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.

Knowledge and understanding of mathematical terms will help in solving many problems in both algebra and geometry courses. An equally important role is given to formulas that display the relationships between mathematical characteristics.

Angle between vectors - terminology explained

In order to formulate a definition of the angle between vectors, it is necessary to find out what the term “vector” means. This concept characterizes a section of a straight line that has a beginning, length and direction. If in front of you there are 2 directed segments that originate at the same point, therefore they form an angle.

That. the term “angle between vectors” defines the degree measure of the smallest angle by which one directed segment should be rotated (relative to the starting point) so that it takes the position/direction of the second directed segment of the line. This statement applies to vectors emanating from one point.

The degree measure of the angle between two directed sections of a straight line originating at the same point is contained in a segment from 0 º up to 180 º. This quantity is denoted as ∠(ā,ū) – the angle between the directed segments ā and ū.

Calculation of the angle between vectors

Calculation of the degree measure of the angle formed by a pair of directed parts of a straight line is carried out using the following formula:

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

∠φ – the desired angle between the given vectors ō and ā,

(ō,ā) – the product of scalars of the directed parts of the line,

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

Determination of the scalar product of directed sections of a straight line

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

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

ō = { o x, o y ), ā = ( a x, a y) or

ō = { o x, o y , o z ), ā = ( a x, a y , a z).

Therefore, to find the value of the numerator - the scalar of directed segments - you should perform the following actions:

(ō,ā) = ō * ā = o x* a x+ o y *a y if the vectors under consideration lie on the plane

(ō,ā) = ō * ā = o x* a x+ o y *a y + o z* a z if the directed sections of the line are located in space.

Determining vector lengths

The length of the directed segment is calculated using the expressions:

|ō| = √ o x 2 + o y 2 or |ō| = √ o x 2 + o y2+ o z 2

|ā| = √ a x 2 + a y 2 or |ā| = √ a x 2 + a y2+ a z 2

That. in the general case of n-dimensional measurement, the expression for determining the degree measure of the angle between directed segments ō = ( o x, o y , … o n) and ā = ( a x, a y , … a n ) looks like this:

φ = arccos (cosφ) = arccos (( o x* a x+ o y *a y + … + o n* a n)/(√ o x 2 + o y 2 + … + o n2*√ a x 2 + a y 2 + … + a n 2)).

An example of calculating the angle between directed segments

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

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

i * u = 3*4 + 4*4 + 0*2 = 28

Then calculate the lengths of the segments:

|ī| = √9 + 16 + 0 = √25 = 5,

|ū| = √16 + 16 + 4 = √36 = 6.

cos (ī,ū) = 28 / 5*6 = 28/30 = 14/15 = 0.9(3).

Using the table of cosine values ​​(Bradis), determine the value of the desired angle:

cos (ī,ū) = 0.9(3) ⇒ ∠(ī,ū) = 21° 6′.

Dot product of vectors

We continue to deal with vectors. At the first lesson Vectors for dummies We looked at the concept of a vector, actions with vectors, vector coordinates and the simplest problems with vectors. If you came to this page for the first time from a search engine, I strongly recommend reading the above introductory article, since in order to master the material you need to be familiar with the terms and notations I use, have basic knowledge about vectors and be able to solve basic problems. This lesson is a logical continuation of the topic, and in it I will analyze in detail typical tasks that use the scalar product of vectors. This is a VERY IMPORTANT activity.. Try not to skip the examples; they come with a useful bonus - practice will help you consolidate the material you have covered and get better at solving common problems in analytical geometry.

Addition of vectors, multiplication of a vector by a number.... It would be naive to think that mathematicians haven't come up with something else. In addition to the actions already discussed, there are a number of other operations with vectors, namely: dot product of vectors, vector product of vectors And mixed product of vectors. The scalar product of vectors is familiar to us from school; the other two products traditionally belong to the course of higher mathematics. The topics are simple, the algorithm for solving many problems is straightforward and understandable. The only thing. There is a decent amount of information, so it is undesirable to try to master and solve EVERYTHING AT ONCE. This is especially true for dummies; believe me, the author absolutely does not want to feel like Chikatilo from mathematics. Well, not from mathematics, of course, either =) More prepared students can use materials selectively, in a certain sense, “get” the missing knowledge; for you I will be a harmless Count Dracula =)

Let's finally open the door and watch with enthusiasm what happens when two vectors meet each other...

