Scalar product vectors

We continue to deal with vectors. In the first lesson Vectors for teapots We looked at the concept of vector, actions with vectors, vector coordinates and simplest tasks with vectors. If you have entered this page for the first time from the search engine, I strongly recommend reading the above introduction article, since it is necessary to navigate the terms used by me, notation by me, to have basic knowledge about the vectors and be able to solve elementary tasks. This lesson is a logical continuation of the topic, and on it, I will define typical tasks in which the scalar product of vectors is used. This is a very important occupation.. Try not to miss the examples, a useful bonus is attached to them - the practice will help you to fix the material passed and "fill the hand" on solving common tasks of analytical geometry.

Addition of vectors, vector multiplication by number .... It would be naive to think that mathematics did not come up with anything else. In addition to the actions already reviewed, there are a number of other operations with vectors, namely: scalar product vectors, vector artwork vectors and mixed vectors. The scalar product of the vectors is familiar to us from school, two other works traditionally refer to the course of higher mathematics. Themes are simple, the algorithm for solving many tasks of stemather and is understandable. The only thing. Information is decent, so it is undesirable to try to master-break everything and immediately. This is especially true of the teapots, believe me, the author does not want to feel chikatilo from mathematics. Well, not from mathematics, of course, too \u003d) more prepared students can use the materials selectively, in a certain sense, "to get" missing knowledge, for you I will be a harmless graph Dracula \u003d)

We will open, finally, the door and passionately see what happens when two versions meet each other ....

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

The concept of a scalar work

First pro Angle between vectors. I think everyone is intuitive that such an angle between vectors, but just in case a little more. Consider free nonzero vectors and. If you postpone these vectors from an arbitrary point, then it turns out the picture that many have already presented mentally:

I confess, here I obradized the situation only at the level of understanding. If you need a strict definition of the angle between vectors, please contact the textbook, for the practical tasks, it is in principle for nothing. Also hereinafter, I will be in places to ignore zero vectors due to their small practical significance. The reservation made specifically for advanced site visitors who can reproach me in theoretical incompleteness of some subsequent statements.

It can take values \u200b\u200bfrom 0 to 180 degrees (from 0 to radians) inclusive. Analytically this fact is recorded in the form of double inequalities: or (in radians).

In the literature, the angle icon often skip and write simply.

Definition: The scalar product of two vectors is called the number equal to the product of these vectors on the cosine of the corner between them:

This is now quite a strict definition.

We focus on essential information:

Designation: The scalar product is denoted by or simply.

The result of the operation is a number: The vector is multiplied by the vector, and the number is obtained. Indeed, if the lengths of the vectors are the numbers, the cosine of the angle - the number, then their work There will also be a number too.

Immediately a couple of warm-up examples:

Example 1.

Decision: We use the formula . In this case:

Answer:

Cosine values \u200b\u200bcan be found in trigonometric table. I recommend to print it - it will be necessary in almost all sections of the tower and will need many times.

Purely from a mathematical point of view the scalar product is dimensionlessly, that is, the result, in this case, is simply the number and that's it. From the point of view of the tasks of physics, the scalar product always has a certain physical meaning, that is, after the result, you need to specify a particular physical unit. The canonical example of calculating the work of force can be found in any textbook (the formula exactly is a scalar product). Work of force is measured in Joules, therefore, the answer will be recorded quite specifically, for example,.

Example 2.

Find if and the angle between the vectors is equal.

This is an example for an independent decision, the answer at the end of the lesson.

The angle between vectors and the value of the scalar product

In Example 1, the scalar product was positive, and in example 2 - negative. Find out what the sign of a scalar product depends. We look at our formula: . The lengths of nonzero vectors are always positive: therefore the sign may depend only on the cosine value.

Note: For a better understanding of the information below, it is better to explore the cosine schedule in the methods Charts and properties of the function. See how the cosine on the segment behaves.

As already noted, the angle between vectors may vary within And the following cases are possible:

1) if angle between vectors acute: (from 0 to 90 degrees), then , I. scalar product will be positive soncedatedThe corner between them is considered zero, and the scalar product will also be positive. Since, the formula is simplified :.

2) if angle between vectors stupid: (from 90 to 180 degrees) , and correspondingly, scalar product negative:. Special case: if vectors directed oppositethen the corner between them is considered departed: (180 degrees). The scalar product is also negative because

Fair and return statements:

1) If, the angle between the data of the vectors is sharp. Alternatively, vectors are coated.

