Angle between vectors in space formula. Dot product of vectors

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

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To consider the concept of an angle between vectors, let’s turn to a graphical illustration: let’s define two vectors a → and b → on a plane or in three-dimensional space, which are non-zero. Let us also set an arbitrary point O and plot the vectors O A → = b → and O B → = b → from it

Definition 1

Angle between vectors a → and b → is the angle between the rays O A and O B.

We will denote the resulting angle as follows: a → , b → ^

Obviously, the angle can take values from 0 to π or from 0 to 180 degrees.

a → , b → ^ = 0 when the vectors are co-directional and a → , b → ^ = π when the vectors are oppositely directed.

Definition 2

The vectors are called perpendicular, if the angle between them is 90 degrees or π 2 radians.

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

The cosine of the angle between two vectors, and hence the angle itself, can usually be determined either using the scalar product of vectors, or using the cosine theorem for a triangle constructed from two given vectors.

According to the definition, the scalar product is a → , b → = a → · b → · cos a → , b → ^ .

If the given vectors a → and b → are non-zero, then we can divide the right and left sides of the equality by the product of the lengths of these vectors, thus obtaining a formula for finding the cosine of the angle between non-zero vectors:

cos a → , b → ^ = a → , b → a → b →

This formula is used when the source data includes the lengths of vectors and their scalar product.

Example 1

Initial data: vectors a → and b →. Their lengths are 3 and 6, respectively, and their scalar product is - 9. It is necessary to calculate the cosine of the angle between the vectors and find the angle itself.

Solution

The initial data is sufficient to apply the formula obtained above, then cos a → , b → ^ = - 9 3 6 = - 1 2 ,

Now let’s determine the angle between the vectors: a → , b → ^ = a r c cos (- 1 2) = 3 π 4

Answer: cos a → , b → ^ = - 1 2 , a → , b → ^ = 3 π 4

More often there are problems where vectors are specified by coordinates in a rectangular coordinate system. For such cases, it is necessary to derive the same formula, but in coordinate form.

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

cos a → , b → ^ = a x b x + a y b y a x 2 + a y 2 b x 2 + b y 2

And the formula for finding the cosine of the angle between vectors in three-dimensional space a → = (a x , a y , a z) , b → = (b x , b y , b z) will look like: cos a → , b → ^ = a x · b x + a y · b y + a z b z a x 2 + a y 2 + a z 2 b x 2 + b y 2 + b z 2

Example 2

Initial data: vectors a → = (2, 0, - 1), b → = (1, 2, 3) in a rectangular coordinate system. It is necessary to determine the angle between them.

Solution

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

cos a → , b → ^ = 2 1 + 0 2 + (- 1) 3 2 2 + 0 2 + (- 1) 2 1 2 + 2 2 + 3 2 = - 1 70 ⇒ a → , b → ^ = a r c cos (- 1 70) = - a r c cos 1 70

You can also determine the angle using the formula:

cos a → , b → ^ = (a → , b →) a → b → ,

but first calculate the lengths of the vectors and the scalar product by coordinates: a → = 2 2 + 0 2 + (- 1) 2 = 5 b → = 1 2 + 2 2 + 3 2 = 14 a → , b → ^ = 2 1 + 0 2 + (- 1) 3 = - 1 cos a → , b → ^ = a → , b → ^ a → b → = - 1 5 14 = - 1 70 ⇒ a → , b → ^ = - a r c cos 1 70

Answer: a → , b → ^ = - a r c cos 1 70

Also common are tasks when the coordinates of three points are given in a rectangular coordinate system and it is necessary to determine some angle. And then, in order to determine the angle between vectors with given coordinates of points, it is necessary to calculate the coordinates of the vectors as the difference between the corresponding points of the beginning and end of the vector.