Definition of the scalar product of vectors.
Properties of the scalar product. Typical tasks

The concept of a dot product

First about angle between vectors. I think everyone intuitively understands what the angle between vectors is, but just in case, a little more detail. Let's consider free nonzero vectors and . If you plot these vectors from an arbitrary point, you will get a picture that many have already imagined mentally:

I admit, here I described the situation only at the level of understanding. If you need a strict definition of the angle between vectors, please refer to the textbook; for practical problems, in principle, it is of no use to us. Also HERE AND HEREIN I will ignore zero vectors in places due to their low practical significance. I made a reservation specifically for advanced site visitors who may reproach me for the theoretical incompleteness of some subsequent statements.

In the literature, the angle symbol is often skipped and simply written.

Definition: The scalar product of two vectors is a NUMBER equal to the product of the lengths of these vectors and the cosine of the angle between them:

Now this is a quite strict definition.

We focus on essential information:

Designation: the scalar product is denoted by or simply.

The result of the operation is a NUMBER: Vector is multiplied by vector, and the result is a number. Indeed, if the lengths of vectors are numbers, the cosine of an angle is a number, then their product will also be a number.

Just a couple of warm-up examples:

Example 1

Solution: We use the formula . In this case:

Answer:

Cosine values ​​can be found in trigonometric table. I recommend printing it out - it will be needed in almost all sections of the tower and will be needed many times.

From a purely mathematical point of view, the scalar product is dimensionless, that is, the result, in this case, is just a number and that’s it. From the point of view of physics problems, a scalar product always has a certain physical meaning, that is, after the result one or another physical unit must be indicated. A canonical example of calculating the work of a force can be found in any textbook (the formula is exactly a scalar product). The work of a force is measured in Joules, therefore, the answer will be written quite specifically, for example, .

Example 2

Find if , and the angle between the vectors is equal to .

This is an example for you to solve on your own, the answer is at the end of the lesson.

Angle between vectors and dot product value

In Example 1 the scalar product turned out to be positive, and in Example 2 it turned out to be negative. Let's find out what the sign of the scalar product depends on. Let's look at our formula: . The lengths of non-zero vectors are always positive: , so the sign can only depend on the value of the cosine.

Note: To better understand the information below, it is better to study the cosine graph in the manual Function graphs and properties. See how the cosine behaves on the segment.

As already noted, the angle between the vectors can vary within , and the following cases are possible:

1) If corner between vectors spicy: (from 0 to 90 degrees), then , And the dot product will be positive co-directed, then the angle between them is considered zero, and the scalar product will also be positive. Since , the formula simplifies: .

2) If corner between vectors blunt: (from 90 to 180 degrees), then , and correspondingly, dot product is negative: . Special case: if the vectors opposite directions, then the angle between them is considered expanded: (180 degrees). The scalar product is also negative, since

The converse statements are also true:

1) If , then the angle between these vectors is acute. Alternatively, the vectors are co-directional.

2) If , then the angle between these vectors is obtuse. Alternatively, the vectors are in opposite directions.

But the third case is of particular interest:

3) If corner between vectors straight: (90 degrees), then scalar product is zero: . The converse is also true: if , then . The statement can be formulated compactly as follows: The scalar product of two vectors is zero if and only if the vectors are orthogonal. Short math notation:

! Note : Let's repeat basics of mathematical logic: A double-sided logical consequence icon is usually read "if and only if", "if and only if". As you can see, the arrows are directed in both directions - “from this follows this, and vice versa - from that follows this.” What, by the way, is the difference from the one-way follow icon? The icon states only that, that “from this follows this”, and it is not a fact that the opposite is true. For example: , but not every animal is a panther, so in this case you cannot use the icon. At the same time, instead of the icon Can use one-sided icon. For example, while solving the problem, we found out that we concluded that the vectors are orthogonal: - such an entry will be correct, and even more appropriate than .

The third case has great practical significance, since it allows you to check whether vectors are orthogonal or not. We will solve this problem in the second section of the lesson.

Properties of the dot product

Let's return to the situation when two vectors co-directed. In this case, the angle between them is zero, , and the scalar product formula takes the form: .

What happens if a vector is multiplied by itself? It is clear that the vector is aligned with itself, so we use the above simplified formula:

The number is called scalar square vector, and are denoted as .

Thus, the scalar square of a vector is equal to the square of the length of the given vector:

From this equality we can obtain a formula for calculating the length of the vector:

So far it seems unclear, but the objectives of the lesson will put everything in its place. To solve the problems we also need properties of the dot product.

