or even three balls. Dexterously it turned out. Now you won't have to juggle at all, since we will consider only spatial vectors, and plane 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, in the same way as in the dot product, involves two vectors... Let these be imperishable letters.
The action itself denoted in the following way: . There are other options, but I'm used to denoting the vector product of vectors that way, in square brackets with a cross.
And immediately question: if in dot product of vectors two vectors are involved, and here, too, two vectors are multiplied, then what is the difference? The obvious difference is, first of all, in the RESULT:
The result of the dot product of vectors is NUMBER:
The vector product of vectors results in a VECTOR:, that is, we multiply the vectors and get a vector again. Closed club. Actually, hence the name of the operation. In different educational literature, the designations can also vary, I will use the letter.
Definition of a cross product
First there will be a definition with a picture, then comments.
Definition: By vector product non-collinear vectors, taken in this order, called VECTOR, length which numerically equal to the area of \u200b\u200bthe parallelogrambuilt 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 are many interesting things!
So, the following essential points can be highlighted:
1) The original vectors, denoted 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 "bh", and not "bh" to "a". The result of vector multiplication is the VECTOR, which is marked in blue. If the vectors are multiplied in reverse order, we get a vector equal in length and opposite in direction (crimson 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 a parallelogram built on 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 \u200b\u200bthe parallelogram.
We recall one of the geometric formulas: the area of \u200b\u200ba parallelogram is equal to the product of adjacent sides by 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 in the formula we are talking about the LENGTH of the vector, and not about the vector itself. What's the practical point? And the meaning is that in problems of analytical geometry, the area of \u200b\u200ba parallelogram is often found through the concept of a vector product:
Let's get the second important formula. The diagonal of the parallelogram (red dotted line) divides it into two equal triangles. Therefore, the area of \u200b\u200ba triangle built on vectors (red shading) can be found by the formula:
4) An equally important fact is that the vector is orthogonal to 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 the lesson about transition to a new basis I spoke in sufficient 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 pinky press to the palm. As a result thumb - the cross product will look up. This is the right-oriented basis (in the figure it is it). Now change the vectors ( index and middle fingers) in some places, as a result, the thumb will unfold, and the cross product will already look down. This is also a right-oriented basis. Perhaps you have a question: what basis has a left orientation? "Assign" to the same fingers left hand vectors, and get the left basis and left orientation of the 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 directions. And this concept should not be considered as something far-fetched or abstract - so, 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 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 in orientation are terrible \u003d)
Cross product of collinear vectors
The definition has been analyzed in detail, it remains to find out what happens when the vectors are collinear. If the vectors are collinear, then they can be located on one straight line and our parallelogram also "folds" into one straight line. The area of \u200b\u200bsuch, 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 
... 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 zero.
A special case is the vector product of a vector by 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, you may need trigonometric tableto find sine values \u200b\u200bfrom it.
Well, let's kindle a fire:
Example 1
a) Find the length of the vector product of vectors if 
b) Find the area of \u200b\u200ba parallelogram built on vectors if 
Decision: 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) By condition, it is required to find length vector (vector product). According to the corresponding formula:
Answer:
Since the question was about the length, then in the answer we indicate the dimension - units.
b) By condition, it is required to find area a parallelogram built on vectors. The area of \u200b\u200bthis parallelogram is numerically equal to the length of the vector product:
Answer:
Please note that the answer about the vector product is out of the question 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 teachers, and the task with good chances will return for revision. Although this is not a particularly strained nagging - if the answer is incorrect, then it seems that the person does not understand simple things and / or does not understand the essence of the task. This moment must 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 into the solution, but in order to shorten the recording, I did not. I hope everyone understands that and is a designation of the same thing.
Popular example for a do-it-yourself solution:
Example 2
Find the area of \u200b\u200ba triangle built on vectors if 
The formula for finding the area of \u200b\u200ba triangle through the cross 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 torture you.
To solve other problems we need:
Vector product properties
We have already considered some of the properties of the cross product, however, I will include them in this list.
For arbitrary vectors and an arbitrary number, the following properties are valid:
1) In other sources of information, this item is usually not highlighted in 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 laws of a vector product. Constants are seamlessly removed outside the vector product. Indeed, what should they do there?
4) - distribution or distributive laws of a vector product. There are no problems with brackets expansion 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 cross product. Let's write our thumbnail: 
(1) According to the associative laws, we move the constants outside the division of the vector product.
(2) We move 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 put some wood on the fire:
Example 4
Calculate the area of \u200b\u200ba triangle built on vectors if 
Decision: The area of \u200b\u200bthe triangle is found by the formula 
... The catch is that the vectors "tse" and "de" are themselves represented as sums of vectors. The algorithm here is standard and is somewhat reminiscent of examples 3 and 4 of the lesson Dot product of vectors... For clarity, let's split the solution into three stages:
1) At the first step, we express the vector product in terms of the vector product, in fact, express the vector in terms of the vector... Not a word about lengths yet!