Example 3

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

Solution

Let's find the coordinates of the vectors from the coordinates of the given points A C → = (7 - 2, - 2 - (- 1)) = (5, - 1) B C → = (7 - 3, - 2 - 2) = (4, - 4)

Now we use the formula to determine the cosine of the angle between vectors on a plane in coordinates: cos A C → , B C → ^ = (A C → , B C →) A C → · B C → = 5 · 4 + (- 1) · (- 4) 5 2 + (- 1) 2 4 2 + (- 4) 2 = 24 26 32 = 3 13

Answer: cos A C → , B C → ^ = 3 13

The angle between vectors can be determined using the cosine theorem. Let us set aside the vectors O A → = a → and O B → = b → from point O, then, according to the cosine theorem in the triangle O A B, the equality will be true:

A B 2 = O A 2 + O B 2 - 2 · O A · O B · cos (∠ A O B) ,

which is equivalent to:

b → - a → 2 = a → + b → - 2 a → b → cos (a → , b →) ^

and from here we derive the formula for the cosine of the angle:

cos (a → , b →) ^ = 1 2 a → 2 + b → 2 - b → - a → 2 a → b →

To apply the resulting formula, we need the lengths of the vectors, which can be easily determined from their coordinates.

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

cos (a → , b →) ^ = a → , b → a → b →

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Scalar product of vectors (hereinafter referred to as SP). Dear friends! The mathematics exam includes a group of problems on solving vectors. We have already considered some problems. You can see them in the “Vectors” category. In general, the theory of vectors is not complicated, the main thing is to study it consistently. Calculations and operations with vectors in the school mathematics course are simple, the formulas are not complicated. Take a look at. In this article we will analyze problems on SP of vectors (included in the Unified State Examination). Now “immersion” in the theory:

H To find the coordinates of a vector, you need to subtract from the coordinates of its endthe corresponding coordinates of its origin

And further:

*Vector length (modulus) is determined as follows:

These formulas must be remembered!!!

Let's show the angle between the vectors:

It is clear that it can vary from 0 to 180 0(or in radians from 0 to Pi).

We can draw some conclusions about the sign of the scalar product. The lengths of vectors have a positive value, this is obvious. This means the sign of the scalar product depends on the value of the cosine of the angle between the vectors.

Possible cases:

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 obtuse (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, that is, when the vectors have opposite directions, the cosine is equal to minus one,and accordingly the result will be negative.

Now the IMPORTANT POINT!

At 90 o, that is, when the vectors are perpendicular to each other, the cosine is equal to zero, and therefore the SP is equal to zero. This fact (consequence, conclusion) is used in solving many problems where we are talking about the relative position of vectors, including in problems included in the open bank of mathematics tasks.

Let us formulate the statement: the scalar product is equal to zero if and only if these vectors lie on perpendicular lines.

So, the formulas for SP vectors:

If the coordinates of the vectors or the coordinates of the points of their beginnings and ends are known, then we can always find the angle between the vectors:

Let's consider the tasks:

27724 Find the scalar product of the vectors a and b.

We can find the scalar product of vectors using one of two formulas:

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

How to find the coordinates of a vector is described in.

We calculate:

Answer: 40

Let's find the coordinates of the vectors and use the formula:

To find the coordinates of a vector, it is necessary to subtract the corresponding coordinates of its beginning from the coordinates of the end of the vector, which means

We calculate the scalar product:

Answer: 40

Find the angle between vectors a and b. Give your answer in degrees.

Let the coordinates of the vectors have the form:

To find the angle between vectors, we use the formula for the scalar product of vectors:

Cosine of the angle between vectors:

Hence:

The coordinates of these vectors are equal:

Let's substitute them into the formula:

The angle between the vectors is 45 degrees.

Answer: 45

Angle between two vectors , :

If the angle between two vectors is acute, then their scalar product is positive; if the angle between the vectors is obtuse, then the scalar product of these vectors is negative. The scalar product of two nonzero vectors is equal to zero if and only if these vectors are orthogonal.

Exercise. Find the angle between the vectors and

Solution. Cosine of the desired angle

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

Angle between a straight line and a plane, intersecting this line and not perpendicular to it, is the angle between the line and its projection onto this plane.

Determining the angle between a line and a plane allows us to conclude that the angle between a line and a plane is the angle between two intersecting lines: the straight line itself and its projection onto the plane. Therefore, the angle between a straight line and a plane is an acute angle.

