Lecture: Vector coordinates; dot product of vectors; angle between vectors

Vector coordinates

So, as mentioned earlier, vectors are a directed segment, which has its own beginning and end. If the beginning and end are represented by some points, then on a plane or in space they have their own coordinates.

If each point has its own coordinates, then we can get the coordinates of the whole vector.

Suppose we have some vector whose beginning and end of the vector have the following designations and coordinates: A (A x; Ay) and B (B x; By)

To get the coordinates of this vector, it is necessary to subtract the corresponding coordinates of the beginning from the coordinates of the end of the vector:

To determine the coordinates of a vector in space, use the following formula:

Dot product of vectors

There are two ways to define the dot product:

• Geometric way. According to him, the dot product is equal to the product of the values ​​of these modules by the cosine of the angle between them.

• Algebraic meaning. From the point of view of algebra, the dot product of two vectors is a certain quantity that is obtained as a result of the sum of the products of the corresponding vectors.

If vectors are given in space, then you should use a similar formula:

Properties:

• If you multiply two identical vectors scalarly, then their dot product will not be negative:

• If the scalar product of two identical vectors turns out to be equal to zero, then these vectors are considered to be zero:

• If a vector is multiplied by itself, then the scalar product will be equal to the square of its modulus:

• The scalar product has a communicative property, that is, the scalar product will not change from the permutation of the vectors:

• The scalar product of nonzero vectors can be zero only if the vectors are perpendicular to each other:

• For the scalar product of vectors, the displacement law is valid in the case of multiplying one of the vectors by a number:

• With the dot product, you can also use the distributive property of multiplication:

Angle between vectors

Vector and dot product makes it easy to calculate the angle between vectors. Let there be given two vectors $ \ overline (a) $ and $ \ overline (b) $, the oriented angle between which is $ \ varphi $. Calculate the values ​​$ x = (\ overline (a), \ overline (b)) $ and $ y = [\ overline (a), \ overline (b)] $. Then $ x = r \ cos \ varphi $, $ y = r \ sin \ varphi $, where $ r = | \ overline (a) | \ cdot | \ overline (b) | $, and $ \ varphi $ is the required angle, that is, the point $ (x, y) $ has a polar angle equal to $ \ varphi $, and therefore $ \ varphi $ can be found as atan2 (y, x).

Area of ​​a triangle

Since the cross product contains the product of two vector lengths by the cosine of the angle between them, the cross product can be used to calculate the area of ​​triangle ABC:

$ S_ (ABC) = \ frac (1) (2) | [\ overline (AB), \ overline (AC)] | $.

A point belonging to a straight line

Let a point $ P $ and a straight line $ AB $ (given by two points $ A $ and $ B $) be given. It is necessary to check whether the point belongs to the line $ AB $.

A point belongs to the straight line $ AB $ if and only if the vectors $ AP $ and $ AB $ are collinear, that is, if $ [\ overline (AP), \ overline (AB)] = 0 $.

Belonging a point to a ray

Let a point $ P $ and a ray $ AB $ (given by two points - the beginning of ray $ A $ and a point on ray $ B $) be given. It is necessary to check whether the point belongs to the ray $ AB $.

To the condition that the point $ P $ belongs to the line $ AB $, it is necessary to add an additional condition - the vectors $ AP $ and $ AB $ are co-directional, that is, they are collinear and their scalar product is non-negative, that is, $ (\ overline (AB), \ overline (AP )) \ ge 0 $.

A point belongs to a line segment

Let a point $ P $ and a segment $ AB $ be given. It is necessary to check if the point belongs to the segment $ AB $.

In this case, the point must belong to both ray $ AB $ and ray $ BA $, so the following conditions must be checked:

$ [\ overline (AP), \ overline (AB)] = 0 $,

$ (\ overline (AB), \ overline (AP)) \ ge 0 $,

$ (\ overline (BA), \ overline (BP)) \ ge 0 $.

Distance from point to line

Let a point $ P $ and a straight line $ AB $ (given by two points $ A $ and $ B $) be given. It is necessary to find the distance from the point of the straight line $ AB $.

Consider a triangle ABP. On the one hand, its area is $ S_ (ABP) = \ frac (1) (2) | [\ overline (AB), \ overline (AP)] | $.

