The angle between the formula vectors. Scalar product vectors

Sections: Mathematics

Type of classes: Studying a new material.

Educational and educational tasks:

- output the formula for calculating the angle between two vectors;

- continue to form skills and skills to apply vectors to solve problems;

- continue to form interest in mathematics by solving problems;

- educate a conscious attitude towards the learning process, instill a sense of responsibility for the quality of knowledge, to carry out self-control over the process of solving and designing exercises.

Ensuring classes:

- Table "Vectors on the plane and in space";

- Quest cards for an individual survey;

- Quest cards for verification;

- Microcalculators.

The student should know:

- A formula for calculating the angle between vectors.

The student must be able to:

- Apply the knowledge gained to solve analytical, geometric and applied tasks.

Motivation of cognitive activities of students.

The teacher reports that today students will learn to calculate the angle between vectors, apply the knowledge gained to solve problems technical mechanics and physics. Most tasks of the discipline "Technical Mechanics" are solved by a vector method. Thus, when studying the topic "Flat system of converging forces", "finding the resulting two forces" applies the formula for calculating the angle between two vectors.

Travel course.
I. Organizational moment.

II. Check your homework.

a) individual survey on cards.

Card 1.

1. Write the properties of the addition of two vectors.

2. With what value m. Vectors I. Will collinearins?

Card 2.

1. What is called the product of the vector to the number?

2. Are Vectors and ?

Card 3.

1. Formulate the definition of a scalar product of two vectors.

2. With what the length of the length of the vectors and will be equal?

Card 4.

1. Write the formulas for calculating the vector coordinates and vector lengths?

2. Whether collinearies Vectors and ?

b) questions for the front survey:

What actions can be performed above the vectors specified by your coordinates?
What vectors are called collinear?
The condition of the collinearity of two non-zero vectors?
Defining the corner between vectors?
Definition of a scalar product of two non-zero vectors?
Required and sufficient condition for the perpendicularity of two vectors?
What is the physical meaning of the scalar product of two vectors?
Write formulas to calculate the scalar product of two vectors through their coordinates on the plane and in space.
Write formulas to calculate the vector length on the plane and in space.

III. Studying a new material.

a) We derive the formula to calculate the angle between the vectors on the plane and in space. By definition of a scalar product of two non-zero vectors:

cos.

Therefore, if, then

the cosine of the angle between nonzero vectors is equal to the scalar product of these vectors divided into the work of their lengths. If the vectors are specified in the rectangular decartular coordinate system on the plane, then the cosine of the angle between them is calculated by the formula:

\u003d (x 1; y 1); \u003d (x 2; y 2)

cos \u003d

In space: \u003d (x 1; y 1; z 1); \u003d (x 2; y 2; z 2)

cos \u003d

Solve tasks:

Task 1: Find the angle between vectors \u003d (1; -2), \u003d (-3; 1).

Arccos \u003d 135 °

Task 2:In the ABC triangle, find the magnitude of the angle in if

A (0; 5; 0), in (4; 3; -8), with (-1; -3; -6).

cos \u003d =

Task 3: Find the angle between vectors and, if a (1; 6),

In (1; 0), c (-2; 3).

cos \u003d = = –

IV. Application of knowledge when solving typical tasks.

Tasks of an analytical nature.

Determine the angle between vectors and, if A (1; -3; -4),

In (-1; 0; 2), with (2; -4; -6), D (1; 1; 1).

Find a scalar product of vectors if, \u003d 30 °.

Under what values \u200b\u200bof the length of the vectors and will be equal?

Calculate the angle between vectors and

Calculate the area of \u200b\u200bthe parallelogram built in the vectors

and .

Tasks of applied character

Find the equal two forces 1 and 2, if \u003d 5h; \u003d 7h, the angle between them \u003d 60 °.

° + .

Calculate the work that the force produces \u003d (6; 2), if its application point, moving straightly, moves from the position A (-1; 3), to position in (3; 4).