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

§ 69. Calculation of the angle between straight lines.

The problem of calculating the angle between two straight lines in space is solved in the same way as on a plane (§ 32). Let us denote by φ the magnitude of the angle between the lines l 1 and l 2, and through ψ - the magnitude of the angle between the direction vectors A And b these straight lines.

Then if

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

hence,

Let the lines be given by their canonical equations

Then the angle φ between the lines is determined using the formula

If one of the lines (or both) is given by non-canonical equations, then to calculate the angle you need to find the coordinates of the direction vectors of these lines, and then use formula (1).

17. Parallel lines, Theorems on parallel lines

Definition. Two lines in a plane are called parallel, if they do not have common points.

Two lines in three-dimensional space are called parallel, if they lie in the same plane and do not have common points.

The angle between two vectors.

From the definition of dot product:

Condition for orthogonality of two vectors:

Condition for collinearity of two vectors:

Follows from Definition 5 - . Indeed, from the definition of the product of a vector and a number, it follows. Therefore, based on the rule of equality of vectors, we write , , , which implies . But the vector resulting from multiplying the vector by the number is collinear to the vector.

Projection of vector onto vector:

Example 4. Given points , , , .

Find the dot product.

Solution. we find using the formula for the scalar product of vectors specified by their coordinates. Because the

, ,

Example 5. Given points , , , .

Find projection.

Solution. Because the

, ,

Based on the projection formula, we have

Example 6. Given points , , , .

Find the angle between the vectors and .

Solution. Note that the vectors

, ,

are not collinear because their coordinates are not proportional:

These vectors are also not perpendicular, since their scalar product is .

Let's find

Corner we find from the formula:

Example 7. Determine at what vectors and collinear.

Solution. In the case of collinearity, the corresponding coordinates of the vectors and must be proportional, that is:

Hence and.

Example 8. Determine at what value of the vector And perpendicular.

Solution. Vector and are perpendicular if their scalar product is zero. From this condition we obtain: . That is, .

Example 9. Find , If , , .

Solution. Due to the properties of the scalar product, we have:

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

Solution. We have: , ,

Finally we have: .

5 B. Vector artwork.

Definition 21.Vector artwork vector by vector is called a vector, or, defined by the following three conditions:

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

It follows that the modulus of the vector product is numerically equal to the area of a parallelogram constructed on vectors and both sides.

2) The vector is perpendicular to each of the vectors and ( ; ), i.e. perpendicular to the plane of a parallelogram constructed on the vectors and .

3) The vector is directed in such a way that if viewed from its end, the shortest turn from vector to vector would be counterclockwise (vectors , , form a right-handed triple).

How to calculate angles between vectors?

When studying geometry, many questions arise on the topic of vectors. The student experiences particular difficulties when it is necessary to find the angles between vectors.

Basic terms

Before looking at angles between vectors, it is necessary to become familiar with the definition of a vector and the concept of an angle between vectors.

A vector is a segment that has a direction, that is, a segment for which its beginning and end are defined.

The angle between two vectors on a plane that have a common origin is the smaller of the angles by the amount by which one of the vectors needs to be moved around the common point until their directions coincide.

Formula for solution

Once you understand what a vector is 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 value of the cosine of the angle. According to the definition, it is equal to the quotient of the scalar product of vectors and the product of their lengths.

The scalar product of vectors is calculated as the sum of the corresponding coordinates of the factor vectors multiplied by each other. The length of a vector, or its modulus, is calculated as the square root of the sum of the squares of its coordinates.

Having received the value of the cosine of the angle, you can calculate the value of the angle itself using a calculator or using a trigonometric table.

Example

Once you figure out how to calculate the angle between vectors, solving the corresponding problem will become simple and clear. As an example, it is worth considering the simple problem of finding the value of an angle.

First of all, it will be more convenient to calculate the values of the vector lengths and their scalar product necessary for the solution. Using the description presented above, we get:

Substituting the obtained values into the formula, we calculate the value of the cosine of the desired angle:

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

To maintain accuracy, the final answer can be left as is, or you can calculate the value of the angle in degrees. According to the Bradis table, its value will be approximately 116 degrees and 70 minutes, and the calculator will show a value of 116.57 degrees.

