Vectors for dummies. Actions with vectors

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Question 1. What is a vector? How are vectors defined?
Answer. We will call a directed segment a vector (Fig. 211). The direction of a vector is determined by specifying its beginning and end. In the drawing, the direction of the vector is marked with an arrow. To designate vectors, we will use lowercase Latin letters a, b, c, ... . You can also designate a vector by specifying its start and end. In this case, the beginning of the vector is placed in the first place. Instead of the word "vector", an arrow or a dash is sometimes placed above the letter designation of the vector. The vector in figure 211 can be denoted as follows:

\(\overline(a)\), \(\overrightarrow(a)\) or \(\overline(AB)\), \(\overrightarrow(AB)\).

Question 2. What vectors are called equally directed (oppositely directed)?
Answer. The vectors \(\overline(AB)\) and \(\overline(CD)\) are said to be equally directed if the half-lines AB and CD are equally directed.
The vectors \(\overline(AB)\) and \(\overline(CD)\) are called oppositely directed if the half-lines AB and CD are oppositely directed.
In Figure 212, the vectors \(\overline(a)\) and \(\overline(b)\) have the same direction, while the vectors \(\overline(a)\) and \(\overline(c)\) have opposite directions.

Question 3. What is the absolute value of a vector?
Answer. The absolute value (or modulus) of a vector is the length of the segment representing the vector. The absolute value of the vector \(\overline(a)\) is denoted by |\(\overline(a)\)|.

Question 4. What is a null vector?
Answer. The beginning of a vector can coincide with its end. Such a vector will be called a zero vector. The zero vector is denoted by zero with a dash (\(\overline(0)\)). No one talks about the direction of the zero vector. The absolute value of the zero vector is considered equal to zero.

Question 5. What vectors are called equal?
Answer. Two vectors are said to be equal if they are combined by a parallel translation. This means that there is a parallel translation that translates the beginning and end of one vector to the beginning and end of another vector, respectively.

Question 6. Prove that equal vectors have the same direction and are equal in absolute value. And vice versa: equally directed vectors that are equal in absolute value are equal.
Answer. With parallel translation, the vector retains its direction, as well as its absolute value. This means that equal vectors have the same direction and are equal in absolute value.
Let \(\overline(AB)\) and \(\overline(CD)\) be equally directed vectors equal in absolute value (Fig. 213). A parallel translation that takes point C to point A combines half-line CD with half-line AB, since they are equally directed. And since the segments AB and CD are equal, then the point D coincides with the point B, i.e. parallel translation translates the vector \(\overline(CD)\) into the vector \(\overline(AB)\). Hence, the vectors \(\overline(AB)\) and \(\overline(CD)\) are equal, as required.

Question 7. Prove that from any point one can draw a vector equal to the given vector, and only one.
Answer. Let CD be a line and the vector \(\overline(CD)\) be a part of line CD. Let AB be the line into which the line CD goes during parallel translation, \(\overline(AB)\) be the vector into which the vector \(\overline(CD)\) goes into during parallel translation, and hence the vectors \(\ overline(AB)\) and \(\overline(CD)\) are equal, and lines AB and CD are parallel (see Fig. 213). As we know, through a point not lying on a given line, it is possible to draw on the plane at most one line parallel to the given one (the axiom of parallel lines). Hence, through the point A one can draw one line parallel to the line CD. Since the vector \(\overline(AB)\) is part of the line AB, it is possible to draw one vector \(\overline(AB)\) through the point A, which is equal to the vector \(\overline(CD)\).

Question 8. What are vector coordinates? What is the absolute value of the vector with coordinates a 1 , a 2 ?
Answer. Let the vector \(\overline(a)\) start at point A 1 (x 1 ; y 1) and end at point A 2 (x 2 ; y 2). The coordinates of the vector \(\overline(a)\) will be the numbers a 1 = x 2 - x 1 , a 2 = y 2 - y 1 . We will put the vector coordinates next to the letter designation of the vector, in this case \(\overline(a)\) (a 1 ; a 2) or just \((\overline(a 1 ; a 2 ))\). The zero vector coordinates are equal to zero.
From the formula expressing the distance between two points in terms of their coordinates, it follows that the absolute value of the vector with coordinates a 1 , a 2 is \(\sqrt(a^2 1 + a^2 2 )\).

Question 9. Prove that equal vectors have respectively equal coordinates, and vectors with respectively equal coordinates are equal.
Answer. Let A 1 (x 1 ; y 1) and A 2 (x 2 ; y 2) be the beginning and end of the vector \(\overline(a)\). Since the vector \(\overline(a")\) equal to it is obtained from the vector \(\overline(a)\) by parallel translation, then its beginning and end will be respectively A" 1 (x 1 + c; y 1 + d ), A" 2 (x 2 + c; y 2 ​​+ d). This shows that both vectors \(\overline(a)\) and \(\overline(a")\) have the same coordinates: x 2 - x 1 , y 2 - y 1 .
Let us now prove the converse assertion. Let the corresponding coordinates of the vectors \(\overline(A 1 A 2 )\) and \(\overline(A" 1 A" 2 )\) be equal. We prove that the vectors are equal.
Let x" 1 and y" 1 be the coordinates of the point A" 1, and x" 2, y" 2 be the coordinates of the point A" 2. By the condition of the theorem x 2 - x 1 \u003d x "2 - x" 1, y 2 - y 1 \u003d y "2 - y" 1. Hence x "2 = x 2 + x" 1 - x 1, y" 2 = y 2 + y" 1 - y 1. Parallel translation given by formulas

x" = x + x" 1 - x 1, y" = y + y" 1 - y 1,

transfers point A 1 to point A" 1 , and point A 2 to point A" 2 , i.e. the vectors \(\overline(A 1 A 2 )\) and \(\overline(A" 1 A" 2 )\) are equal, as required.