2) If, the angle between the data vectors is stupid. Alternatively, vectors are directed opposite.

But the third case is of particular interest:

3) if angle between vectors straight: (90 degrees), then scalar product is zero:. The opposite is also true: if, then. Compact statement is formulated as follows: The scalar product of two vectors is zero if and only if these vectors are orthogonal. Short mathematical recording:

! Note : Repeat basics of mathematical logic: A two-way logical consequence icon is usually read "if and only then", "in that and only in the case." As you can see, the arrows are directed in both sides - "This follows from this, and back - from that, it follows." What, by the way, the difference from the unilateral following icon? Icon approves only thatthat "This follows from this", and not the fact that the opposite is right. For example: but not every beast is a panther, so in this case it is impossible to use the icon. At the same time, instead of icon can Use one-way icon. For example, solving the task, we found out that we concluded that vectors are orthogonal: - Such a record will be correct, and even more relevant than .

The third case has great practical significance.because it allows you to check, orthogonal vectors or not. We solve this task in the second section of the lesson.

Properties of a scalar piece

Let's return to the situation when two versions soncedated. In this case, the angle between them is zero, and the formula of the scalar product takes the form :.

And what will happen if the vector is multiplied to yourself? It is clear that the vector is coated with himself, so we use the above simplified formula:

The number is called scalar square Vector, and referred to as.

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

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

While it seems uninterrupted, but the tasks of the lesson will all disappear into place. To solve problems, we will also need properties of a scalar piece.

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

1) - Movement or commutative The law of the scalar work.

2) - Distribution or distributive The law of the scalar work. Simply, you can reveal brackets.

3) - Breathable or associative The law of the scalar work. The constant can be taken out of the scalar product.

Often, all sorts of properties (which are still needed!) Perceived by students as unnecessary trash, which only needs to be sent and immediately after the exam is safely forgotten. It would seem that there is important here, everything and so from the first class know that the work does not change from the permutation of multipliers :. Must warn, in higher mathematics with a similar approach it is easy to block firewood. So, for example, the transition property is not fair for algebraic matrices. It is wrong for vector art vectors. Therefore, in any properties that you will meet in the course of higher mathematics, at least, it is better to delve to understand what you can do, but why it is impossible.

Example 3.

.

Decision:First, clarify the situation with the vector. What is it at all? The sum of the vectors is a completely defined vector, which is indicated through. Geometric interpretation of actions with vectors can be found in the article Vectors for teapots. The same parsley with a vector is the sum of the vectors and.

So, by condition, it is required to find a scalar product. In theory, you need to apply the working formula But the trouble is that we are unknown by the length of the vectors and the angle between them. But in the condition given similar parameters for vectors, so we will go different ways:

(1) We substitute the expression of vectors.

(2) Reveal brackets according to the rule of multiplication of polynomials, you can find a spell in the article. Complex numbers or Integrating a fractional rational function. I will not repeat \u003d) By the way, to reveal the brackets to us all the distribution property of the scalar product. We have right.

(3) In the first and last term, the scalar squares of the vectors are compact: . In the second, we use the rearrangement of the scalar product :.

(4) We give similar terms :.

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

(6) We substitute these conditions , and carefully carry out final calculations.

Answer:

The negative value of the scalar product states the fact that the angle between vectors is blunt.

Task typical, here is an example for an independent solution:

Example 4.

Find a scalar product of vectors and, if you know that .

Now another common task is just a new vector length formula. The designations here will be a bit coincide, so for clarity I will rewrite it with another letter:

Example 5.

Find the length of the vector if .

Decision It will be as follows:

(1) We supply the expression of the vector.

(2) Using the length formula:, while as the vector "VE", we have an integer expression.

(3) We use the Summer Summary Summary Formula. Please note how it works curious here: "In fact, this is a square of the difference, and, in fact, it is." Those who wish can rearrange the vectors in places: - It turned out the same with the accuracy of the alkalis.

(4) Further is already familiar from the two previous tasks.

Answer:

If you are talking about the length, do not forget to specify the dimension - "units".

Example 6.

Find the length of the vector if .

This is an example for an independent solution. Complete solution and answer at the end of the lesson.