Calculating an angle in n-dimensional space

When considering two vectors in three-dimensional space, it is much more difficult to understand which angle we are talking about if they do not lie in the same plane. To simplify perception, you can draw two intersecting segments that form the smallest angle between them, this will be the desired one. Even though there is a third coordinate in the vector, the process of how angles between vectors are calculated will not change. Calculate the scalar product and moduli of the vectors; the arc cosine of their quotient will be the answer to this problem.

In geometry, there are often problems with spaces that have more than three dimensions. But for them, the algorithm for finding the answer looks similar.

Difference between 0 and 180 degrees

One of the common mistakes when writing an answer to a problem designed to calculate the angle between vectors is the decision to write that the vectors are parallel, that is, the desired angle is equal to 0 or 180 degrees. This answer is incorrect.

Having received the angle value of 0 degrees as a result of the solution, the correct answer would be to designate the vectors as codirectional, that is, the vectors will have the same direction. If 180 degrees are obtained, the vectors will be oppositely directed.

Specific vectors

Having found the angles between the vectors, you can find one of the special types, in addition to the co-directional and opposite-directional ones described above.

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

How to find the angle between vectors?

help me please! I know the formula, but I can’t calculate it ((
vector a (8; 10; 4) vector b (5; -20; -10)

Alexander Titov

The angle between vectors specified by their coordinates is found using a standard algorithm. First you need to find the scalar product of vectors a and b: (a, b) = x1x2 + y1y2 + z1z2. We substitute the coordinates of these vectors here and calculate:
(a,b) = 8*5 + 10*(-20) = 4*(-10) = 40 - 200 - 40 = -200.
Next, we determine the lengths of each vector. The length or modulus of a vector is the square root of the sum of the squares of its coordinates:
|a| = root of (x1^2 + y1^2 + z1^2) = root of (8^2 + 10^2 + 4^2) = root of (64 + 100 + 16) = root of 180 = 6 roots of 5
|b| = root of (x2^2 + y2^2 + z2^2) = root of (5^2 + (-20)^2 + (-10)^2) = root of (25 + 400 + 100) = root of 525 = 5 roots of 21.
We multiply these lengths. We get 30 roots out of 105.
And finally, we divide the scalar product of vectors by the product of the lengths of these vectors. We get -200/(30 roots of 105) or
- (4 roots of 105) / 63. This is the cosine of the angle between the vectors. And the angle itself is equal to the arc cosine of this number
f = arccos(-4 roots of 105) / 63.
If I counted everything correctly.

How to calculate the sine of the angle between vectors using the coordinates of the vectors

Mikhail Tkachev

Let's multiply these vectors. Their scalar product is equal to the product of the lengths of these vectors and the cosine of the angle between them.
The angle is unknown to us, but the coordinates are known.
Let's write it down mathematically like this.
Let the vectors a(x1;y1) and b(x2;y2) be given
Then

A*b=|a|*|b|*cosA

CosA=a*b/|a|*|b|

Let's talk.
a*b-scalar product of vectors is equal to the sum of the products of the corresponding coordinates of the coordinates of these vectors, i.e. equal to x1*x2+y1*y2

|a|*|b|-product of vector lengths is equal to √((x1)^2+(y1)^2)*√((x2)^2+(y2)^2).

This means that the cosine of the angle between the vectors is equal to:

CosA=(x1*x2+y1*y2)/√((x1)^2+(y1)^2)*√((x2)^2+(y2)^2)

Knowing the cosine of an angle, we can calculate its sine. Let's discuss how to do this:

If the cosine of an angle is positive, then this angle lies in 1 or 4 quadrants, which means its sine is either positive or negative. But since the angle between the vectors is less than or equal to 180 degrees, then its sine is positive. We reason similarly if the cosine is negative.