On the other hand, its area is $ S_ (ABP) = \ frac (1) (2) h | AB | $, where $ h $ is the height dropped from the point $ P $, that is, the distance from $ P $ to $ AB $. From where $ h = | [\ overline (AB), \ overline (AP)] | / | AB | $.

Point to beam distance

Let a point $ P $ and a ray $ AB $ (given by two points - the beginning of ray $ A $ and a point on ray $ B $) be given. It is necessary to find the distance from the point to the ray, that is, the length of the shortest segment from the point $ P $ to any point on the ray.

This distance is equal to either the length $ AP $, or the distance from the point $ P $ to the line $ AB $. Which of the cases takes place is easy to determine by the relative position of the beam and the point. If the angle PAB is acute, that is, $ (\ overline (AB), \ overline (AP))> 0 $, then the answer will be the distance from the point $ P $ to the straight line $ AB $, otherwise the answer will be the length of the segment $ AB $.

Distance from point to line

Let a point $ P $ and a segment $ AB $ be given. It is necessary to find the distance from $ P $ to the segment $ AB $.

If the base of the perpendicular dropped from $ P $ to line $ AB $ falls on the segment $ AB $, which can be verified by the conditions

$ (\ overline (AP), \ overline (AB)) \ ge 0 $,

$ (\ overline (BP), \ overline (BA)) \ ge 0 $,

then the answer is the distance from point $ P $ to line $ AB $. Otherwise, the distance will be equal to $ \ min (AP, BP) $.

Dot product of vectors

We continue to deal with vectors. In the first lesson Vectors for dummies we examined the concept of a vector, actions with vectors, coordinates of a vector and the simplest tasks with vectors. If you have come to this page for the first time from a search engine, I highly recommend reading the above introductory article, because to master the material, you need to navigate in the terms and notations I use, have basic knowledge of vectors and be able to solve elementary problems. This lesson is a logical continuation of the topic, and in it I will analyze in detail typical tasks in which the dot product of vectors is used. This is a VERY IMPORTANT activity.... Try not to skip examples, they are accompanied by a useful bonus - practice will help you consolidate the material you have covered and get your hands on the solution to 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 anything else. In addition to the actions already considered, 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 are traditionally related to the course of higher mathematics. The topics are simple, the algorithm for solving many problems is stereotyped and understandable. The only thing. There is a decent amount of information, so it is undesirable to try to master, solve EVERYTHING AT ONCE. This is especially true for teapots, believe me, the author does not want to feel like Chikatilo from mathematics at all. Well, and not from mathematics, of course, too =) More prepared students can use the materials selectively, in a sense, "get" the missing knowledge, for you I will be a harmless Count Dracula =)

Finally, let us open the door a little and see with enthusiasm what happens when two vectors meet each other….

Determination of the dot product of vectors.
Dot product properties. Typical tasks

Dot product concept

First about angle between vectors... I think everyone intuitively understands what the angle between vectors is, but just in case, a little more in detail. Consider free nonzero vectors and. If you postpone these vectors from an arbitrary point, you get a picture that many have already imagined in their minds:

I confess that here I have outlined the situation only at the level of understanding. If you need a strict definition of the angle between the vectors, please refer to the textbook, but for practical problems we, in principle, do not need it. Also HERE AND FURTHER I will in places ignore zero vectors due to their low practical significance. I made a reservation specifically for advanced site visitors who can reproach me for the theoretical incompleteness of some of the following statements.

can take values ​​from 0 to 180 degrees (from 0 to radians) inclusive. Analytically, this fact is written in the form of a double inequality: or (in radians).

In the literature, the angle icon is often overlooked and written simply.

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

This is already quite a strict definition.

We focus on essential information:

Designation: dot product is denoted by or simply.

The result of the operation is a NUMBER: The vector is multiplied by the 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:

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

From a purely mathematical point of view, the dot 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, 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 the dot product). The work of force is measured in Joules, therefore, and the answer will be written down quite specifically, for example,.

Example 2

Find if , and the angle between the vectors is.

This is an example for a do-it-yourself solution, the answer is at the end of the tutorial.

Angle between vectors and dot product value

In Example 1, the dot product turned out to be positive, and in Example 2, it turned out to be negative. Let us find out what the sign of the dot product depends on. We look at our formula: ... The lengths of nonzero 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 a segment.

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

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

2) If injection between vectors blunt: (from 90 to 180 degrees), then , and correspondingly, dot product is negative:. Special case: if vectors opposite direction, then the angle between them is considered deployed: (180 degrees). The dot 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 codirectional.

