Find the scalar product if you know what. Dot product of vectors

Scalar product 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 vector lengths are positive value, It 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, cosine 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 open bank math assignments.

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

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 of 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, we do not need it. 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 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, the scalar product always has a certain physical meaning, that is, after the result you need to indicate one or another physical unit. 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: For a better understanding of 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: . A special case: if 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.” By the way, what 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 expressions for 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 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:

Negative meaning 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 at new formula vector length. 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:

On final stage 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:

Inverse values 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 paragraph, 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 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 full information, so that 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 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, we omit from the beginning and end of the vector 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