Let - the speed of the material point, the force acting on it. What is the power, developing by force, if \u003d 5h, \u003d 3.5 m / s;

Vi. Summing up the classes.

VII. Homework:

G.N. Yakovlev, Geometry, §22, p. 3, p. 191

No. 5.22, No. 5.27, p. 192.

Vector length, angle between vectors - these concepts are naturally applicable and intuitive when determining the vector as a segment of a certain direction. Below will learn to determine the angle between vectors in three-dimensional space, its cosine and consider the theory on the examples.

Yandex.rtb R-A-339285-1

To consider the concept of angle between vectors, we turn to the graphic illustration: we set two vectors on the plane or in three-dimensional space. We also set an arbitrary point O and postpone the vectors O A → \u003d B → and O B → \u003d B →

Definition 1.

Angle between vectors A → and B → called the angle between the rays about a and about V.

The resulting angle will be referred to as follows: A →, B → ^

Obviously, the angle has the ability to take values \u200b\u200bfrom 0 to π or from 0 to 180 degrees.

a →, B → ^ \u003d 0, when the vectors are coated and A →, B → ^ \u003d π, when the vectors are oppositely directed.

Definition 2.

Vectors are called perpendicularIf the angle between them is 90 degrees or π 2 radians.

If at least one of the vectors is zero, then the angle of A →, B → ^ is not defined.

The cosine of the angle between two vectors, which means the actual angle, usually can be determined or using a scalar product of vectors, or by means of the cosine theorem for a triangle, built on the basis of two vector data.

According to the definition, the scalar product is A →, B → \u003d A → · B → · COS A →, B → ^.

If the predetermined vectors are → and b → nonzero, then we can split the right and left parts of equality to the product of the lengths of these vectors, thus obtaining the formula for finding the cosine of the angle between nonzero vectors:

cOS A →, B → ^ \u003d A →, B → A → B →

This formula is used when there are vectors and their scalar product among the source data.

Example 1.

Source data: Vectors A → and B →. The lengths of them are 3 and 6, respectively, and their scalar product is equal to 9. It is necessary to calculate the cosine of the angle between vectors and find the angle itself.

Decision

The source data is sufficient to apply the formula obtained above, then COS A →, B → ^ \u003d - 9 3 · 6 \u003d - 1 2,

Now we define the angle between the vectors: a →, b → ^ \u003d a r c cos (- 1 2) \u003d 3 π 4

Answer: COS A →, B → ^ \u003d - 1 2, A →, B → ^ \u003d 3 π 4

Tasks are more common, where the vectors are specified by the coordinates in the rectangular coordinate system. For such cases, it is necessary to derive the same formula, but in coordinate form.

The length of the vector is defined as the root square from the sum of the squares of its coordinates, and the scalar product of the vectors is equal to the amount of products of the corresponding coordinates. Then the formula for finding the cosine angle between the vectors on the plane A → \u003d (A x, a y), B → \u003d (b x, b y) looks like this:

cos a →, b → ^ \u003d a x · b x + a y · b y a x 2 + a y 2 · b x 2 + b y 2

A formula for finding a cosine angle between vectors in three-dimensional space A → \u003d (AX, AY, AZ), B → \u003d (BX, BZ) will be viewed: COS A →, B → ^ \u003d AX · BX + AY · BY + AZ · BZAX 2 + AY 2 + AZ 2 · BX 2 + BY 2 + BZ 2

Example 2.

Source data: Vectors A → \u003d (2, 0, - 1), B → \u003d (1, 2, 3) in a rectangular coordinate system. It is necessary to determine the angle between them.