2) If, then the angle between the given vectors is obtuse. Alternatively, the vectors are oppositely directed.

But the third case is of particular interest:

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

! Note : repeat foundations of mathematical logic: the double-sided logical consequence icon is usually read "then and only then", "if and only if". As you can see, the arrows are directed in both directions - "from this follows this, and vice versa - from what follows from this." By the way, what is the difference from the one-way follow icon? The icon claims only that that "this follows from this", and it is not a fact that the opposite is true. For example: but not every animal is a panther, so the icon cannot be used in this case. At the same time, instead of the icon can use one-way icon. For example, 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 is of great practical importance. since it allows you to check if vectors are orthogonal or not. We will solve this problem in the second section of the lesson.

Dot product properties

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

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

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

In this way, the scalar square of a vector is equal to the square of the length of the given vector:

From this equality, you can get a formula for calculating the length of a vector:

While it seems obscure, but the tasks of the lesson will put everything in its place. To solve problems, we also need dot product properties.

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

1) - displaceable or commutative scalar product law.

2) - distribution or distributive scalar product law. Simply, you can expand the parentheses.

3) - combination or associative scalar product law. The constant can be taken out from the dot product.

Often, all kinds of properties (which also need to be proved!) Are perceived by students as unnecessary trash, which just needs to be memorized and safely forgotten right after the exam. It would seem that what is important here, everyone knows from the first grade that the product does not change from the rearrangement of the factors:. I must warn you, in higher mathematics with this approach, it is easy to break wood. So, for example, the displacement property is not valid for algebraic matrices... It is also not true for vector product of vectors... Therefore, at least it is better to delve into any properties that you come across in the course of higher mathematics in order to understand what can and cannot be done.

Example 3

.

Solution: First, let's clarify the situation with the vector. What is this anyway? The sum of vectors and is a well-defined vector, which is denoted by. The 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 vectors and.

So, by condition it is required to find the dot 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'll go the other way:

(1) Substitute vector expressions.

(2) We expand the brackets according to the rule of multiplication of polynomials, a vulgar tongue twister can be found in the article Complex numbers or Integration of a fractional rational function... I will not repeat myself =) By the way, the distribution property of the scalar product allows us to expand the brackets. We have the right.

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

(4) We give similar terms:.

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

(6) We substitute these conditions , and CAREFULLY make the final calculations.

Answer:

The negative value of the dot product states the fact that the angle between the vectors is obtuse.

The task is typical, here is an example for an independent solution:

Example 4

Find the dot 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 designations here will overlap a bit, 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) Supply a vector expression.

(2) We use the formula of length:, while the whole expression acts as a vector "ve".

(3) We use the school formula for the square of the sum. Note how it works curiously here: - in fact, it is the square of the difference, and, in fact, it is. Those interested can rearrange the vectors in places: - it turned out the same up to the rearrangement of the terms.

(4) The rest 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 a do-it-yourself solution. Complete solution and answer at the end of the tutorial.

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

And we will swap the parts:

What is the meaning of this formula? If you know the lengths of two vectors and their dot product, then you can calculate the cosine of the angle between these vectors, and, therefore, the angle itself.

Is the dot product a number? Number. Are the lengths of the vectors numbers? Numbers. Hence, the fraction is also a certain 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 technique was used - the elimination of irrationality in the denominator. In order to eliminate irrationality, I multiplied the numerator and denominator by.

So if , then:

The values ​​of inverse trigonometric functions can be found by trigonometric table... Although this rarely happens. In problems of analytical geometry, some kind of clumsy bear appears much more often, 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 dimension - radians and degrees. Personally, in order to knowingly “clear all the questions”, I prefer to indicate both that and that (unless, of course, by the condition, it is required to present the answer only in radians or only in degrees).

Now you will be able to cope with a more difficult task on your own:

Example 7 *

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

The task is not even so difficult as multi-step.
Let's analyze the solution algorithm:

1) According to the condition, it is required to find the angle between the vectors and, therefore, you need to use the formula .

2) Find the dot 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 end of the solution coincides with Example No. 7 - we know the number, which means that it is easy to find the angle itself:

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

The second section of the lesson focuses on the same dot 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 dot product of vectors and, if

This is an example for a do-it-yourself solution. Here you can use the associativity of the operation, that is, do not count, but immediately move the triple out of the scalar product and multiply by it last. Solution and answer at the end of the lesson.