SinA=√(1-cos^2A)=√(1-((x1*x2+y1*y2)/√((x1)^2+(y1)^2)*√((x2)^2+( y2)^2))^2)

That's it)))) good luck figuring it out)))

Dmitry Levishchev

The fact that it is impossible to directly sine is not true.
In addition to the formula:
(a,b)=|a|*|b|*cos A
There is also this one:
||=|a|*|b|*sin A
That is, instead of the scalar product, you can take the module of the vector product.

Sections: Mathematics

Type of lesson: learning new material.

Educational tasks:

– derive a formula for calculating the angle between two vectors;

– continue to develop skills in applying vectors to solve problems;

– continue to develop interest in mathematics through problem solving;

– cultivate a conscious attitude towards the learning process, instill a sense of responsibility for the quality of knowledge, exercise self-control over the process of solving and designing exercises.

Providing classes:

– table “Vectors on the plane and in space”;

– task cards for individual questioning;

– task cards for test work;

- microcalculators.

The student must know:

– formula for calculating the angle between vectors.

The student must be able to:

– apply the acquired knowledge to solving analytical, geometric and applied problems.

Motivation of cognitive activity of students.

The teacher reports that today in class students will learn to calculate the angle between vectors and apply the acquired knowledge to solve problems of technical mechanics and physics. Most of the problems in the discipline “Technical Mechanics” are solved by the vector method. Thus, when studying the topic “Plane system of converging forces”, “Finding the resultant of two forces”, the formula for calculating the angle between two vectors is used.

Progress of the lesson.
I. Organizational moment.

II. Checking homework.

a) Individual survey using cards.

Card 1.

1. Write the properties of addition of two vectors.

2. At what value m vectors and will they be collinear?

Card 2.

1. What is called the product of a vector and a number?

2. Are the vectors and ?

Card 3.

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

2. At what value of the length of vectors and will they be equal?

Card 4.

1. Write down formulas for calculating vector coordinates and vector length?

2. Are the vectors and ?

b) Questions for frontal survey:

What actions can be performed on vectors given their coordinates?
What vectors are called collinear?
Condition for collinearity of two non-zero vectors?
Determining the angle between vectors?
Definition of scalar product of two non-zero vectors?
Necessary and sufficient condition for two vectors to be perpendicular?
What is the physical meaning of the scalar product of two vectors?
Write down formulas for calculating the scalar product of two vectors through their coordinates on the plane and in space.
Write down formulas for calculating the length of a vector on the plane and in space.

III. Learning new material.

a) Let us derive a formula for calculating the angle between vectors on the plane and in space. By definition of the scalar product of two non-zero vectors:

cos

Therefore, if and , then

the cosine of the angle between non-zero vectors and is equal to the scalar product of these vectors divided by the product of their lengths. If the vectors are specified in a rectangular Cartesian coordinate system on a plane, then the cosine of the angle between them is calculated by the formula:

= (x 1 ; y 1); = (x 2 ; y 2)

cos =

In space: = (x 1; y 1; z 1); = (x 2 ; y 2 ; z 2)

cos =

Solve problems:

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

Arccos = 135°

Task 2: In triangle ABC, find the size of angle B if

A (0; 5; 0), B (4; 3; -8), C (-1; -3; -6).

cos = =

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

B (1; 0), C (-2; 3).

cos = = = –

IV. Application of knowledge in solving typical problems.

TASKS OF AN ANALYTICAL CHARACTER.

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

B (-1; 0; 2), C (2; -4; -6), D (1; 1; 1).

Find the scalar product of vectors if , = 30°.

At what values of the vector lengths and will they be equal?

Calculate the angle between vectors and

Calculate the area of a parallelogram constructed using vectors

And .

APPLIED TASKS

Find the resultant of two forces 1 and 2, if = 5H; = 7H, angle between them = 60°.

° + .

Calculate the work performed by force = (6; 2), if its point of application, moving rectilinearly, moves from position A (-1; 3) to position B (3; 4).

Let be the speed of the material point and let be the force acting on it. What is the power developed by the force if = 5H, = 3.5 m/s;

VI. Summing up the lesson.

VII. Homework:

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

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