(1) Substitute vector expressions.
(2) Using distributive laws, we expand the brackets according to the rule of multiplication of polynomials.
(3) Using associative laws, we move all constants outside the vector products. With a little experience, actions 2 and 3 can be performed simultaneously.
(4) The first and last terms are equal to zero (zero vector) due to a pleasant property. In the second term, we use the anti-commutativity property of the vector product:
(5) We present similar terms.
As a result, the vector was expressed in terms of the 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 resembles Example 3: 
3) Find the area of \u200b\u200bthe required triangle: 
Stages 2-3 decisions could be completed in one line.
Answer:
The considered problem is quite common in test papers, here is an example for an independent solution:
Example 5
Find if
A short solution and answer at the end of the tutorial. Let's see how careful you were when studying the previous examples ;-)
Vector product of vectors in coordinates
given in an orthonormal basis, expressed by the formula:
The formula is really simple: in the top line of the determinant we write the 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 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 be swapped: 
Example 10
Check if the following space vectors are collinear:
and)
b) 
Decision: The check is based on one of the statements in this lesson: if vectors are collinear, then their cross product is equal to zero (zero vector): 
.
a) Find the cross product: 
So the vectors are not collinear.
b) Find the cross 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 not many tasks where a 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:
So they lined up with a little train and are waiting, they can’t wait to be figured out.
First, again the definition and the picture:
Definition: Mixed work non-coplanar vectors, taken in this orderis called parallelepiped volume, built on the given vectors, supplied with a “+” sign if the basis is right, and a “-” sign if the basis is left.
Let's complete the drawing. Lines invisible to us are drawn with a dotted line: 
Diving into the definition:
2) Vectors are 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 an obvious fact: the mixed product of vectors is a NUMBER:. In educational literature, the design may be somewhat different, I am used to denote a mixed work through, and the result of calculations by the letter "pe".
By definition mixed product is the volume of a parallelepiped, built on vectors (the figure is drawn with red vectors and black lines). That is, the number is equal to the volume of this parallelepiped.
Note : the drawing is schematic.
4) Let's not bother anew with the concept of base and space orientation. The meaning of the final part is that a minus sign can be added to the volume. In simple words, a mixed work can be negative:.
The formula for calculating the volume of a parallelepiped built on vectors follows directly from the definition.
7.1. Definition of a cross product
Three non-coplanar vectors a, b and c, taken in the indicated order, form a right triplet if from the end of the third vector c the shortest rotation from the first vector a to the second vector b is seen counterclockwise, and the left, if clockwise (see Fig. . sixteen).
The vector product of a vector a by a vector b is a vector c, which:
1. Perpendicular to vectors a and b, that is, c ^ a and c ^ b;
2. Has a length numerically equal to the area of \u200b\u200ba parallelogram built on vectors a andbas on the sides (see fig. 17), ie.
3. Vectors a, b and c form a right-hand triplet.
The cross product is denoted a x b or [a, b]. The definition of a vector product directly implies the following relations between the vectors i, 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, thati хj \u003d k.
1) k ^ i, k ^ j;
2) | k | \u003d 1, but | i x j| \u003d | i | | J | sin (90 °) \u003d 1;
3) vectors i, j and k form a right-hand triplet (see Fig. 16).
7.2. Vector product properties
1. When the factors are rearranged, the vector product changes sign; a хb \u003d (b хa) (see Fig. 19).
Vectors a хb and b ha are collinear, have the same moduli (the parallelogram area remains unchanged), but opposite directions (triplets a, b, a хb and a, b, b x a of opposite orientation). That is a xb = -(b xa).
2. The vector product possesses the combinatory property with respect to the scalar factor, ie, l (а хb) \u003d (l а) х b \u003d а х (l b).
Let l\u003e 0. Vector l (a xb) is perpendicular to vectors a and b. Vector ( la) x bis also perpendicular to vectors a and b(vectors a, land lie in the same plane). Hence the vectors l(a xb) and ( la) x bcollinear. Obviously, their directions coincide. Have the same length:
therefore l(a хb) \u003d la xb. It is proved similarly for l<0.
3. Two nonzero vectors a and bcollinear if and only if their cross product is equal to the zero vector, i.e., a || b<=>a xb \u003d 0.
In particular, i * i \u003d j * j \u003d k * k \u003d 0.
4. The vector product has the distribution property:
(a + b) xc \u003d a xc + b xc.
We will accept it without proof.
7.3. Cross product expression in terms of coordinates
We will use the cross product table of vectors i, jand 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 not, the third vector is taken with a minus sign.
Let two vectors a \u003d a x i + a y j + a z kand b \u003d b x i + b y j + b z k ... Find the cross product of these vectors, multiplying them as polynomials (according to the properties of the cross product):
The resulting formula can be written even shorter:
since the right-hand 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 vector work
Establishing collinear vectors
Finding the area of \u200b\u200ba parallelogram and a triangle
According to the definition of the vector product of vectors andand b | a xb | \u003d | a | * | b | sin g, that is, S pairs \u003d | a x b |. And, therefore, D S \u003d 1/2 | a x b |.