Decision

To solve the problem, we can immediately apply the formula:

cOS A →, B → ^ \u003d 2 · 1 + 0 · 2 + (- 1) · 3 2 2 + 0 2 + (- 1) 2 · 1 2 + 2 2 + 3 2 \u003d - 1 70 ⇒ A →, B → ^ \u003d Arc COS (- 1 70) \u003d - Arc COS 1 70

You can also determine the angle by the formula:

cOS A →, B → ^ \u003d (A →, B →) A → · B →,

but pre-calculate the lengths of the vectors and the scalar product by coordinates: a → \u003d 2 2 + 0 2 + (- 1) 2 \u003d 5 b → \u003d 1 2 + 2 2 + 3 2 \u003d 14 A →, B → ^ \u003d 2 · 1 + 0 · 2 + (- 1) · 3 \u003d - 1 COS A →, B → ^ \u003d A →, B → ^ A → · B → \u003d - 1 5 · 14 \u003d - 1 70 ⇒ A →, B → ^ \u003d - Arc COS 1 70

Answer: A →, B → ^ \u003d - A R C COS 1 70

The tasks are also common when the coordinates of three points are specified in the rectangular coordinate system and it is necessary to determine any angle. And then, in order to determine the angle between vectors with the specified turn coordinates, it is necessary to calculate the coordinates of the vectors as the difference in the corresponding points of the beginning and end of the vector.

Example 3.

Source data: On the plane in the rectangular coordinate system, the points A (2, - 1), B (3, 2), C (7, - 2) are set. It is necessary to determine the cosine of the angle between the vectors A C → and B C →.

Decision

We will find the coordinates of the vectors by the coordinates of the specified points a C → \u003d (7 - 2, - 2 - (- 1)) \u003d (5, - 1) b c → \u003d (7 - 3, 2 - 2) \u003d (4, - 4)

Now we use the formula to determine the cosine of the angle between the vectors on the plane in the coordinates: COS AC →, BC → ^ \u003d (AC →, BC →) AC → BC → \u003d 5 · 4 + (- 1) · (- 4) 5 2 + (- 1) 2 · 4 2 + (- 4) 2 \u003d 24 26 · 32 \u003d 3 13

Answer: COS A C →, B C → ^ \u003d 3 13

The angle between vectors can be determined by the cosine theorem. I will postpone from the point o vectors O A → \u003d A → and O B → \u003d B →, then, according to the cosine theorem in the triangle O A B, the equality will be faithful:

A b 2 \u003d o a 2 + o b 2 - 2 · o a · o b · cos (∠ a o b),

what is equivalent to:

b → - A → 2 \u003d A → + B → 2 · A → · B → · COS (A →, B →) ^

and from here, we bring the cosine formula of the corner:

cOS (A →, B →) ^ \u003d 1 2 · A → 2 + B → 2 - B → - A → 2 A → · B →

To use the resulting formula, we need the lengths of the vectors that are easy to determine their coordinates.

Although this method takes place, still use the formula more often:

cOS (A →, B →) ^ \u003d A →, B → A → B →

If you notice a mistake in the text, please select it and press Ctrl + Enter

The angle between two vectors,:

If the angle between two vectors is sharp, then their scalar product is positive; If the angle between the vectors is stupid, then the scalar product of these vectors is negative. The scalar product of two non-zero vectors is zero, then and only if these vectors are orthogonal.

The task. Find the angle between vectors and

Decision. Cosinus of the articulated corner

16. Calculation of the angle between straight, straight and plane

The angle between the straight and planeCrossing this direct and non-perpendicular to it is the angle between the straight and its projection on this plane.

The definition of the angle between the straight and plane allows you to conclude that the angle between the straight and plane is the angle between two intersecting straight: the most direct and its projection on the plane. Consequently, the angle between the straight and plane is a sharp angle.

The angle between perpendicular direct and the plane is considered to be equal, and the angle between parallel direct and the plane is either not determined at all, or considered to be equal.

§ 69. Calculation of angle between straight.

The task of calculating the angle between two direct in the space is solved as well as on the plane (§ 32). Denote by φ the magnitude of the angle between straight l. 1 I. l. 2, and through ψ - the magnitude of the angle between the guide vectors but and b. These direct.