Question 10. Define the sum of vectors.
Answer. The sum of the vectors \(\overline(a)\) and \(\overline(b)\) with coordinates a 1 , a 2 and b 1 , b 2 is the vector \(\overline(c)\) with coordinates a 1 + b 1 , a 2 + b a 2 , i.e.

\(\overline(a) (a 1 ; a 2) + \overline(b)(b 1 ; b 2) = \overline(c) (a 1 + b 1 ; a 2 + b 2)\).

The vector \(\overrightarrow(AB)\) can be viewed as moving a point from position \(A\) (start of movement) to position \(B\) (end of movement). That is, the trajectory of movement in this case is not important, only the beginning and end are important!

\(\blacktriangleright\) Two vectors are collinear if they lie on the same line or on two parallel lines.
Otherwise, the vectors are called non-collinear.

\(\blacktriangleright\) Two collinear vectors are said to be codirectional if their directions are the same.
If their directions are opposite, then they are called oppositely directed.

Rules for adding collinear vectors:

co-directional end first. Then their sum is a vector whose beginning coincides with the beginning of the first vector, and the end coincides with the end of the second (Fig. 1).

\(\blacktriangleright\) To add two opposite directions vector, you can postpone the second vector from start first. Then their sum is a vector, the beginning of which coincides with the beginning of both vectors, the length is equal to the difference in the lengths of the vectors, the direction coincides with the direction of the longer vector (Fig. 2).

Rules for adding non-collinear vectors \(\overrightarrow (a)\) and \(\overrightarrow(b)\) :

\(\blacktriangleright\) Triangle rule (Fig. 3).

It is necessary to postpone the vector \(\overrightarrow (b)\) from the end of the vector \(\overrightarrow (a)\) . Then the sum is a vector whose beginning coincides with the beginning of the vector \(\overrightarrow (a)\) , and whose end coincides with the end of the vector \(\overrightarrow (b)\) .

\(\blacktriangleright\) Parallelogram rule (Fig. 4).

It is necessary to postpone the vector \(\overrightarrow (b)\) from the beginning of the vector \(\overrightarrow (a)\) . Then the sum \(\overrightarrow (a)+\overrightarrow (b)\) is a vector coinciding with the diagonal of the parallelogram built on the vectors \(\overrightarrow (a)\) and \(\overrightarrow (b)\) (the beginning of which coincides with the beginning of both vectors).

\(\blacktriangleright\) To find the difference of two vectors \(\overrightarrow(a)-\overrightarrow(b)\), you need to find the sum of the vectors \(\overrightarrow (a)\) and \(-\overrightarrow(b)\) : \(\overrightarrow(a)-\overrightarrow(b)=\overrightarrow(a)+(-\overrightarrow(b))\)(Fig. 5).

Task 1 #2638

Task level: More difficult than the exam

Given a right triangle \(ABC\) with a right angle \(A\) , point \(O\) is the center of the circumscribed circle around the given triangle. Vector coordinates \(\overrightarrow(AB)=\(1;1\)\), \(\overrightarrow(AC)=\(-1;1\)\). Find the sum of the coordinates of the vector \(\overrightarrow(OC)\) .

Because the triangle \(ABC\) is right-angled, then the center of the circumscribed circle lies in the middle of the hypotenuse, i.e. \(O\) is the middle of \(BC\) .

notice, that \(\overrightarrow(BC)=\overrightarrow(AC)-\overrightarrow(AB)\), hence, \(\overrightarrow(BC)=\(-1-1;1-1\)=\(-2;0\)\).

Because \(\overrightarrow(OC)=\dfrac12 \overrightarrow(BC)\), then \(\overrightarrow(OC)=\(-1;0\)\).

Hence, the sum of the coordinates of the vector \(\overrightarrow(OC)\) is equal to \(-1+0=-1\) .

Answer: -1

Task 2 #674

Task level: More difficult than the exam

\(ABCD\) is a quadrilateral whose sides contain the vectors \(\overrightarrow(AB)\) , \(\overrightarrow(BC)\) , \(\overrightarrow(CD)\) , \(\overrightarrow(DA) \) . Find the length of the vector \(\overrightarrow(AB) + \overrightarrow(BC) + \overrightarrow(CD) + \overrightarrow(DA)\).

\(\overrightarrow(AB) + \overrightarrow(BC) = \overrightarrow(AC)\), \(\overrightarrow(AC) + \overrightarrow(CD) = \overrightarrow(AD)\), then
\(\overrightarrow(AB) + \overrightarrow(BC) + \overrightarrow(CD) + \overrightarrow(DA) = \overrightarrow(AC) + \overrightarrow(CD) + \overrightarrow(DA)= \overrightarrow(AD) + \overrightarrow(DA) = \overrightarrow(AD) - \overrightarrow(AD) = \vec(0)\).
The null vector has length equal to \(0\) .

A vector can be thought of as a displacement, then \(\overrightarrow(AB) + \overrightarrow(BC)\)- move from \(A\) to \(B\) , and then from \(B\) to \(C\) - in the end it is a move from \(A\) to \(C\) .

With this interpretation, it becomes clear that \(\overrightarrow(AB) + \overrightarrow(BC) + \overrightarrow(CD) + \overrightarrow(DA) = \vec(0)\), because as a result, here we moved from the point \(A\) to the point \(A\) , that is, the length of such a movement is equal to \(0\) , which means that the vector of such a movement itself is \(\vec(0)\) .