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

Example 15

Find the lengths of vectors , if

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

Find the vector:

And its length according to the trivial formula :

The dot product is out of the question here at all!

As out of business it is when calculating the length of a vector:
Stop. Why not take advantage of the obvious property of the vector length? What about the length of the vector? This vector is 5 times longer than the vector. The direction is opposite, but it doesn't matter, because the talk is about length. Obviously, the length of the vector is equal to the product module numbers per vector length:
- the sign of the module "eats" a possible minus of the number.

In this way:

Answer:

The formula for the cosine of the angle between vectors, which are given by coordinates

Now we have complete information to express the previously derived formula for the cosine of the angle between vectors in terms of the coordinates of the vectors:

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

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

Example 16

Three vertices of the triangle are given. Find (vertex angle).

Solution: According to the condition, the drawing is not required to be performed, but still:

The required angle is marked with a green arc. We immediately recall the school designation of the angle: - special attention to average the letter - this is the vertex of the corner we need. For brevity, it could also be written 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 desirable to learn how to carry out the analysis performed mentally.

Find vectors:

Let's calculate the dot product:

And the lengths of the vectors:

Cosine of an angle:

This is the order of completing the task that I recommend to teapots. More advanced readers can write computations "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 corner itself:

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

Answer:

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

Those who have enjoyed the process can calculate the angles and make sure that the canonical equality is true

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 a do-it-yourself solution. Complete solution and answer at the end of the tutorial

A short final section will be devoted to projections, in which the scalar product is also "mixed":

Vector-to-vector projection. The projection of the vector to the coordinate axes.
Direction cosines of a vector

Consider vectors and:

We project the vector onto the vector, for this we omit from the beginning and end of the vector perpendiculars per vector (green dotted lines). Imagine rays of light falling perpendicular to 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, the PROJECTION IS A NUMBER.

This NUMBER is denoted as follows:, "large vector" denotes a vector WHICH THE project, "small subscript vector" denotes a vector ON THE which is being projected.

The record itself reads like this: "the projection of the vector" a "onto the vector" bh "".

What happens if the vector "bs" is "too short"? We draw a straight line containing the vector "be". And the vector "a" will be projected already on the direction of the vector "bh", simply - on the straight line containing the vector "be". The same 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 "bh".

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

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

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

Let's postpone these vectors from one point:

Obviously, when the vector moves, its projection does not change.

Definition 1

The scalar product of vectors is a number equal to the product of the dyn of these vectors and the cosine of the angle between them.

The notation of the product of vectors a → and b → has the form a →, b →. Let's convert to the formula:

a →, b → = a → b → cos a →, b → ^. a → and b → denote the lengths of vectors, a →, b → ^ denote the angle between given vectors. If at least one vector is zero, that is, it has a value of 0, then the result will also be zero, a →, b → = 0

When multiplying the vector by itself, we get the square of its length:

a →, b → = a → b → cos a →, a → ^ = a → 2 cos 0 = a → 2

Definition 2

Scalar multiplication of a vector by itself is called a scalar square.

Calculated by the formula:

a →, b → = a → b → cos a →, b → ^.

The notation a →, b → = a → b → cos a →, b → ^ = a → npa → b → = b → npb → a → shows that npb → a → is the numerical projection of a → on b →, npa → a → is the projection of b → onto a →, respectively.

Let us formulate the definition of a product for two vectors:

The scalar product of two vectors a → by b → is called the product of the length of the vector a → by the projection b → by the direction a → or the product of the length b → by the projection a → respectively.

Dot product in coordinates

Calculation of the dot product can be performed through the coordinates of vectors in a given plane or in space.

The scalar product of two vectors on a plane, in three-dimensional space, is called the sum of the coordinates of the given vectors a → and b →.

When calculating the scalar product of the given vectors a → = (a x, a y), b → = (b x, b y) in the Cartesian system, use:

a →, b → = a x b x + a y b y,

for three-dimensional space, the following expression applies:

a →, b → = a x b x + a y b y + a z b z.

In fact, this is the third definition of the dot product.

Let's prove it.

Proof 1

For the proof, we use a →, b → = a → b → cos a →, b → ^ = ax bx + ay by for vectors a → = (ax, ay), b → = (bx, by) on Cartesian system.