Determination of the moment of force relative to a point
Let a force be applied at point A F \u003d ABlet it go ABOUT- some point in space (see Fig. 20).
It is known from physics that moment of force F relative to point ABOUT vector is called M,which goes through the point ABOUTand:
1) perpendicular to the plane passing through the points O, A, B;
2) numerically equal to the product of force per shoulder
3) forms a right triplet with vectors OA and A B.
Therefore, M \u003d OA x F.
Finding the linear speed of rotation
Speed vpoint M of a rigid body rotating with an angular velocity waround a fixed axis, is determined by the Euler formula v \u003d w хr, where r \u003d ОМ, where О is some fixed point of the axis (see Fig. 21).
A MIXED PRODUCT OF THREE VECTORS AND ITS PROPERTIES
Mixed work three vectors are called a number equal to. Denoted 
... Here the first two vectors are multiplied by vector and then the resulting vector is multiplied scalar by the third vector. Obviously, such a product is a certain number.
Consider the properties of the mixed product.
- Geometric meaning mixed work. The mixed product of 3 vectors, up to a sign, is equal to the volume of a parallelepiped built on these vectors, as on edges, i.e. ...
Thus, and
.
Evidence... Let's set aside vectors from the common origin and build a parallelepiped on them. We denote and note that. By the definition of the dot product
Assuming that and denoting by h the height of the parallelepiped, we find.
Thus, for
If, then and. Consequently, .
Combining both of these cases, we get or.
In particular, it follows from the proof of this property that if the triple of vectors is right, then it is a mixed product, and if it is left, then.
- For any vectors,, the equality
The proof of this property follows from property 1. Indeed, it is easy to show that and. Moreover, the signs "+" and "-" are taken simultaneously, because the angles between vectors and and and are both acute or obtuse.
- Upon permutation of any two factors, the mixed product changes sign.
Indeed, if we consider a mixed work, then, for example, or
- Mixed product if and only if one of the factors is zero or the vectors are coplanar.
Evidence.
Thus, a necessary and sufficient condition for the coplanarity of 3 vectors is the equality to zero of their mixed product. In addition, this implies that three vectors form a basis in space if.
If vectors are given in coordinate form, then it can be shown that their mixed product is found by the formula:
.Thus, the mixed product is equal to the determinant of the third order, in which the first line contains the coordinates of the first vector, the second line contains the coordinates of the second vector, and the third line contains the third vector.
Examples.
ANALYTICAL GEOMETRY IN SPACE
The equation F (x, y, z) \u003d 0 defines in space Oxyz some surface, i.e. locus of points whose coordinates x, y, z satisfy this equation. This equation is called the equation of the surface, and x, y, z - current coordinates.
However, often the surface is not given by an equation, but as a set of points in space that have this or that property. In this case, it is required to find the equation of the surface based on its geometric properties.
PLANE.
NORMAL PLANE VECTOR.
EQUATION OF A PLANE PASSING THROUGH A GIVEN POINT
Consider an arbitrary plane σ in space. Its position is determined by specifying a vector perpendicular to this plane and some fixed point M 0(x 0, y 0, z 0) lying in the plane σ.
A vector perpendicular to the plane σ is called normal vector of this plane. Let the vector have coordinates.
Let us derive the equation of the plane σ passing through this point M 0 and having a normal vector. To do this, take an arbitrary point on the plane σ M (x, y, z) and consider a vector.
For any point MÎ σ is a vector. Therefore, their scalar product is equal to zero. This equality is the condition that the point MÎ σ. It is valid for all points of this plane and is violated as soon as the point M will be outside the plane σ.
If we denote by the radius vector of the point M, Is the radius vector of the point M 0, then the equation can also be written as
This equation is called vector equation of the plane. Let's write it down in coordinate form. Since then
So, we got the equation of the plane passing through this point. Thus, in order to compose the equation of the plane, you need to know the coordinates of the normal vector and the coordinates of some point lying on the plane.
Note that the equation of the plane is an equation of the 1st degree relative to the current coordinates x, y and z.
Examples.
GENERAL EQUATION OF THE PLANE
It can be shown that any equation of the first degree with respect to Cartesian coordinates x, y, z is an equation of some plane. This equation is written as:
Ax + By + Cz + D=0
and called general equation plane, and the coordinates A, B, C here are the coordinates of the normal vector of the plane.
Consider particular cases of the general equation. Let us find out how the plane is located relative to the coordinate system if one or more coefficients of the equation vanish.
A is the length of the line cut by the plane on the axis Ox... Similarly, one can show that b and c - the lengths of the segments cut off by the plane in question on the axes Oy and Oz.
It is convenient to use the plane equation in line segments to construct planes.