Answer: 0

Task 3 #1805

Task level: More difficult than the exam

Given a parallelogram \(ABCD\) . The diagonals \(AC\) and \(BD\) intersect at the point \(O\) . Let, then \(\overrightarrow(OA) = x\cdot\vec(a) + y\cdot\vec(b)\)

\[\overrightarrow(OA) = \frac(1)(2)\overrightarrow(CA) = \frac(1)(2)(\overrightarrow(CB) + \overrightarrow(BA)) = \frac(1)( 2)(\overrightarrow(DA) + \overrightarrow(BA)) = \frac(1)(2)(-\vec(b) - \vec(a)) = - \frac(1)(2)\vec (a) - \frac(1)(2)\vec(b)\]\(\Rightarrow\) \(x = - \frac(1)(2)\) , \(y = - \frac(1)(2)\) \(\Rightarrow\) \(x + y = - one\) .

Answer: -1

Task 4 #1806

Task level: More difficult than the exam

Given a parallelogram \(ABCD\) . The points \(K\) and \(L\) lie on the sides \(BC\) and \(CD\), respectively, and \(BK:KC = 3:1\) , and \(L\) is the midpoint \ (CD\) . Let be \(\overrightarrow(AB) = \vec(a)\), \(\overrightarrow(AD) = \vec(b)\), then \(\overrightarrow(KL) = x\cdot\vec(a) + y\cdot\vec(b)\), where \(x\) and \(y\) are some numbers. Find the number equal to \(x + y\) .

\[\overrightarrow(KL) = \overrightarrow(KC) + \overrightarrow(CL) = \frac(1)(4)\overrightarrow(BC) + \frac(1)(2)\overrightarrow(CD) = \frac (1)(4)\overrightarrow(AD) + \frac(1)(2)\overrightarrow(BA) = \frac(1)(4)\vec(b) - \frac(1)(2)\vec (a)\]\(\Rightarrow\) \(x = -\frac(1)(2)\) , \(y = \frac(1)(4)\) \(\Rightarrow\) \(x + y = -0 ,25\) .

Answer: -0.25

Task 5 #1807

Task level: More difficult than the exam

Given a parallelogram \(ABCD\) . The points \(M\) and \(N\) lie on the sides \(AD\) and \(BC\) respectively, where \(AM:MD = 2:3\) and \(BN:NC = 3): one\) . Let be \(\overrightarrow(AB) = \vec(a)\), \(\overrightarrow(AD) = \vec(b)\), then \(\overrightarrow(MN) = x\cdot\vec(a) + y\cdot\vec(b)\)

\[\overrightarrow(MN) = \overrightarrow(MA) + \overrightarrow(AB) + \overrightarrow(BN) = \frac(2)(5)\overrightarrow(DA) + \overrightarrow(AB) + \frac(3 )(4)\overrightarrow(BC) = - \frac(2)(5)\overrightarrow(AD) + \overrightarrow(AB) + \frac(3)(4)\overrightarrow(BC) = -\frac(2 )(5)\vec(b) + \vec(a) + \frac(3)(4)\vec(b) = \vec(a) + \frac(7)(20)\vec(b)\ ]\(\Rightarrow\) \(x = 1\) , \(y = \frac(7)(20)\) \(\Rightarrow\) \(x\cdot y = 0.35\) .

Answer: 0.35

Task 6 #1808

Task level: More difficult than the exam

Given a parallelogram \(ABCD\) . The point \(P\) lies on the diagonal \(BD\) , the point \(Q\) lies on the side \(CD\) , where \(BP:PD = 4:1\) , and \(CQ:QD = 1:9 \) . Let be \(\overrightarrow(AB) = \vec(a)\), \(\overrightarrow(AD) = \vec(b)\), then \(\overrightarrow(PQ) = x\cdot\vec(a) + y\cdot\vec(b)\), where \(x\) and \(y\) are some numbers. Find the number equal to \(x\cdot y\) .

\[\begin(gathered) \overrightarrow(PQ) = \overrightarrow(PD) + \overrightarrow(DQ) = \frac(1)(5)\overrightarrow(BD) + \frac(9)(10)\overrightarrow( DC) = \frac(1)(5)(\overrightarrow(BC) + \overrightarrow(CD)) + \frac(9)(10)\overrightarrow(AB) =\\ = \frac(1)(5) (\overrightarrow(AD) + \overrightarrow(BA)) + \frac(9)(10)\overrightarrow(AB) = \frac(1)(5)(\overrightarrow(AD) - \overrightarrow(AB)) + \frac(9)(10)\overrightarrow(AB) = \frac(1)(5)\overrightarrow(AD) + \frac(7)(10)\overrightarrow(AB) = \frac(1)(5) \vec(b) + \frac(7)(10)\vec(a)\end(gathered)\]

\(\Rightarrow\) \(x = \frac(7)(10)\) , \(y = \frac(1)(5)\) \(\Rightarrow\) \(x\cdot y = 0, fourteen\) . and \(ABCO\) is a parallelogram; \(AF \parallel BE\) and \(ABOF\) – parallelogram \(\Rightarrow\) \[\overrightarrow(BC) = \overrightarrow(AO) = \overrightarrow(AB) + \overrightarrow(BO) = \overrightarrow(AB) + \overrightarrow(AF) = \vec(a) + \vec(b)\ ]\(\Rightarrow\) \(x = 1\) , \(y = 1\) \(\Rightarrow\) \(x + y = 2\) .