Vectors should be postponed

O A → = a → = a x, a y and O B → = b → = b x, b y.

Then the length of the vector A B → will be equal to A B → = O B → - O A → = b → - a → = (b x - a x, b y - a y).

Consider a triangle O A B.

A B 2 = O A 2 + O B 2 - 2 O A O B cos (∠ A O B) is true based on the cosine theorem.

By the condition, it can be seen that O A = a →, O B = b →, A B = b → - a →, ∠ A O B = a →, b → ^, therefore, we write the formula for finding the angle between vectors differently

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

Then it follows from the first definition that b → - a → 2 = a → 2 + b → 2 - 2 (a →, b →), hence (a →, b →) = 1 2 (a → 2 + b → 2 - b → - a → 2).

Applying the formula for calculating the length of vectors, we get:
a →, b → = 1 2 ((a 2 x + ay 2) 2 + (b 2 x + by 2) 2 - ((bx - ax) 2 + (by - ay) 2) 2) = = 1 2 (a 2 x + a 2 y + b 2 x + b 2 y - (bx - ax) 2 - (by - ay) 2) = = ax bx + ay by

Let us prove the equalities:

(a →, b →) = a → b → cos (a →, b → ^) = = a x b x + a y b y + a z b z

- respectively for vectors of three-dimensional space.

The scalar product of vectors with coordinates says that the scalar square of a vector is equal to the sum of the squares of its coordinates in space and on a plane, respectively. a → = (a x, a y, a z), b → = (b x, b y, b z) and (a →, a →) = a x 2 + a y 2.

Dot product and its properties

There are dot product properties that are applicable for a →, b →, and c →:

1. commutativity (a →, b →) = (b →, a →);
2. distributivity (a → + b →, c →) = (a →, c →) + (b →, c →), (a → + b →, c →) = (a →, b →) + (a → , c →);
3. the combination property (λ a →, b →) = λ (a →, b →), (a →, λ b →) = λ (a →, b →), λ is any number;
4. the scalar square is always greater than zero (a →, a →) ≥ 0, where (a →, a →) = 0 in the case when a → is zero.

Example 1

The properties are explicable thanks to the definition of the dot product on the plane and the properties when adding and multiplying real numbers.

Prove the commutativity property (a →, b →) = (b →, a →). From the definition we have that (a →, b →) = a y b y + a y b y and (b →, a →) = b x a x + b y a y.

By the commutativity property, the equalities a x b x = b x a x and a y b y = b y a y are true, so a x b x + a y b y = b x a x + b y a y.

It follows that (a →, b →) = (b →, a →). Q.E.D.

Distributivity is valid for any numbers:

(a (1) → + a (2) → +.. + a (n) →, b →) = (a (1) →, b →) + (a (2) →, b →) +. ... ... + (a (n) →, b →)

and (a →, b (1) → + b (2) → +.. + b (n) →) = (a →, b (1) →) + (a →, b (2) →) + ... ... ... + (a →, b → (n)),

hence we have

(a (1) → + a (2) → +.. + a (n) →, b (1) → + b (2) → +... + b (m) →) = (a ( 1) →, b (1) →) + (a (1) →, b (2) →) +. ... ... + (a (1) →, b (m) →) + + (a (2) →, b (1) →) + (a (2) →, b (2) →) +. ... ... + (a (2) →, b (m) →) +. ... ... + + (a (n) →, b (1) →) + (a (n) →, b (2) →) +. ... ... + (a (n) →, b (m) →)

Dot product with examples and solutions

Any problem of such a plan is solved using properties and formulas concerning the dot product:

1. (a →, b →) = a → b → cos (a →, b → ^);
2. (a →, b →) = a → n p a → b → = b → n p b → a →;
3. (a →, b →) = a x b x + a y b y or (a →, b →) = a x b x + a y b y + a z b z;
4. (a →, a →) = a → 2.

Let's look at some solution examples.

Example 2

The length of a → is 3, the length of b → is 7. Find the dot product if the angle is 60 degrees.

Solution

By condition, we have all the data, so we calculate by the formula:

(a →, b →) = a → b → cos (a →, b → ^) = 3 7 cos 60 ° = 3 7 1 2 = 21 2

Answer: (a →, b →) = 21 2.

Example 3

Given vectors a → = (1, - 1, 2 - 3), b → = (0, 2, 2 + 3). What is the dot product.