We continue to squeeze the useful things from the scalar product. Again let's look at our formula . According to the rule of proportion to reset the length of the vectors in the denominator of the left side:

And parts will change places:

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

Scalar product is a number? Number. Vector length - numbers? Numbers. So, the fraction is also some number. And if the cosine of the corner is known: , it is easy to find an angle itself using reverse function: .

Example 7.

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

Decision: We use the formula:

At the final stage of calculations, the technical reception was used - the elimination of irrationality in the denominator. In order to eliminate the irrationality, I Domnoved the Nizer and the denominator on.

So, if , then:

Values \u200b\u200bof inverse trigonometric functions can be found by trigonometric table. Although it happens rarely. In the tasks of the analytical geometry, some kind of vague bear seems to appear much more often, and the angle value has to find approximately using the calculator. Actually, we will still repeat such a picture.

Answer:

Again, do not forget to indicate the dimension - radians and degrees. Personally, I will be sure to "remove all questions", I prefer to indicate both that (if, by condition, of course, it is not necessary to present the answer only in radians or only in degrees).

Now you can cope with a more complex task:

Example 7 *

Danies - the lengths of the vectors, and the angle between them. Find the angle between vectors ,.

The task is not even so complicated as multiple.
We will analyze the solution algorithm:

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

2) Find a scalar product (see examples number 3, 4).

3) we find the length of the vector and the length of the vector (see examples number 5, 6).

4) The ending of the decision coincides with example number 7 - we know the number, and therefore it is easy to find an angle itself:

A brief 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.

Scalar product of vectors,
asked coordinates in the orthonormal basis

Answer:

What to say, to deal with coordinates is much more pleasant.

Example 14.

Find a scalar product of vectors and if

This is an example for an independent solution. Here you can use the associativity of the operation, that is, do not count, but immediately bring the top three outside the scalar product and upgrade to it last. Solution and answer at the end of the lesson.

In the conclusion of the paragraph provocative example on calculating the length of the vector:

Example 15.

Find length vectors , if a

Decision:the method of the previous section appears again: but there is another road:

Find a vector:

And its length in the trivial formula :

The scalar product is not here at all.

Not as it does not, when calculating the length of the vector:
Stop. Do not take advantage of the obvious property of the vector length? What can be said about the length of the vector? This vector is longer vector 5 times. The direction is the opposite, but it does not play a role, because talk about length. Obviously, the length of the vector is equal to the work module Numbers for the length of the vector:
- The sign of the module "eats" a possible minus number.

In this way:

Answer:

The cosine formula of the angle between the vectors that are set by the coordinates

Now we have complete information to be previously derived from the Cosine Cosine formula between the vectors through the coordinates of the vectors:

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

Cosine angle between space vectors defined in the orthonormal basis formula is expressed:

Example 16.

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

Decision:By condition, the drawing is not required, but still:

The desired angle is marked with a green arc. Immediately remember the school designation of the corner: - special attention to middle The letter is the top of the corner you need. For brevity, it was also possible to record simply.

It is clear from the drawing that the triangle angle coincides with the angle between vectors and, in other words: .

The analysis is preferably learning to perform mentally.

Find vectors:

We calculate the scalar product:

And the length of the vectors:

Cosine corner:

It is this procedure for performing the task that recommend teapots. More prepared readers can record the calculations of the "one line":

Here is an example of a "bad" cosine value. The value obtained is not final, so there is no particular sense to get rid of irrationality in the denominator.

Find the angle itself:

If you look at the drawing, the result is quite believable. To check the angle can also be measured and the transporter. Do not damage the monitor coating \u003d)

Answer:

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

Those who have enjoyed the process can calculate the angles, and make sure the justice of canonical equality

Example 17.

The space is given by the triangle coordinates of their vertices. Find the angle between the parties and

This is an example for an independent solution. Complete solution and answer at the end of the lesson

The small final section will be devoted to the projections in which the scalar product is also "involved":

Vector projection on vector. Projection of vector on coordinate axes.
Cosine guides vector

Consider vectors and:

Sprogit vector on the vector, for this, out of the beginning and end of the vector omit perpendiculary on the vector (green dashed lines). Imagine that the rays of light perpendicularly fall into the vector. Then the segment (red line) will be a "shadow" of the vector. In this case, the vector projection on the vector is the length of the segment. That is, the projection is a number.