Answer: 2

High school students who are preparing for the exam in mathematics and at the same time count on getting decent scores must definitely repeat the topic "Rules for adding and subtracting several vectors." As can be seen from many years of practice, such tasks are included in the certification test every year. If a graduate has difficulties with tasks from the “Geometry on a Plane” section, for example, in which it is required to apply the rules of addition and subtraction of vectors, he should definitely repeat or re-understand the material in order to successfully pass the exam.

The educational project "Shkolkovo" offers a new approach to preparing for the certification test. Our resource is built in such a way that students can identify the most difficult sections for themselves and fill in knowledge gaps. Shkolkovo specialists have prepared and systematized all the necessary material to prepare for the certification test.

In order for the USE tasks, in which it is necessary to apply the rules of addition and subtraction of two vectors, to not cause difficulties, we recommend that you first of all refresh the basic concepts in your memory. Students can find this material in the "Theoretical Reference" section.

If you have already remembered the vector subtraction rule and the basic definitions on this topic, we suggest that you consolidate your knowledge by completing the appropriate exercises that were selected by the specialists of the Shkolkovo educational portal. For each problem, the site presents a solution algorithm and gives the correct answer. The Vector Addition Rules topic contains various exercises; after completing two or three relatively easy tasks, students can successively move on to more difficult ones.

To hone their own skills in such tasks, for example, as schoolchildren have the opportunity online, being in Moscow or any other city in Russia. If necessary, the task can be saved in the "Favorites" section. Thanks to this, you can quickly find examples of interest and discuss the algorithms for finding the correct answer with the teacher.

Standard definition: "A vector is a directed line segment." This is usually the limit of a graduate's knowledge of vectors. Who needs some kind of "directed segments"?

But in fact, what are vectors and why are they?
Weather forecast. "Wind northwest, speed 18 meters per second." Agree, the direction of the wind (where it blows from) and the module (that is, the absolute value) of its speed also matter.

Quantities that have no direction are called scalars. Mass, work, electric charge are not directed anywhere. They are characterized only by a numerical value - “how many kilograms” or “how many joules”.

Physical quantities that have not only an absolute value, but also a direction are called vector quantities.

Speed, force, acceleration - vectors. For them, it is important "how much" and it is important "where". For example, the free fall acceleration is directed towards the Earth's surface, and its value is 9.8 m/s 2 . Momentum, electric field strength, magnetic field induction are also vector quantities.

You remember that physical quantities are denoted by letters, Latin or Greek. The arrow above the letter indicates that the quantity is a vector:

Here is another example.
The car is moving from A to B. The end result is its movement from point A to point B, i.e. movement by a vector .

Now it is clear why a vector is a directed segment. Pay attention, the end of the vector is where the arrow is. Vector length is called the length of this segment. Designated: or

So far, we have been working with scalar quantities, according to the rules of arithmetic and elementary algebra. Vectors are a new concept. This is another class of mathematical objects. They have their own rules.

Once upon a time, we didn’t even know about numbers. Acquaintance with them began in elementary grades. It turned out that numbers can be compared with each other, added, subtracted, multiplied and divided. We learned that there is a number one and a number zero.
Now we get to know vectors.

The concepts of "greater than" and "less than" do not exist for vectors - after all, their directions can be different. You can only compare the lengths of vectors.

But the concept of equality for vectors is.
Equal are vectors that have the same length and the same direction. This means that the vector can be moved parallel to itself to any point in the plane.
single is called a vector whose length is 1 . Zero - a vector whose length is equal to zero, that is, its beginning coincides with the end.

It is most convenient to work with vectors in a rectangular coordinate system - the one in which we draw function graphs. Each point in the coordinate system corresponds to two numbers - its x and y coordinates, abscissa and ordinate.
The vector is also given by two coordinates:

Here, the coordinates of the vector are written in brackets - in x and in y.
They are easy to find: the coordinate of the end of the vector minus the coordinate of its beginning.

If the vector coordinates are given, its length is found by the formula

Vector addition

There are two ways to add vectors.

one . parallelogram rule. To add the vectors and , we place the origins of both at the same point. We complete the parallelogram and draw the diagonal of the parallelogram from the same point. This will be the sum of the vectors and .

Remember the fable about the swan, cancer and pike? They tried very hard, but they never moved the cart. After all, the vector sum of the forces applied by them to the cart was equal to zero.

2. The second way to add vectors is the triangle rule. Let's take the same vectors and . We add the beginning of the second to the end of the first vector. Now let's connect the beginning of the first and the end of the second. This is the sum of the vectors and .

By the same rule, you can add several vectors. We attach them one by one, and then connect the beginning of the first to the end of the last.

Imagine that you are going from point A to point B, from B to C, from C to D, then to E and then to F. The end result of these actions is a move from A to F.

When adding vectors and we get:

Vector subtraction

The vector is directed opposite to the vector . The lengths of the vectors and are equal.

Now it is clear what subtraction of vectors is. The difference of the vectors and is the sum of the vector and the vector .

Multiply a vector by a number

Multiplying a vector by a number k results in a vector whose length is k times different from the length . It is codirectional with the vector if k is greater than zero, and directed oppositely if k is less than zero.

Dot product of vectors

Vectors can be multiplied not only by numbers, but also by each other.

The scalar product of vectors is the product of the lengths of vectors and the cosine of the angle between them.

Pay attention - we multiplied two vectors, and we got a scalar, that is, a number. For example, in physics, mechanical work is equal to the scalar product of two vectors - force and displacement:

If the vectors are perpendicular, their dot product is zero.
And this is how the scalar product is expressed in terms of the coordinates of the vectors and:

From the formula for the scalar product, you can find the angle between the vectors:

This formula is especially convenient in stereometry. For example, in problem 14 of the Profile USE in mathematics, you need to find the angle between intersecting lines or between a line and a plane. Problem 14 is often solved several times faster than the classical one.

