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 delved into 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, we 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 required 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

The formula is really simple: we write the coordinate vectors in the top line of the determinant, we “pack” the coordinates of the vectors into 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.

First, let's remember what a vector product is.

Remark 1

vector art for $\vec(a)$ and $\vec(b)$ is $\vec(c)$, which is some third vector $\vec(c)= ||$, and this vector has special properties:

• The scalar of the resulting vector is the product of $|\vec(a)|$ and $|\vec(b)|$ times the sine of the angle $\vec(c)= ||= |\vec(a)| \cdot |\vec(b)|\cdot \sin α \left(1\right)$;
• All $\vec(a), \vec(b)$ and $\vec(c)$ form a right triple;
• The resulting vector is orthogonal to $\vec(a)$ and $\vec(b)$.

If there are some coordinates for vectors ($\vec(a)=\(x_1; y_1; z_1\)$ and $\vec(b)= \(x_2; y_2; z_2\)$), then their vector product in the Cartesian coordinate system can be determined by the formula:

$ = \(y_1 \cdot z_2 - y_2 \cdot z_1; z_1 \cdot x_2 - z_2 \cdot x_1; x_2 \cdot y_2 - x_2 \cdot y_1\)$

The easiest way to remember this formula is to write it in the form of a determinant:

$ = \begin(array) (|ccc|) i & j & k \\ x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ \end(array)$.

This formula is quite convenient to use, but to understand how to use it, you first need to familiarize yourself with the topic of matrices and their determinants.

Parallelogram area, whose sides are defined by two vectors $\vec(a)$ and $vec(b)$ is equal to to the scalar of the cross product of the given two vectors.

This ratio is quite easy to derive.

Recall the formula for finding the area of ​​an ordinary parallelogram, which can be characterized by its segments $a$ and $b$:

$S = a \cdot b \cdot \sin α$

In this case, the lengths of the sides are equal to the scalar values ​​of the vectors $\vec(a)$ and $\vec(b)$, which is quite suitable for us, that is, the scalar of the vector product of these vectors will be the area of ​​the figure under consideration.

Example 1

Given vectors $\vec(c)$ with coordinates $\(5;3; 7\)$ and a vector $\vec(g)$ with coordinates $\(3; 7;10 \)$ in Cartesian coordinates. Find the area of ​​the parallelogram formed by $\vec(c)$ and $\vec(g)$.

Solution:

Find the vector product for these vectors:

$ = \begin(array) (|ccc|) i & j & k \\ 5 & 3 & 7 \\ 3 & 7 & 10 \\ \end(array)= i \cdot \begin(array) (|cc |) 3 & 7 \\ 7 & 10 \\ \end(array) - j \cdot \begin(array) (|cc|) 5 & 7 \\ 3 & 10 \\ \end(array) + k \cdot \begin(array) (|cc|) 5 & 3 \\ 3 & 7 \\ \end(array) = i \cdot (3 \cdot 10 - 49) - j \cdot (50 -21) + k \cdot (35-9) = -19i -29j + 26k=\(- 19; 29; 26\)$.

Now let's find the modular value for the resulting directional segment, it is the value of the area of ​​the constructed parallelogram:

$S= \sqrt(|19|^2 + |29|^2 + |26|^2) = \sqrt(1878) ≈ 43.34$.

This line of reasoning is valid not only for finding the area in a 3-dimensional space, but also for a two-dimensional one. Check out the next question on this topic.

Example 2

Calculate the area of ​​the parallelogram if its generating segments are given by vectors $\vec(m)$ with coordinates $\(2; 3\)$ and $\vec(d)$ with coordinates $\(-5; 6\)$.

Solution:

This problem is a particular example of problem 1, solved above, but both vectors lie in the same plane, which means that the third coordinate, $z$, can be taken as zero.

To sum up the above, the area of ​​the parallelogram will be:

$S = \begin(array) (||cc||) 2 & 3\\ -5 & 6 \\ \end(array) = \sqrt(12 + 15) =3 \sqrt3$.

Example 3

Given vectors $\vec(a) = 3i – j + k; \vec(b)=5i$. Find the area of ​​the parallelogram they form.

$[ \vec(a) \times \vec(b)] = (3i - j + k) \times 5i = 15 - 5 + $

Let's simplify according to the given table for unit vectors:

Figure 1. Decomposition of a vector in terms of a basis. Author24 - online exchange of student papers

$[ \vec(a) \times \vec(b)] = 5 k + 5 j$.

Calculation time:

$S = \sqrt(|-5|^2 + |5|^2) = 5\sqrt(2)$.

The previous problems were about vectors whose coordinates are given in the Cartesian coordinate system, but consider also the case if the angle between the basis vectors differs from $90°$:

Example 4

The vector $\vec(d) = 2a + 3b$, $\vec(f)= a – 4b$, the lengths of $\vec(a)$ and $\vec(b)$ are equal to each other and equal to one, and the angle between $\vec(a)$ and $\vec(b)$ is 45°.

Solution:

Let's calculate the vector product $\vec(d) \times \vec(f)$:

$[\vec(d) \times \vec(f) ]= (2a + 3b) \times (a - 4b) = 2 - 8 + 3 - 12 $.

For vector products, according to their properties, the following is true: $$ and $$ are equal to zero, $ = - $.

Let's use this to simplify:

$[\vec(d) \times \vec(f) ]= -8 + 3 = -8 - 3 = -11$.

Now let's use the formula $(1)$ :

$[\vec(d) \times \vec(f) ] = |-11 | = 11 \cdot |a| \cdot |b| \cdot \sin α = 11 \cdot 1 \cdot 1 \cdot \frac12=5.5$.

The area of ​​a parallelogram built on vectors is equal to the product of the lengths of these vectors and the angle of the angle that lies between them.

It is good when the lengths of these same vectors are given according to the conditions. However, it also happens that it is possible to apply the formula for the area of ​​a parallelogram built on vectors only after calculations on coordinates.
If you are lucky, and the lengths of the vectors are given according to the conditions, then you just need to apply the formula, which we have already analyzed in detail in the article. The area will be equal to the product of the modules and the sine of the angle between them:

Consider an example of calculating the area of ​​a parallelogram built on vectors.

A task: The parallelogram is built on the vectors and . Find the area if , and the angle between them is 30°.
Let's express the vectors in terms of their values:

Perhaps you have a question - where did the zeros come from? It is worth remembering that we are working with vectors, and for them . also note that if we get an expression as a result, then it will be converted to. Now let's do the final calculations:

Let's return to the problem when the lengths of the vectors are not specified in the conditions. If your parallelogram lies in the Cartesian coordinate system, then you need to do the following.

Calculation of the lengths of the sides of a figure given by coordinates

To begin with, we find the coordinates of the vectors and subtract the corresponding start coordinates from the end coordinates. Let's assume the coordinates of the vector a (x1;y1;z1), and the vector b (x3;y3;z3).
Now we find the length of each vector. To do this, each coordinate must be squared, then add the results and extract the root from a finite number. According to our vectors, the following calculations will be made:


Now we need to find the dot product of our vectors. To do this, their respective coordinates are multiplied and added.

Given the lengths of the vectors and their scalar product, we can find the cosine of the angle lying between them .
Now we can find the sine of the same angle:
Now we have all the necessary quantities, and we can easily find the area of ​​a parallelogram built on vectors using the already known formula.