Solution

In this example, the formula for calculating by coordinates is considered, since they are specified in the problem statement:

(a →, b →) = ax bx + ay by + az bz = = 1 0 + (- 1) 2 + (2 + 3) (2 + 3) = = 0 - 2 + ( 2 - 9) = - 9

Answer: (a →, b →) = - 9

Example 4

Find the dot product A B → and A C →. Points A (1, - 3), B (5, 4), C (1, 1) are given on the coordinate plane.

Solution

To begin with, the coordinates of the vectors are calculated, since the coordinates of the points are given by the condition:

A B → = (5 - 1, 4 - (- 3)) = (4, 7) A C → = (1 - 1, 1 - (- 3)) = (0, 4)

Substituting into the formula using coordinates, we get:

(A B →, A C →) = 4 0 + 7 4 = 0 + 28 = 28.

Answer: (A B →, A C →) = 28.

Example 5

Given vectors a → = 7 m → + 3 n → and b → = 5 m → + 8 n →, find their product. m → is equal to 3 and n → is equal to 2 units, they are perpendicular.

Solution

(a →, b →) = (7 m → + 3 n →, 5 m → + 8 n →). Applying the distributive property, we get:

(7 m → + 3 n →, 5 m → + 8 n →) = = (7 m →, 5 m →) + (7 m →, 8 n →) + (3 n →, 5 m →) + (3 n →, 8 n →)

We take out the coefficient for the sign of the product and get:

(7 m →, 5 m →) + (7 m →, 8 n →) + (3 n →, 5 m →) + (3 n →, 8 n →) = = 7 5 (m →, m →) + 7 8 (m →, n →) + 3 5 (n →, m →) + 3 8 (n →, n →) = = 35 (m →, m →) + 56 (m →, n →) + 15 (n →, m →) + 24 (n →, n →)

By the commutativity property we transform:

35 (m →, m →) + 56 (m →, n →) + 15 (n →, m →) + 24 (n →, n →) = 35 (m →, m →) + 56 (m →, n →) + 15 (m →, n →) + 24 (n →, n →) = 35 (m →, m →) + 71 (m →, n → ) + 24 (n →, n →)

As a result, we get:

(a →, b →) = 35 (m →, m →) + 71 (m →, n →) + 24 (n →, n →).

Now let's apply the formula for the dot product with a predetermined angle:

(a →, b →) = 35 (m →, m →) + 71 (m →, n →) + 24 (n →, n →) = = 35 m → 2 + 71 m → n → cos (m →, n → ^) + 24 n → 2 = = 35 3 2 + 71 3 2 cos π 2 + 24 2 2 = 411.

Answer: (a →, b →) = 411

If there is a numerical projection.

Example 6

Find the dot product a → and b →. Vector a → has coordinates a → = (9, 3, - 3), projection b → with coordinates (- 3, - 1, 1).

Solution

By hypothesis, the vectors a → and the projection b → are oppositely directed, because a → = - 1 3 · n p a → b → →, so the projection b → corresponds to the length n p a → b → →, and with the sign "-":

n p a → b → → = - n p a → b → → = - (- 3) 2 + (- 1) 2 + 1 2 = - 11,

Substituting into the formula, we get the expression:

(a →, b →) = a → n p a → b → → = 9 2 + 3 2 + (- 3) 2 (- 11) = - 33.

Answer: (a →, b →) = - 33.

Problems with a known dot product, where it is necessary to find the length of a vector or a numerical projection.

Example 7

What value should λ take for a given scalar product a → = (1, 0, λ + 1) and b → = (λ, 1, λ) will be equal to -1.

Solution

The formula shows that it is necessary to find the sum of the products of coordinates:

(a →, b →) = 1 λ + 0 1 + (λ + 1) λ = λ 2 + 2 λ.

Given we have (a →, b →) = - 1.

To find λ, we calculate the equation:

λ 2 + 2 λ = - 1, hence λ = - 1.

Answer: λ = - 1.

The physical meaning of the dot product

Mechanics deals with the application of the dot product.

When working A with a constant force F → the body moved from point M to N, you can find the product of the lengths of the vectors F → and M N → with the cosine of the angle between them, which means that the work is equal to the product of the vectors of force and displacement:

A = (F →, M N →).