For arbitrary vectors and any number, the following properties are true:

1) – commutative or commutative scalar product law.

2) – distribution or distributive scalar product law. Simply, you can open the brackets.

3) – associative or associative scalar product law. The constant can be derived from the scalar product.

Often, all kinds of properties (which also need to be proven!) are perceived by students as unnecessary rubbish, which only needs to be memorized and safely forgotten immediately after the exam. It would seem that what is important here, everyone already knows from the first grade that rearranging the factors does not change the product: . I must warn you that in higher mathematics it is easy to mess things up with such an approach. So, for example, the commutative property is not true for algebraic matrices. It is also not true for vector product of vectors. Therefore, at a minimum, it is better to delve into any properties that you come across in a higher mathematics course in order to understand what you can do and what you cannot do.

Example 3

.

Solution: First, let's clarify the situation with the vector. What is this anyway? The sum of vectors is a well-defined vector, which is denoted by . A geometric interpretation of actions with vectors can be found in the article Vectors for dummies. The same parsley with a vector is the sum of the vectors and .

So, according to the condition, it is required to find the scalar product. In theory, you need to apply the working formula , but the trouble is that we do not know the lengths of the vectors and the angle between them. But the condition gives similar parameters for vectors, so we will take a different route:

(1) Substitute the expressions of the vectors.

(2) We open the brackets according to the rule for multiplying polynomials; a vulgar tongue twister can be found in the article Complex numbers or Integrating a Fractional-Rational Function. I won’t repeat myself =) By the way, the distributive property of the scalar product allows us to open the brackets. We have the right.

(3) In the first and last terms we compactly write the scalar squares of the vectors: . In the second term we use the commutability of the scalar product: .

(4) We present similar terms: .

(5) In the first term we use the scalar square formula, which was mentioned not so long ago. In the last term, accordingly, the same thing works: . We expand the second term according to the standard formula .

(6) Substitute these conditions , and CAREFULLY carry out the final calculations.

Answer:

A negative value of the scalar product states the fact that the angle between the vectors is obtuse.

The problem is typical, here is an example for solving it yourself:

Example 4

Find the scalar product of vectors and if it is known that .

Now another common task, just for the new formula for the length of a vector. The notation here will be a little overlapping, so for clarity I’ll rewrite it with a different letter:

Example 5

Find the length of the vector if .

Solution will be as follows:

(1) We supply the expression for the vector .

(2) We use the length formula: , and the whole expression ve acts as the vector “ve”.

(3) We use the school formula for the square of the sum. Notice how it works here in a curious way: – in fact, it is the square of the difference, and, in fact, that’s how it is. Those who wish can rearrange the vectors: - the same thing happens, up to the rearrangement of the terms.

(4) What follows is already familiar from the two previous problems.

Answer:

Since we are talking about length, do not forget to indicate the dimension - “units”.

Example 6

Find the length of the vector if .

This is an example for you to solve on your own. Full solution and answer at the end of the lesson.

We continue to squeeze useful things out of the dot product. Let's look at our formula again . Using the rule of proportion, we reset the lengths of the vectors to the denominator of the left side:

Let's swap the parts:

What is the meaning of this formula? If the lengths of two vectors and their scalar product are known, then we can calculate the cosine of the angle between these vectors, and, consequently, the angle itself.

Is a dot product a number? Number. Are vector lengths numbers? Numbers. This means that a fraction is also a number. And if the cosine of the angle is known: , then using the inverse function it is easy to find the angle itself: .

Example 7

Find the angle between the vectors and if it is known that .

Solution: We use the formula:

At the final stage of the calculations, a technical technique was used - eliminating irrationality in the denominator. In order to eliminate irrationality, I multiplied the numerator and denominator by .

So if , That:

The values ​​of inverse trigonometric functions can be found by trigonometric table. Although this happens rarely. In problems of analytical geometry, much more often some clumsy bear like , and the value of the angle has to be found approximately using a calculator. Actually, we will see such a picture more than once.

Answer:

Again, do not forget to indicate the dimensions - radians and degrees. Personally, in order to obviously “resolve all questions”, I prefer to indicate both (unless the condition, of course, requires presenting the answer only in radians or only in degrees).

Now you can independently cope with a more complex task:

Example 7*

Given are the lengths of the vectors and the angle between them. Find the angle between the vectors , .

The task is not so much difficult as it is multi-step.
Let's look at the solution algorithm:

1) According to the condition, you need to find the angle between the vectors and , so you need to use the formula .