In the school curriculum in mathematics, only the scalar product of vectors is studied.
It turns out that, in addition to the scalar, there is also a vector product, when a vector is obtained as a result of multiplying two vectors. Who passes the exam in physics, knows what the Lorentz force and the Ampère force are. The formulas for finding these forces include exactly vector products.

Vectors are a very useful mathematical tool. You will be convinced of this in the first course.

Finally, I got my hands on an extensive and long-awaited topic analytical geometry. First, a little about this section of higher mathematics…. Surely you now remembered the school geometry course with numerous theorems, their proofs, drawings, etc. What to hide, an unloved and often obscure subject for a significant proportion of students. Analytic geometry, oddly enough, may seem more interesting and accessible. What does the adjective "analytical" mean? Two stamped mathematical turns immediately come to mind: “graphic method of solution” and “analytical method of solution”. Graphic method, of course, is associated with the construction of graphs, drawings. Analytical same method involves problem solving predominantly through algebraic operations. In this regard, the algorithm for solving almost all problems of analytical geometry is simple and transparent, often it is enough to accurately apply the necessary formulas - and the answer is ready! No, of course, it will not do without drawings at all, besides, for a better understanding of the material, I will try to bring them in excess of the need.

The open course of lessons in geometry does not claim to be theoretical completeness, it is focused on solving practical problems. I will include in my lectures only what, from my point of view, is important in practical terms. If you need a more complete reference on any subsection, I recommend the following quite accessible literature:

1) A thing that, no joke, is familiar to several generations: School textbook on geometry, the authors - L.S. Atanasyan and Company. This school locker room hanger has already withstood 20 (!) reissues, which, of course, is not the limit.

2) Geometry in 2 volumes. The authors L.S. Atanasyan, Bazylev V.T.. This is literature for higher education, you will need first volume. Rarely occurring tasks may fall out of my field of vision, and the tutorial will be of invaluable help.

Both books are free to download online. In addition, you can use my archive with ready-made solutions, which can be found on the page Download higher mathematics examples.

Of the tools, I again offer my own development - software package on analytical geometry, which will greatly simplify life and save a lot of time.

It is assumed that the reader is familiar with basic geometric concepts and figures: point, line, plane, triangle, parallelogram, parallelepiped, cube, etc. It is advisable to remember some theorems, at least the Pythagorean theorem, hello repeaters)

And now we will sequentially consider: the concept of a vector, actions with vectors, vector coordinates. Further I recommend reading the most important article Dot product of vectors, as well as Vector and mixed product of vectors. The local task will not be superfluous - Division of the segment in this regard. Based on the above information, you can equation of a straight line in a plane with the simplest examples of solutions, which will allow learn how to solve problems in geometry. The following articles are also helpful: Equation of a plane in space, Equations of a straight line in space, Basic problems on the line and plane , other sections of analytic geometry. Naturally, standard tasks will be considered along the way.

The concept of a vector. free vector

First, let's repeat the school definition of a vector. Vector called directed a segment for which its beginning and end are indicated:

In this case, the beginning of the segment is the point , the end of the segment is the point . The vector itself is denoted by . Direction is essential, if you rearrange the arrow to the other end of the segment, you get a vector, and this is already completely different vector. It is convenient to identify the concept of a vector with the movement of a physical body: you must admit that entering the doors of an institute or leaving the doors of an institute are completely different things.

It is convenient to consider individual points of a plane, space as the so-called zero vector. Such a vector has the same end and beginning.

!!! Note: Here and below, you can assume that the vectors lie in the same plane or you can assume that they are located in space - the essence of the material presented is valid for both the plane and space.

Designations: Many immediately drew attention to a stick without an arrow in the designation and said that they also put an arrow at the top! That's right, you can write with an arrow: , but admissible and record that I will use later. Why? Apparently, such a habit has developed from practical considerations, my shooters at school and university turned out to be too diverse and shaggy. In educational literature, sometimes they don’t bother with cuneiform at all, but highlight the letters in bold: , thereby implying that this is a vector.

That was the style, and now about the ways of writing vectors:

1) Vectors can be written in two capital Latin letters:
etc. While the first letter necessarily denotes the start point of the vector, and the second letter denotes the end point of the vector.

2) Vectors are also written in small Latin letters:
In particular, our vector can be redesignated for brevity by a small Latin letter .

Length or module non-zero vector is called the length of the segment. The length of the null vector is zero. Logically.

The length of a vector is denoted by the modulo sign: ,

How to find the length of a vector, we will learn (or repeat, for whom how) a little later.

That was elementary information about the vector, familiar to all schoolchildren. In analytic geometry, the so-called free vector.

If it's quite simple - vector can be drawn from any point:

We used to call such vectors equal (the definition of equal vectors will be given below), but from a purely mathematical point of view, this is the SAME VECTOR or free vector. Why free? Because in the course of solving problems you can “attach” one or another “school” vector to ANY point of the plane or space you need. This is a very cool property! Imagine a directed segment of arbitrary length and direction - it can be "cloned" an infinite number of times and at any point in space, in fact, it exists EVERYWHERE. There is such a student's proverb: Each lecturer in f ** u in the vector. After all, it’s not just a witty rhyme, everything is almost correct - a directed segment can be attached there too. But do not rush to rejoice, students themselves suffer more often =)

So, free vector- This a bunch of identical directional segments. The school definition of a vector, given at the beginning of the paragraph: “A directed segment is called a vector ...”, implies specific a directed segment taken from a given set, which is attached to a certain point in the plane or space.