Then, if

ψ 90 ° (Fig. 206.6), then φ \u003d 180 ° - ψ. Obviously, in both cases, the equality cos φ \u003d | cos ψ |. By formula (1) § 20 we have

hence,

Let the direct are given by their canonical equations

Then the angle φ between direct is determined using the formula

If one of the direct (or both) is specified not by canonic equations, then to calculate the angle, it is necessary to find the coordinates of the guide vectors of these strains, and then use the formula (1).

17. Parallel straight, theorems on parallel direct

Definition. Two straight planes are called parallelIf they do not have common points.

Two straight in three-dimensional space are called parallelIf they lie in the same plane and do not have common points.

The angle between two vectors.

From the definition of a scalar product:

The condition of the orthogonality of two vectors:

The condition of the collinearity of the two vectors:

It follows from the definition 5 -. Indeed, from the definition of the product of the vector to the number, follows. Therefore, based on the rule of equality of vectors, write down, where it follows . But the vector, resulting from the multiplication of the vector by the number, collinearin vector.

Vector projection on vector:

Example 4.. Danities are given ,,,

Find a scalar product.

Decision. We will find by the formula of the scalar product of the vectors specified by their coordinates. Insofar as

, ,

Example 5.Danities are given ,,,

Find a projection.

Decision. Insofar as

, ,

Based on the projection formula, we have

Example 6.Danities are given ,,,

Find the angle between vectors and.

Decision. Note that the vector

, ,

are not collinear, because their coordinates are not proportional:

These vectors are also perpendicular, as their scalar product.

Find

Angle Find out of the formula:

Example 7. Determine at what vectors and Collinear.

Decision. In the case of collinearity, the corresponding vectors coordinates And should be proportional to, that is:

From here.

Example 8.. Determine with what value of the vector and Perpendicular.

Decision. Vector and perpendicular if their scalar product is zero. From this condition we get :. That is, .

Example 9.. To find , if a , , .

Decision. By virtue of the properties of a scalar product, we have:

Example 10.. Find the angle between the vectors and, where and - single vectors and angle between vectors and equal to 120 °.

Decision. We have: , ,

Finally we have: .

5 B. Vector art.

Definition 21..Vector work Vector on the vector is called vector, or, as defined by the following three conditions:

1) The vector module is equal to, where - the angle between vectors and, i.e. .

It follows that the vector product module is numerically equal to the area of \u200b\u200bthe parallelogram, built in the versions and as on the sides.

2) the vector is perpendicular to each of the vectors and (;), i.e. Perpendicular to the plane of the parallelogram built in the versions and.

3) The vector is directed so that if you look from its end, the shortest rotation from the vector to the vector would be counterclockwise (vectors, form the right troika).

How to calculate the angles between vectors?

When studying geometry, many questions arise on the topic of vectors. Special difficulties learning tests, if necessary, find corners between vectors.

Major terms

Before viewing the angles between the vectors, it is necessary to familiarize themselves with the definition of the vector and the concept of the angle between the vectors.

The vector is called a segment having a direction, that is, a segment for which its beginning and end is determined.

The angle between two vectors on the plane having a common principle is called smaller from the corners, the value of which is required to move one of the vectors around the common point, to the position when their directions coincide.

Formula for solution

Realizing that the vector represents and how its angle is determined, you can calculate the angle between the vectors. The solution formula for this is quite simple, and the result of its application will be the cosine value of the angle. According to the definition, it is equal to a private scalar product of vectors and their lengths.

The scalar product of the vectors is considered as a sum of multiplicated by each other corresponding to the coordinates of the factors vectors. Vector length, or its module, is calculated as square root From the sum of the squares of its coordinates.

Having obtained the value of the cosine of the angle, calculate the magnitude of the corner itself using the calculator or using the trigonometric table.