This number is indicated as follows:, "Large vector" indicate vector WHICH THE projection, "a small substrate vector" indicate vector ON THE which is projected.

The record itself is read like this: "The projection of the vector" A "on the BE vector."

What happens if the BE vector is "too short"? We carry out a straight line containing the BE vector. And the vector "A" will be projected already on the direction of the vector "BE"Simply - on a straight line containing the BE vector. The same happens if the vector "A" is postponed in the thirtieth of the kingdom - it is still easily disroached to a straight line containing the BE vector.

If corner between vectors acute (as in the figure), then

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

If corner between vectors stupid(In the figure, mentally rearrange the arrow of the vector), then (the same length, but taken with a minus sign).

I will postpone these vectors from one point:

Obviously, when moving the vector, his projection does not change

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

If in the task and length of the vectors, and the angle between them is presented "on a saucer with a blue drive", then the condition of the problem and its solution look like this:

Example 1.Vast vectors. Find a scalar product of vectors if their lengths and angle between them are presented in the following meaning:

Another definition is also defined, fully equivalent to definition 1.

Definition 2.. The scalar product of the vectors is called the number (scalar), equal to the length of the length of these these vectors on the projection of another vector on the axis determined by the first of the specified vectors. Formula according to definition 2:

The task with the use of this formula is solved after the next important theoretical point.

Determination of a scalar product of vectors through coordinates

The same number can be obtained if the variable vectors are set by their coordinates.

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

On surface

If two versions and on the plane are defined by their two cartesian rectangular coordinates

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

.

Example 2.Find the numerical magnitude of the vector projection on the axis parallel to the vector.

Decision. We find a scalar product of vectors, folding the pairwise works of their coordinates:

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

We find the length of the vector as a square root from the sum of the squares of its coordinate:

.

We compile equation and solve it:

Answer. The desired numerical value is minus 8.

In space

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

,

the scalar product of these vectors is also equal to the sum of the pairing works of their respective coordinates, only the coordinates are already three:

.

The task of finding a scalar product with the considered method - after parsing the properties of the scalar product. Because the task will need to determine which angle form variable vectors.

Properties of the scalar product of vectors

Algebraic properties

1. (move property: From change in places of variable vectors, the magnitude of their scalar product does not change).

2. (cOMBULATOR TELEPHONE NUMBER PROPORT: The scalar product of the vector multiplied by some multiplier, and another vector, equal to the scalar product of these vectors, multiplied by the same factor).

3. (distribution relative to the sum of the vectors property: The scalar product of the sum of two vectors on the third vector is equal to the sum of the scalar works of the first vector on the third vector and the second vector on the third vector).

4. (scalar Square Vector More Zero), if it is a nonzero vector, and, if - zero vector.

Geometric properties

In the definitions of the surchated operation, we already concerned the concept of angle between two vectors. It's time to clarify this concept.

The figure above shows two vector, which are shown to the general beginning. And the first thing to pay attention to: there are two angle between these vectors - φ 1 and φ 2 . Which of these angles appears in the definitions and properties of the scalar product of the vectors? The amount of the considered angles is equal to 2 π And therefore, the cosines of these angles are equal. The definition of the scalar product includes only the cosine of the angle, and not the meaning of its expression. But only one angle is considered in the properties. And this is one of two angles that does not exceed π , that is, 180 degrees. In the picture, this angle is indicated as φ 1 .

1. Two vectors call orthogonal and the angle between these vectors - direct (90 degrees or π / 2) if scalar product of these vectors is zero :

.

Orthodalism in vector algebra is the perpendicularity of two vectors.

2. Two non-zero vector constitute sharp corner (from 0 to 90 degrees, or, which is the same - less π scalar product positively .

3. Two non-zero vectors constitute obtuse angle (from 90 to 180 degrees, or that the same thing is more π / 2) if and only when they scalar product negative .

Example 3. Vectors are given in the coordinates:

.

Calculate scalar works of all pairs of these vectors. What angle (sharp, straight, stupid) form these pairs of vectors?

Decision. Calculate will be the addition of the works of the relevant coordinates.

Received a negative number, so the vectors form a stupid angle.

Received a positive number, so the vectors form a sharp angle.

Received zero, so the vectors form a straight corner.

Received a positive number, so the vectors form a sharp angle.

.

Received a positive number, so the vectors form a sharp angle.