It should be noted that from the point of view of physics, the concept of a free vector is generally incorrect, and the point of application matters. Indeed, a direct blow of the same force on the nose or on the forehead is enough to develop my stupid example entails different consequences. However, not free vectors are also found in the course of vyshmat (do not go there :)).

Actions with vectors. Collinearity of vectors

In the school geometry course, a number of actions and rules with vectors are considered: addition according to the triangle rule, addition according to the parallelogram rule, the rule of the difference of vectors, multiplication of a vector by a number, the scalar product of vectors, etc. As a seed, we repeat two rules that are especially relevant for solving problems of analytical geometry.

Rule of addition of vectors according to the rule of triangles

Consider two arbitrary non-zero vectors and :

It is required to find the sum of these vectors. Due to the fact that all vectors are considered free, we postpone the vector from end vector :

The sum of vectors is the vector . For a better understanding of the rule, it is advisable to put a physical meaning into it: let some body make a path along the vector , and then along the vector . Then the sum of the vectors is the vector of the resulting path starting at the point of departure and ending at the point of arrival. A similar rule is formulated for the sum of any number of vectors. As they say, the body can go its way strongly zigzag, or maybe on autopilot - along the resulting sum vector.

By the way, if the vector is postponed from start vector , then we get the equivalent parallelogram rule addition of vectors.

First, about the collinearity of vectors. The two vectors are called collinear if they lie on the same line or on parallel lines. Roughly speaking, we are talking about parallel vectors. But in relation to them, the adjective "collinear" is always used.

Imagine two collinear vectors. If the arrows of these vectors are directed in the same direction, then such vectors are called co-directional. If the arrows look in different directions, then the vectors will be oppositely directed.

Designations: collinearity of vectors is written with the usual parallelism icon: , while detailing is possible: (vectors are co-directed) or (vectors are directed oppositely).

work of a nonzero vector by a number is a vector whose length is equal to , and the vectors and are co-directed at and oppositely directed at .

The rule for multiplying a vector by a number is easier to understand with a picture:

We understand in more detail:

1) Direction. If the multiplier is negative, then the vector changes direction to the opposite.

2) Length. If the factor is contained within or , then the length of the vector decreases. So, the length of the vector is twice less than the length of the vector . If the modulo multiplier is greater than one, then the length of the vector increases in time.

3) Please note that all vectors are collinear, while one vector is expressed through another, for example, . The reverse is also true: if one vector can be expressed in terms of another, then such vectors are necessarily collinear. Thus: if we multiply a vector by a number, we get collinear(relative to original) vector.

4) The vectors are codirectional. The vectors and are also codirectional. Any vector of the first group is opposite to any vector of the second group.

What vectors are equal?

Two vectors are equal if they are codirectional and have the same length. Note that co-direction implies that the vectors are collinear. The definition will be inaccurate (redundant) if you say: "Two vectors are equal if they are collinear, co-directed and have the same length."

From the point of view of the concept of a free vector, equal vectors are the same vector, which was already discussed in the previous paragraph.

Vector coordinates on the plane and in space

The first point is to consider vectors on a plane. Draw a Cartesian rectangular coordinate system and set aside from the origin single vectors and :

Vectors and orthogonal. Orthogonal = Perpendicular. I recommend slowly getting used to the terms: instead of parallelism and perpendicularity, we use the words respectively collinearity and orthogonality.

Designation: orthogonality of vectors is written with the usual perpendicular sign, for example: .

The considered vectors are called coordinate vectors or orts. These vectors form basis on surface. What is the basis, I think, is intuitively clear to many, more detailed information can be found in the article Linear (non) dependence of vectors. Vector basis.In simple words, the basis and the origin of coordinates define the entire system - this is a kind of foundation on which a full and rich geometric life boils.

Sometimes the constructed basis is called orthonormal basis of the plane: "ortho" - because the coordinate vectors are orthogonal, the adjective "normalized" means unit, i.e. the lengths of the basis vectors are equal to one.

Designation: the basis is usually written in parentheses, inside which in strict order basis vectors are listed, for example: . Coordinate vectors it is forbidden swap places.

Any plane vector the only way expressed as:
, where - numbers, which are called vector coordinates in this basis. But the expression itself called vector decompositionbasis .

Dinner served:

Let's start with the first letter of the alphabet: . The drawing clearly shows that when decomposing the vector in terms of the basis, the ones just considered are used:
1) the rule of multiplication of a vector by a number: and ;
2) addition of vectors according to the triangle rule: .

Now mentally set aside the vector from any other point on the plane. It is quite obvious that his corruption will "relentlessly follow him." Here it is, the freedom of the vector - the vector "carries everything with you." This property, of course, is true for any vector. It's funny that the basis (free) vectors themselves do not have to be set aside from the origin, one can be drawn, for example, at the bottom left, and the other at the top right, and nothing will change from this! True, you don’t need to do this, because the teacher will also show originality and draw you a “pass” in an unexpected place.

Vectors , illustrate exactly the rule for multiplying a vector by a number, the vector is co-directed with the basis vector , the vector is directed opposite to the basis vector . For these vectors, one of the coordinates is equal to zero, it can be meticulously written as follows:


And the basis vectors, by the way, are like this: (in fact, they are expressed through themselves).

And finally: , . By the way, what is vector subtraction, and why didn't I tell you about the subtraction rule? Somewhere in linear algebra, I don't remember where, I noted that subtraction is a special case of addition. So, the expansions of the vectors "de" and "e" are calmly written as a sum: . Follow the drawing to see how well the good old addition of vectors according to the triangle rule works in these situations.