Example

After you figure out how to calculate the angle between vectors, the solution of the corresponding task will become simple and understandable. As an example, it is worth considering an impending task about finding an angle value.

First of all, it will be more convenient to calculate the value of the vectors and their scalar product necessary to solve. Taking advantage of the description presented above, we obtain:

Substitting the obtained values \u200b\u200bin the formula, calculate the value of the cosine of the artificial angle:

This number is not one of the five common cosine values, so to obtain the value of the angle, will have to use the calculator or trigonometric table of the brady. But before getting the angle between vectors, the formula can be simplified to get rid of an excess negative sign:

The final response to preserve accuracy can be left in this form, and you can calculate the angle value in degrees. According to the Bradys table, its value will be approximately 116 degrees and 70 minutes, and the calculator will show the value of 116.57 degrees.

Calculating angle in N-dimensional space

When considering two vectors in three-dimensional space, it is much more difficult to understand what kind of corner is much more difficult if they are not lying in the same plane. To simplify perception, you can draw two intersecting segments that form the smallest corner between them, it will be the desired. Despite the presence of a third coordinate in the vector, the process of how the angles between vectors are calculated, will not change. Calculate the scalar product and modules of vectors, the arcsinus of them is private and will respond to this task.

In geometry, problems are often found and with spaces that have more than three measurements. But for them, the algorithm of finding the answer looks likewise.

The difference between 0 and 180 degrees

One of the common errors when writing an answer to the task calculated to calculate the angle between vectors is to write down that the vectors are parallel, that is, the desired angle turned out to be 0 or 180 degrees. This answer is incorrect.

Having received the value of the angle of 0 degrees according to the solution, the correct answer will be the designation of the vectors as coincided, that is, the vectors will coincide the direction. In the case of obtaining 180 degrees, the vectors will be the oppositely directed.

Specific vectors

Finding the corners between the vectors, one of the special types can be found, in addition to the above-described and oppositely directed.

Several vectors of parallel one plane are called compartment.
Vectors, the same in length and direction, are called equal.
Vectors lying on one straight line, regardless of direction, are referred to as collinear.
If the length of the vector is zero, that is, its beginning and the end coincide, it is called zero, and if a unit, then a single one.

How to find the corner between vectors?

help me please! I know the formula, but it does not work out ((
Vector A (8; 10; 4) Vector B (5; -20; -10)

Alexander Titov

The angle between the vectors specified by its coordinates is located according to the standard algorithm. First you need to find a scalar product of vectors a and b: (a, b) \u003d x1x2 + y1y2 + z1z2. We substitute here the coordinates of these vectors and believe:
(a, b) \u003d 8 * 5 + 10 * (- 20) \u003d 4 * (- 10) \u003d 40 - 200 - 40 \u003d -200.
Next, determine the lengths of each of the vectors. The length or module of the vector is the root square from the sum of the squares of its coordinates:
| A | \u003d root of (x1 ^ 2 + y1 ^ 2 + z1 ^ 2) \u003d root of (8 ^ 2 + 10 ^ 2 + 4 ^ 2) \u003d root of (64 + 100 + 16) \u003d root of 180 \u003d 6 roots from five
| b | \u003d root of (x2 ^ 2 + y2 ^ 2 + z2 ^ 2) \u003d root of (5 ^ 2 + (-20) ^ 2 + (-10) ^ 2) \u003d root of (25 + 400 + 100) \u003d root Of 525 \u003d 5 roots of 21.
Moving these lengths. We get 30 roots out of 105.
Finally, we divide the scalar product of vectors on the work of the lengths of these vectors. We get, -200 / (30 roots out of 105) or
- (4 root out of 105) / 63. This is a cosine of the angle between vectors. And the corner itself is equal to the Arkkosinus from this number
F \u003d Arccos (-4 root out of 105) / 63.
If I considered everything correctly.