Example 8

The movement of a material point by 3 meters under the action of a force equal to 5 ntons is directed at an angle of 45 degrees relative to the axis. Find A.

Solution

Since work is the product of the force vector and displacement, it means that, based on the condition F → = 5, S → = 3, (F →, S → ^) = 45 °, we get A = (F →, S →) = F → S → cos (F →, S → ^) = 5 3 cos (45 °) = 15 2 2.

Answer: A = 15 2 2.

Example 9

A material point, moving from M (2, - 1, - 3) to N (5, 3 λ - 2, 4) under the force F → = (3, 1, 2), performed work equal to 13 J. Calculate the length of movement.

Solution

For the given coordinates of the vector M N → we have M N → = (5 - 2, 3 λ - 2 - (- 1), 4 - (- 3)) = (3, 3 λ - 1, 7).

Using the formula for finding work with vectors F → = (3, 1, 2) and MN → = (3, 3 λ - 1, 7), we obtain A = (F ⇒, MN →) = 3 3 + 1 (3 λ - 1) + 2 7 = 22 + 3 λ.

By hypothesis, it is given that A = 13 J, which means 22 + 3 λ = 13. Hence λ = - 3, hence M N → = (3, 3 λ - 1, 7) = (3, - 10, 7).

To find the length of displacement M N →, apply the formula and substitute the values:

M N → = 3 2 + (- 10) 2 + 7 2 = 158.

Answer: 158.

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There will also be tasks for an independent solution, to which you can see the answers.

If in the problem both the lengths of the vectors and the angle between them are presented "on a silver platter", then the condition of the problem and its solution look like this:

Example 1. Given vectors. Find the dot product of vectors if their lengths and the angle between them are represented by the following values:

Another definition is also valid, which is completely equivalent to Definition 1.

Definition 2... The scalar product of vectors is a number (scalar) equal to the product of the length of one of these vectors by the projection of the other vector onto the axis determined by the first of the indicated vectors. Formula according to Definition 2:

We will solve the problem using this formula after the next important theoretical point.

Determining the dot product of vectors in terms of coordinates

The same number can be obtained if the vectors being multiplied are given by their coordinates.

Definition 3. The dot product of vectors is a number equal to the sum of the pairwise products of their respective coordinates.

On surface

If two vectors and on the plane are defined by their two Cartesian rectangular coordinates

then the scalar product of these vectors is equal to the sum of the pairwise products of their respective coordinates:

.

Example 2. Find the numerical value of the vector's projection onto an axis parallel to the vector.

Solution. We find the dot product of vectors by adding the pairwise products of their coordinates:

Now we need to equate the resulting scalar product to the product of the length of the vector and the projection of the vector onto the axis parallel to the vector (in accordance with the formula).

We find the length of the vector as the square root of the sum of the squares of its coordinates:

.

We draw up an equation and solve it:

Answer. The desired numerical value is minus 8.

In space

If two vectors and in space are defined by their three Cartesian rectangular coordinates

,

then the scalar product of these vectors is also equal to the sum of the pairwise products of their corresponding coordinates, only there are already three coordinates:

.

The problem of finding the dot product by the considered method is after parsing the properties of the dot product. Because in the task it will be necessary to determine what angle the multiplied vectors form.

Vector dot product properties

Algebraic properties

1. (displacement property: the magnitude of their dot product does not change from the swapping of the vectors being multiplied).

2. (multiplier combinatory property: the dot product of a vector multiplied by some factor and another vector is equal to the dot product of these vectors multiplied by the same factor).

3. (distributional property with respect to the sum of vectors: the dot product of the sum of two vectors by the third vector is equal to the sum of the dot products of the first vector by the third vector and the second vector by the third vector).

4. (scalar square of vector is greater than zero), if is a nonzero vector, and, if, is a zero vector.

Geometric properties

In the definitions of the operation under study, we have already touched on the concept of the angle between two vectors. It's time to clarify this concept.

In the picture above, two vectors are visible, which are brought to a common origin. And the first thing to pay attention to: there are two angles between these vectors - φ 1 and φ 2 ... Which of these angles appears in the definitions and properties of the dot product of vectors? The sum of the considered angles is 2 π and therefore the cosines of these angles are equal. The definition of the dot product includes only the cosine of an angle, not the value of its expression. But in properties only one corner is considered. And this is the one of two angles that does not surpass π , that is, 180 degrees. In the figure, this angle is designated as φ 1 .