Considered decomposition of the form sometimes called a vector decomposition in the system ort(i.e. in the system of unit vectors). But this is not the only way to write a vector, the following option is common:

Or with an equals sign:

The basis vectors themselves are written as follows: and

That is, the coordinates of the vector are indicated in parentheses. In practical tasks, all three recording options are used.

I doubted whether to speak, but still I will say: vector coordinates cannot be rearranged. Strictly in first place write down the coordinate that corresponds to the unit vector , strictly in second place write down the coordinate that corresponds to the unit vector . Indeed, and are two different vectors.

We figured out the coordinates on the plane. Now consider vectors in three-dimensional space, everything is almost the same here! Only one more coordinate will be added. It is difficult to perform three-dimensional drawings, so I will limit myself to one vector, which for simplicity I will postpone from the origin:

Any 3d space vector the only way expand in an orthonormal basis:
, where are the coordinates of the vector (number) in the given basis.

Example from the picture: . Let's see how the vector action rules work here. First, multiplying a vector by a number: (red arrow), (green arrow) and (magenta arrow). Secondly, here is an example of adding several, in this case three, vectors: . The sum vector starts at the starting point of departure (the beginning of the vector ) and ends up at the final point of arrival (the end of the vector ).

All vectors of three-dimensional space, of course, are also free, try to mentally postpone the vector from any other point, and you will understand that its expansion "remains with it."

Similarly to the plane case, in addition to writing versions with brackets are widely used: either .

If one (or two) coordinate vectors are missing in the expansion, then zeros are put instead. Examples:
vector (meticulously ) – write down ;
vector (meticulously ) – write down ;
vector (meticulously ) – write down .

Basis vectors are written as follows:

Here, perhaps, is all the minimum theoretical knowledge necessary for solving problems of analytical geometry. Perhaps there are too many terms and definitions, so I recommend dummies to re-read and comprehend this information again. And it will be useful for any reader to refer to the basic lesson from time to time for better assimilation of the material. Collinearity, orthogonality, orthonormal basis, vector decomposition - these and other concepts will be often used in what follows. I note that the materials of the site are not enough to pass a theoretical test, a colloquium on geometry, since I carefully encrypt all theorems (besides without proofs) - to the detriment of the scientific style of presentation, but a plus for your understanding of the subject. For detailed theoretical information, I ask you to bow to Professor Atanasyan.

Now let's move on to the practical part:

The simplest problems of analytic geometry.
Actions with vectors in coordinates

The tasks that will be considered, it is highly desirable to learn how to solve them fully automatically, and the formulas memorize, don't even remember it on purpose, they will remember it themselves =) This is very important, since other problems of analytical geometry are based on the simplest elementary examples, and it will be annoying to spend extra time eating pawns. You do not need to fasten the top buttons on your shirt, many things are familiar to you from school.

The presentation of the material will follow a parallel course - both for the plane and for space. For the reason that all the formulas ... you will see for yourself.

How to find a vector given two points?

If two points of the plane and are given, then the vector has the following coordinates:

If two points in space and are given, then the vector has the following coordinates:

I.e, from the coordinates of the end of the vector you need to subtract the corresponding coordinates vector start.

Exercise: For the same points, write down the formulas for finding the coordinates of the vector. Formulas at the end of the lesson.

Example 1

Given two points in the plane and . Find vector coordinates

Decision: according to the corresponding formula:

Alternatively, the following notation could be used:

Aesthetes will decide like this:

Personally, I'm used to the first version of the record.

Answer:

According to the condition, it was not required to build a drawing (which is typical for problems of analytical geometry), but in order to explain some points to dummies, I will not be too lazy:

Must be understood difference between point coordinates and vector coordinates:

Point coordinates are the usual coordinates in a rectangular coordinate system. I think everyone knows how to plot points on the coordinate plane since grade 5-6. Each point has a strict place on the plane, and they cannot be moved anywhere.

The coordinates of the same vector is its expansion with respect to the basis , in this case . Any vector is free, therefore, if desired or necessary, we can easily postpone it from some other point of the plane (renaming it, for example, through , to avoid confusion). Interestingly, for vectors, you can not build axes at all, a rectangular coordinate system, you only need a basis, in this case, an orthonormal basis of the plane.

The records of point coordinates and vector coordinates seem to be similar: , and sense of coordinates absolutely different, and you should be well aware of this difference. This difference, of course, is also true for space.

Ladies and gentlemen, we fill our hands:

Example 2

a) Given points and . Find vectors and .
b) Points are given and . Find vectors and .
c) Given points and . Find vectors and .
d) Points are given. Find Vectors .

Perhaps enough. These are examples for an independent decision, try not to neglect them, it will pay off ;-). Drawings are not required. Solutions and answers at the end of the lesson.

What is important in solving problems of analytical geometry? It is important to be EXTREMELY CAREFUL in order to avoid the masterful “two plus two equals zero” error. I apologize in advance if I made a mistake =)

How to find the length of a segment?

The length, as already noted, is indicated by the modulus sign.

If two points of the plane and are given, then the length of the segment can be calculated by the formula

If two points in space and are given, then the length of the segment can be calculated by the formula

Note: The formulas will remain correct if the corresponding coordinates are swapped: and , but the first option is more standard

Example 3

Decision: according to the corresponding formula:

Answer:

For clarity, I will make a drawing

Line segment - it's not a vector, and you can't move it anywhere, of course. In addition, if you complete the drawing to scale: 1 unit. \u003d 1 cm (two tetrad cells), then the answer can be checked with a regular ruler by directly measuring the length of the segment.