2) Find the scalar product (see Examples No. 3, 4).

3) Find the length of the vector and the length of the vector (see Examples No. 5, 6).

4) The ending of the solution coincides with Example No. 7 - we know the number , which means it’s easy to find the angle itself:

A short solution and answer at the end of the lesson.

The second section of the lesson is devoted to the same scalar product. Coordinates. It will be even easier than in the first part.

Dot product of vectors,
given by coordinates in an orthonormal basis

Answer:

Needless to say, dealing with coordinates is much more pleasant.

Example 14

Find the scalar product of vectors and if

This is an example for you to solve on your own. Here you can use the associativity of the operation, that is, do not count , but immediately take the triple outside the scalar product and multiply it by it last. The solution and answer are at the end of the lesson.

At the end of the section, a provocative example on calculating the length of a vector:

Example 15

Find the lengths of vectors , If

Solution: The method of the previous section suggests itself again: but there is another way:

Let's find the vector:

And its length according to the trivial formula :

The dot product is not relevant here at all!

It is also not useful when calculating the length of a vector:
Stop. Shouldn't we take advantage of the obvious property of vector length? What can you say about the length of the vector? This vector is 5 times longer than the vector. The direction is opposite, but this does not matter, because we are talking about length. Obviously, the length of the vector is equal to the product module numbers per vector length:
– the modulus sign “eats” the possible minus of the number.

Thus:

Answer:

Formula for the cosine of the angle between vectors that are specified by coordinates

Now we have complete information to use the previously derived formula for the cosine of the angle between vectors express through vector coordinates:

Cosine of the angle between plane vectors and , specified in an orthonormal basis, expressed by the formula:
.

Cosine of the angle between space vectors, specified in an orthonormal basis, expressed by the formula:

Example 16

Given three vertices of a triangle. Find (vertex angle).

Solution: According to the conditions, the drawing is not required, but still:

The required angle is marked with a green arc. Let us immediately remember the school designation of an angle: – special attention to average letter - this is the vertex of the angle we need. For brevity, you could also write simply .

From the drawing it is quite obvious that the angle of the triangle coincides with the angle between the vectors and, in other words: .

It is advisable to learn how to perform the analysis mentally.

Let's find the vectors:

Let's calculate the scalar product:

And the lengths of the vectors:

Cosine of angle:

This is exactly the order of completing the task that I recommend for dummies. More advanced readers can write the calculations “in one line”:

Here is an example of a “bad” cosine value. The resulting value is not final, so there is little point in getting rid of irrationality in the denominator.

Let's find the angle itself:

If you look at the drawing, the result is quite plausible. To check, the angle can also be measured with a protractor. Do not damage the monitor cover =)

Answer:

In the answer we do not forget that asked about the angle of a triangle(and not about the angle between the vectors), do not forget to indicate the exact answer: and the approximate value of the angle: , found using a calculator.

Those who have enjoyed the process can calculate the angles and verify the validity of the canonical equality

Example 17

A triangle is defined in space by the coordinates of its vertices. Find the angle between the sides and

This is an example for you to solve on your own. Full solution and answer at the end of the lesson

A short final section will be devoted to projections, which also involve a scalar product:

Projection of a vector onto a vector. Projection of a vector onto coordinate axes.
Direction cosines of a vector

Consider the vectors and :

Let's project the vector onto the vector; to do this, from the beginning and end of the vector we omit perpendiculars to vector (green dotted lines). Imagine that rays of light fall perpendicularly onto the vector. Then the segment (red line) will be the “shadow” of the vector. In this case, the projection of the vector onto the vector is the LENGTH of the segment. That is, PROJECTION IS A NUMBER.

This NUMBER is denoted as follows: , “large vector” denotes the vector WHICH project, “small subscript vector” denotes the vector ON which is projected.

The entry itself reads like this: “projection of vector “a” onto vector “be”.”

What happens if the vector "be" is "too short"? We draw a straight line containing the vector “be”. And vector “a” will be projected already to the direction of the vector "be", simply - to the straight line containing the vector “be”. The same thing will happen if the vector “a” is postponed in the thirtieth kingdom - it will still be easily projected onto the straight line containing the vector “be”.

If the angle between vectors spicy(as in the picture), then

If the vectors orthogonal, then (the projection is a point whose dimensions are considered zero).

If the angle between vectors blunt(in the figure, mentally rearrange the vector arrow), then (the same length, but taken with a minus sign).

Let us plot these vectors from one point:

Obviously, when a vector moves, its projection does not change