1. Two vectors are called orthogonal and the angle between these vectors is a straight line (90 degrees or π / 2) if the dot product of these vectors is zero :

.

Orthogonality in vector algebra is the perpendicularity of two vectors.

2. Two nonzero vectors make up sharp corner (from 0 to 90 degrees, or, which is the same - less π dot product is positive .

3. Two nonzero vectors make up obtuse angle (from 90 to 180 degrees, or, which is the same - more π / 2) if and only if their dot product is negative .

Example 3. The vectors are given in coordinates:

.

Calculate the dot products of all pairs of given vectors. What angle (acute, straight, obtuse) do these pairs of vectors form?

Solution. We will calculate by adding the products of the corresponding coordinates.

Received a negative number, so the vectors form an obtuse angle.

We got a positive number, so the vectors form an acute angle.

We got zero, so the vectors form a right angle.

We got a positive number, so the vectors form an acute angle.

.

We got a positive number, so the vectors form an acute angle.

For self-test, you can use online calculator Dot product of vectors and the cosine of the angle between them .

Example 4. The lengths of two vectors and the angle between them are given:

.

Determine at what value of the number the vectors and are orthogonal (perpendicular).

Solution. We multiply the vectors according to the rule of multiplying polynomials:

Now let's calculate each term:

.

Let's compose an equation (equality of the product to zero), give similar terms and solve the equation:

Answer: we got the meaning λ = 1.8, for which the vectors are orthogonal.

Example 5. Prove that the vector orthogonal (perpendicular) to the vector

Solution. To check the orthogonality, we multiply the vectors and as polynomials, substituting instead the expression given in the problem statement:

.

To do this, you need to multiply each term (term) of the first polynomial by each term of the second and add the resulting products:

.

As a result, the fraction is reduced at the expense. The result is the following:

Conclusion: as a result of multiplication, we got zero, therefore, the orthogonality (perpendicularity) of the vectors is proved.

Solve the problem yourself, and then see the solution

Example 6. Given the lengths of the vectors and, and the angle between these vectors is π /4 . Determine at what value μ vectors and are mutually perpendicular.

For self-test, you can use online calculator Dot product of vectors and the cosine of the angle between them .

Matrix representation of dot product of vectors and product of n-dimensional vectors

Sometimes it is advantageous for clarity to represent the two vectors being multiplied in the form of matrices. Then the first vector is represented as a row matrix, and the second - as a column matrix:

Then the scalar product of vectors will be product of these matrices :

The result is the same as that obtained by the method that we have already considered. One single number is obtained, and the product of the row matrix by the column matrix is ​​also one single number.

It is convenient to represent the product of abstract n-dimensional vectors in matrix form. So, the product of two four-dimensional vectors will be the product of a row matrix with four elements and a column matrix also with four elements, the product of two five-dimensional vectors will be the product of a row matrix with five elements and a column matrix also with five elements, and so on.

Example 7. Find dot products of pairs of vectors

,

using matrix representation.

Solution. The first pair of vectors. We represent the first vector as a row matrix, and the second as a column matrix. We find the dot product of these vectors as the product of the row matrix by the column matrix:

Similarly, we represent the second pair and find:

As you can see, the results are the same as those of the same pairs from example 2.

Angle between two vectors

The derivation of the formula for the cosine of the angle between two vectors is very beautiful and concise.

To express the dot product of vectors

(1)

in coordinate form, we first find the scalar product of the unit vectors. The dot product of a vector by itself by definition:

What is written in the formula above means: the dot product of a vector by itself is equal to the square of its length... The cosine of zero is equal to one, so the square of each ort will be equal to one:

Since vectors

are pairwise perpendicular, then the pairwise products of unit vectors will be equal to zero:

Now let's do the multiplication of vector polynomials:

We substitute in the right side of the equality the values ​​of the corresponding scalar products of the unit vectors:

We get the formula for the cosine of the angle between two vectors:

Example 8. Given three points A(1;1;1), B(2;2;1), C(2;1;2).

Find the corner.

Solution. Find the coordinates of the vectors:

,

.

According to the formula for the cosine of an angle, we get:

Hence, .

For self-test, you can use online calculator Dot product of vectors and the cosine of the angle between them .

Example 9. Two vectors are given

Find the sum, difference, length, dot product and angle between them.

2.Difference