How to calculate the sine corner between the vectors coordinates

Mikhail Tkachev

Multiply these vector. Their scalar product is equal to the product of the lengths of these vectors on the cosine of the corner between them.
The angle is unknown to us, but the coordinates are known.
Mathematically write it so.
Let the vector a (x1; y1) and b (x2; y2)
Then

A * B \u003d | A | * | B | * COSA

Cosa \u003d A * B / | A | * | B |

We argue.
A * B-scalar product of vectors equal to the amount of products of the corresponding coordinates of the coordinates of these vectors, i.e. equal to x1 * x2 + y1 * y2

| A | * | B | - Manufacture of vectors, equal to √ ((x1) ^ 2 + (y1) ^ 2) * √ ((x2) ^ 2 + (y2) ^ 2).

So, the cosine of the angle between vectors is equal to:

Cosa \u003d (x1 * x2 + y1 * y2) / √ ((x1) ^ 2 + (y1) ^ 2) * √ ((x2) ^ 2 + (y2) ^ 2)

Knowing the cosine of the angle, we can calculate its sinus. We argue how to do it:

If the cosine of the angle is positive, then this angle lies in 1 or 4 quarters, which means its sinus or positive or negative. But since the angle between vectors is less than or equal to 180 degrees, then its sinus is positive. Similarly, we argue if the cosine is negative.

Sina \u003d √ (1-cos ^ 2a) \u003d √ (1 - ((x1 * x2 + y1 * y2) / √ ((x1) ^ 2 + (y1) ^ 2) * √ ((x2) ^ 2 + ( y2) ^ 2)) ^ 2)

Like this)))) good luck to figure out)))

Dmitry Levishev

The fact that directly sine can not be is not true.
In addition to formula:
(A, B) \u003d | A | * | B | * COS A
There is also such:
|| \u003d | A | * | B | * SIN A
That is, instead of a scalar product, you can take the vector art module.

Knowledge and understanding of mathematical terms will help in solving many tasks as a course of algebra and geometry. An equally important role is given to formulas that displays the relationship between mathematical characteristics.

The angle between vectors - explanation of terminology

In order to formulate the definition of the angle between vectors, it is necessary to find out what implies the term "vector". This concept characterizes a straight line that has start, length and direction. If you are depicted 2 directed segments that originate in the same point, therefore they form an angle.

So The term "the angle between vectors" determines the degree of the smallest angle to which one directional segment should be turned (relative to the starting point) so that it takes the position / direction of the second directional portion. This statement applies to the vector vectors from one point.

The degree of the corner between the two directed areas of the straight, originated at one point is concluded in the segment from 0 º up to 180. º. Denotes this amount As ∠ (ā, ū) - the angle between the directed segments ā and ū.

Calculation of the corner between vectors

The calculation of the degree measure of the angle formed by a pair of directed parts of the line is made using the following formula:

cosφ \u003d (ō, ā) / | ō | · ā |, ⇒ φ \u003d arccos (cosφ).

∠φ - the desired angle between the specified vectors ō and ā,

(ō, ā) - the work of the regiments of the directed parts of the line,

| ō | · | ā | - The product of the lengths of the given directed segments.

Determination of a scalar product of directed areas

How to use this formula and determine the value of the numerator and denominator of the presented relationship?

Depending on the coordinate system (decartian or three-dimensional space), in which the specified vectors are located, each directional segment has the following parameters:

ō = { o. x o. y), ā \u003d ( a X., a. y) or

ō = { o. x o. Y. O. z), ā \u003d ( a X., a. Y. , A. z).

Consequently, to find the value of the numerator - the scalalar of the directed segments - such actions should be made:

(ō,ā) = ō * ā = o. x * a X.+ o. Y. * A. Y, if the vector under consideration lie on the plane

(ō,ā) = ō * ā = o. x * a X.+ o. Y. * A. Y +. o. z * a. z, If the directions are directly located in space.