For self-test you can use online calculator Scalar product of vectors and cosine corners between them .

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

.

Determine, with what the value of the number of vectors and orthogonal (perpendicular).

Decision. Move the vectors according to the rule of multiplication of polynomials:

Now we calculate each term:

.

Make an equation (equality of the work of zero), we present similar members and solving the equation:

Answer: We got a value λ \u003d 1.8, in which the vectors are orthogonal.

Example 5.Prove that vector orthogonal (perpendicular) vector

Decision. To check orthogonality, variable vectors and as polynomials, substituting instead of its expression given in the Terk Condition:

.

To do this, each member (the term) of the first polynomials multiply to each member of the second and the obtained works are folded:

.

In the result resulting, the fraction is reduced at the expense. The following result is obtained:

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

Solve the task yourself, and then see the decision

Example 6. The lengths of the vectors are given and, and the angle between these vectors is equal π /four . Determine with what value μ Vectors and mutually perpendicular.

For self-test you can use online calculator Scalar product of vectors and cosine corners between them .

Matrix representation of the scalar product of vectors and the product of n-dimensional vectors

Sometimes a winning for clarity is the representation of two variable vectors in the form of matrices. Then the first vector is represented as a matrix string, and the second - in the form of a column matrix:

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

The result is the same as the method obtained, which we have already considered. We received one single number, and the product of the matrix string on the column matrix is \u200b\u200balso one single number.

In matrix form it is convenient to represent the product of abstract n-dimensional vectors. Thus, the product of two four-dimensional vectors will be the product of the matrix string with four elements on the column matrix also with four elements, the product of two five-dimensional vectors - the product of the matrix string with five elements on the column matrix also with five elements and so on.

Example 7. Find scalar works of steam vectors

,

using a matrix representation.

Decision. First pair of vectors. We present the first vector in the form of a matrix string, and the second - in the form of a column matrix. We find a scalar product of these vectors as a product of a matrix string on a column matrix:

Similarly, we present the second pair and find:

As we can see, the results turned out the same as the same pairs from Example 2.

Corner between two vectors

The output of the cosine formula of the corner between two vectors is very beautiful and brief.

To express a scalar product of vectors

(1)

in coordinate form, we will preliminarily find the scalar product of ort. The scalar product of the vector on itself by definition:

What is recorded in the formula above means: the scalar product of the vector on 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

parly perpendicular, then pair works of orts will be zero:

Now perform multiplication of vector polynomials:

We substitute the equality of the values \u200b\u200bof the corresponding scalar works of orthops:

We obtain the cosine formula of the corner between two vectors:

Example 8.Three points are given A.(1;1;1), B.(2;2;1), C.(2;1;2).

Find an angle.

Decision. We find the coordinates of the vectors:

,

.

According to the cosine formula, we get:

Hence, .

For self-test you can use online calculator Scalar product of vectors and cosine corners between them .

Example 9.Dana two vectors

Find amount, difference, length, scalar product and angle between them.

2. Dism

I. The scalar product is drawn to zero in that and only in the case when at least one of the vectors is zero or if the vectors are perpendicular. In fact, if either, or that.

Back, if the variable vectors are not zero, then because of the condition

when it follows:

Since the direction of the zero vector is uncertain, the zero vector can be considered perpendicular to any vector. Therefore, the specified property of the scalar product can be formulated in short: the scalar product is drawn to zero in that and only the case when the vectors are perpendicular.

II. The scalar product has a property of moving:

This property directly follows from the definition:

because the various designations of the same corner.

III. The distributional law is extremely important. Its use is as large as in the usual arithmetic or algebra, where it is formulated as: to multiply the amount, you need to multiply each well and folded the obtained works, i.e..

Obviously, multiplication of multivalued numbers in arithmetic or polynomials in algebra is based on this property of multiplication.

This law has the same major importance in the vector algebra, since on the basis of it we can apply to vectors the usual multiplication rule of polynomials.

We prove that for any three vectors A, B, with equality

According to the second definition of the scalar product, expressed by the formula, we obtain:

Applying now property of 2 projections from § 5, we find:

q.E.D.

IV. The scalar product has the accuracy of the combination relative to the numerical factor; This property is expressed as follows:

i.e. to multiply the scalar product of vectors by the number, it is enough to multiply by this number one of the factors.