Yes, the solution is short, but there are a couple of important points in it that I would like to clarify:

First, in the answer we set the dimension: “units”. The condition does not say WHAT it is, millimeters, centimeters, meters or kilometers. Therefore, the general formulation will be a mathematically competent solution: “units” - abbreviated as “units”.

Secondly, let's repeat the school material, which is useful not only for the considered problem:

pay attention to important technical trick – taking the multiplier out from under the root. As a result of the calculations, we got the result and good mathematical style involves taking the multiplier out from under the root (if possible). The process looks like this in more detail: . Of course, leaving the answer in the form will not be a mistake - but it is definitely a flaw and a weighty argument for nitpicking on the part of the teacher.

Here are other common cases:

Often a sufficiently large number is obtained under the root, for example. How to be in such cases? On the calculator, we check if the number is divisible by 4:. Yes, split completely, thus: . Or maybe the number can be divided by 4 again? . Thus: . The last digit of the number is odd, so dividing by 4 for the third time is clearly not possible. Trying to divide by nine: . As a result:
Ready.

Conclusion: if under the root we get a completely non-extractable number, then we try to take out the factor from under the root - on the calculator we check whether the number is divisible by: 4, 9, 16, 25, 36, 49, etc.

In the course of solving various problems, roots are often found, always try to extract factors from under the root in order to avoid a lower score and unnecessary troubles with finalizing your solutions according to the teacher's remark.

Let's repeat the squaring of the roots and other powers at the same time:

The rules for actions with degrees in a general form can be found in a school textbook on algebra, but I think that everything or almost everything is already clear from the examples given.

Task for an independent solution with a segment in space:

Example 4

Given points and . Find the length of the segment.

Solution and answer at the end of the lesson.

How to find the length of a vector?

If a plane vector is given, then its length is calculated by the formula.

If a space vector is given, then its length is calculated by the formula .

These formulas (as well as the formulas for the length of a segment) are easily derived using the notorious Pythagorean theorem.

Before proceeding to the subject of the article, we recall the basic concepts.

Definition 1

Vector- a straight line segment characterized by a numerical value and direction. A vector is denoted by a lowercase Latin letter with an arrow at the top. If there are specific boundary points, the designation of the vector looks like two uppercase Latin letters (marking the boundaries of the vector) also with an arrow on top.

Definition 2

Zero vector- any point of the plane, denoted as zero with an arrow above.

Definition 3

Vector length- a value equal to or greater than zero, which determines the length of the segment that makes up the vector.

Definition 4

Collinear vectors- lying on one line or on parallel lines. Vectors that do not fulfill this condition are called non-collinear.

Definition 5

Input: vectors a → and b →. To perform the addition operation on them, it is necessary to postpone the vector from an arbitrary point A B →, equal to the vector a →; from the received point undefined - vector In C →, equal to the vector b →. By connecting the points undefined and C , we get a segment (vector) A C →, which will be the sum of the original data. Otherwise, the described vector addition scheme is called triangle rule.

Geometrically, vector addition looks like this:

For non-collinear vectors:

For collinear (codirectional or opposite) vectors:

Taking the scheme described above as a basis, we get the opportunity to perform the operation of adding more than 2 vectors: adding each subsequent vector in turn.

Definition 6

Input: vectors a → , b → , c →, d → . From an arbitrary point A on the plane, it is necessary to set aside a segment (vector) equal to the vector a →; then, from the end of the resulting vector, a vector equal to the vector b →; further - the subsequent vectors are postponed according to the same principle. The end point of the last postponed vector will be point B , and the resulting segment (vector) A B →- the sum of all the initial data. The described scheme for adding several vectors is also called polygon rule .

Geometrically, it looks like this:

Definition 7

A separate scheme of action for vector subtraction no, because in fact the difference of vectors a → and b → is the sum of the vectors a → and - b → .

Definition 8

To perform the action of multiplying a vector by a certain number k, the following rules must be taken into account:
- if k > 1, then this number will stretch the vector by k times;
- if 0< k < 1 , то это число приведет к сжатию вектора в 1k times;
- if k< 0 , то это число приведет к смене направления вектора при одновременном выполнении одного из первых двух правил;
- if k = 1 , then the vector remains the same;
- if one of the factors is a zero vector or a number equal to zero, the result of multiplication will be a zero vector.

Initial data:
1) vector a → and the number k = 2;
2) vector b → and number k = - 1 3 .

Geometrically, the result of multiplication in accordance with the above rules will look like this:

The operations on vectors described above have properties, some of which are obvious, while others can be justified geometrically.

Input: vectors a → , b → , c → and arbitrary real numbers λ and μ.

The properties of commutativity and associativity make it possible to add vectors in an arbitrary order.

The listed properties of operations allow to carry out the necessary transformations of vector-numerical expressions similarly to the usual numerical ones. Let's look at this with an example.

Example 1

Task: simplify the expression a → - 2 (b → + 3 a →)
Decision
- using the second distributive property, we get: a → - 2 (b → + 3 a →) = a → - 2 b → - 2 (3 a →)
- use the associative property of multiplication, the expression will take the following form: a → - 2 b → - 2 (3 a →) = a → - 2 b → - (2 3) a → = a → - 2 b → - 6 a →
- using the commutativity property, we swap the terms: a → - 2 b → - 6 a → = a → - 6 a → - 2 b →
- then, according to the first distribution property, we get: a → - 6 a → - 2 b → = (1 - 6) a → - 2 b → = - 5 a → - 2 b → A brief record of the solution will look like so: a → - 2 (b → + 3 a →) = a → - 2 b → - 2 3 a → = 5 a → - 2 b →
Answer: a → - 2 (b → + 3 a →) = - 5 a → - 2 b →

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