Determination of vectors

The length of the directional segment is calculated using expressions:

|ō| = √ o. x 2 +. o. Y 2 or | ō | \u003d √ o. x 2 +. o. Y 2 +. o. z 2.

| ā | \u003d √ a x 2 + a. Y 2 or | ā | \u003d √ a. x 2 +. a. Y 2 +. a. z 2.

So In the general case of n-dimensional measurement, the expression to determine the degree of the angle between the directed segments ō \u003d ( o. x o. Y. ... O. n) and ā \u003d ( a X., a. Y. ... A. n) looks like this:

φ \u003d arccos (cosφ) \u003d ArcCOS (( o. x * a X.+ o. Y. * A. y + ... + o. N * a. N) / (√ o. x 2 +. o. Y 2 + ... + o. N 2 * √ a. x 2 +. a. Y 2 + ... + a. N 2)).

An example of calculating the angle between directional segments

According to the conditions, the vectors ī \u003d (3; 4; 0) and ū \u003d (4; 4; 2) are given. What is the degree of a measure of an angle formed by these segments?

Determine the scalar of vectors ī and ū. For this:

i * U \u003d 3 * 4 + 4 * 4 + 0 * 2 \u003d 28

After calculating the length of the segments:

| ī | \u003d √9 + 16 + 0 \u003d √25 \u003d 5,

| ū ū | \u003d √16 + 16 + 4 \u003d √36 \u003d 6.

cOS (ī, ū) \u003d 28/5 * 6 \u003d 28/30 \u003d 14/15 \u003d 0.9 (3).

Taking advantage of the table of cosine (bradys) values, determine the magnitude of the original angle:

cOS (ī, ū) \u003d 0.9 (3) ⇒ ∠ (ī, ū) \u003d 21 ° 6 '.

Scalar product of vectors (hereinafter referred to as the SP text). Dear friends! The composition of the Mathematics exam includes a group of tasks for the solution of vectors. Some tasks we have already considered. You can see them in the category "Vectors". In general, the theory of vectors is simple, the main thing is to study it. Calculations and actions with vectors in school course Mathematics simple, formulas are not complex. Look at. In this article we will analyze the tasks of the vectors (included in the USE). Now "immersion" to the theory:

C. to find the coordinates of the vector, you need to subtract from the coordinatesrelevant coordinates of its start

And further:

* The length of the vector (module) is defined as follows:

These formulas must be remembered !!!

Let's show the angle between the vectors:

It is clear that it may vary in the range from 0 to 180 0 (or in radians from 0 to PI).

We can make some conclusions about the sign of the scalar product. The lengths of the vectors are positive, it is obvious. So the scalar product mark depends on the cosine value between the vectors.

Cases are possible:

1. If the angle between the vectors is acute (from 0 0 to 90 0), then the cosine of the angle will have a positive value.

2. If the angle between the vectors is stupid (from 90 0 to 180 0), then the cosine of the angle will have a negative value.

* At zero degrees, that is, when the vectors have the same direction, the cosine is equal to one and, accordingly, the result will be positive.

At 180 o, i.e., when the vectors have opposite directions, the cosine is minus one, And accordingly, the result will be negative.

Now an important point!

At 90 o, i.e., when the vectors are perpendicular to each other, the cosine is zero, and therefore the joint venture is zero. This fact (consequence, conclusion) is used when solving many tasks, where we are talking about mutual location vectors, including in the tasks of open Bank Tasks in mathematics.

We formulate the assertion: the scalar product is zero if and only if these vectors lie on perpendicular direct.

So, the formula of the joint ventures:

If the coordinates of the vectors or coordinates of their points started and ends, you can always find the angle between vectors:

Consider the tasks:

27724 Find a scalar product of vectors A and b.

We can find a scalar product of vectors for one of two formulas:

The angle between vectors is unknown, but we can easily find the coordinates of the vectors and then use the first formula. Since the beginning of both vectors coincide with the origin of the coordinates, the coordinates of these vectors are equal to the coordinates of their ends, that is