First of all, we need to understand the concept of a vector itself. In order to introduce the definition of a geometric vector, let us remember what a segment is. Let us introduce the following definition.
Definition 1
A segment is a part of a line that has two boundaries in the form of points.
A segment can have 2 directions. To denote the direction, we will call one of the boundaries of the segment its beginning, and the other boundary its end. The direction is indicated from its beginning to the end of the segment.
Definition 2
A vector or directed segment will be a segment for which it is known which of the boundaries of the segment is considered the beginning and which is its end.
Designation: In two letters: $\overline(AB)$ – (where $A$ is its beginning, and $B$ is its end).
In one small letter: $\overline(a)$ (Fig. 1).
Let us now introduce directly the concept of vector lengths.
Definition 3
The length of the vector $\overline(a)$ will be the length of the segment $a$.
Notation: $|\overline(a)|$
The concept of vector length is associated, for example, with such a concept as the equality of two vectors.
Definition 4
We will call two vectors equal if they satisfy two conditions: 1. They are codirectional; 1. Their lengths are equal (Fig. 2).
In order to define vectors, enter a coordinate system and determine the coordinates for the vector in the entered system. As we know, any vector can be decomposed in the form $\overline(c)=m\overline(i)+n\overline(j)$, where $m$ and $n$ are real numbers, and $\overline(i )$ and $\overline(j)$ are unit vectors on the $Ox$ and $Oy$ axis, respectively.
Definition 5
We will call the expansion coefficients of the vector $\overline(c)=m\overline(i)+n\overline(j)$ the coordinates of this vector in the introduced coordinate system. Mathematically:
$\overline(c)=(m,n)$
How to find the length of a vector?
In order to derive a formula for calculating the length of an arbitrary vector given its coordinates, consider the following problem:
Example 1
Given: vector $\overline(α)$ with coordinates $(x,y)$. Find: the length of this vector.
Let us introduce a Cartesian coordinate system $xOy$ on the plane. Let us set aside $\overline(OA)=\overline(a)$ from the origins of the introduced coordinate system. Let us construct projections $OA_1$ and $OA_2$ of the constructed vector on the $Ox$ and $Oy$ axes, respectively (Fig. 3).
The vector $\overline(OA)$ we have constructed will be the radius vector for point $A$, therefore, it will have coordinates $(x,y)$, which means
$=x$, $[OA_2]=y$
Now we can easily find the required length using the Pythagorean theorem, we get
$|\overline(α)|^2=^2+^2$
$|\overline(α)|^2=x^2+y^2$
$|\overline(α)|=\sqrt(x^2+y^2)$
Answer: $\sqrt(x^2+y^2)$.
Conclusion: To find the length of a vector whose coordinates are given, it is necessary to find the root of the square of the sum of these coordinates.
Sample tasks
Example 2
Find the distance between points $X$ and $Y$, which have the following coordinates: $(-1.5)$ and $(7.3)$, respectively.
Any two points can be easily associated with the concept of a vector. Consider, for example, the vector $\overline(XY)$. As we already know, the coordinates of such a vector can be found by subtracting from the coordinates end point($Y$) the corresponding coordinates of the starting point ($X$). We get that
Finally, I got my hands on this vast and long-awaited topic. analytical geometry . First a little about this section higher mathematics…. Surely you now remember a 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. Analytical geometry, oddly enough, may seem more interesting and accessible. What does the adjective “analytical” mean? Two cliched mathematical phrases immediately come to mind: “graphical solution method” and “ analytical method solutions". Graphical method , of course, is associated with the construction of graphs and drawings. Analytical same method involves solving problems mainly through algebraic operations. In this regard, the algorithm for solving almost all problems of analytical geometry is simple and transparent; it is often enough to carefully apply necessary formulas- and the answer is ready! No, of course, we won’t be able to do this without drawings at all, and besides, for a better understanding of the material, I will try to cite them beyond necessity.
The newly opened course of lessons on geometry does not pretend to be theoretically complete; 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 more complete help on any subsection, I recommend the following quite accessible literature:
1) A thing that, no joke, several generations are familiar with: School textbook on geometry, authors - L.S. Atanasyan and Company. This school locker room hanger has already gone through 20 (!) reprints, which, of course, is not the limit.
2) Geometry in 2 volumes. Authors L.S. Atanasyan, Bazylev V.T.. This is literature for high school, you will need first volume. Rarely encountered tasks may fall out of my sight, and tutorial will provide invaluable assistance.
Both books can be downloaded for free online. In addition, you can use my archive with ready-made solutions, which can be found on the page Download examples in higher mathematics.
Among the tools, I again propose my own development - software package in analytical geometry, which will greatly simplify life and save a lot of time.
It is assumed that the reader is familiar with the 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 to repeaters)
And now we will consider sequentially: the concept of a vector, actions with vectors, vector coordinates. I recommend reading further the most important article Dot product of vectors, and also Vector and mixed product of vectors. It will not be superfluous local problem– Division of a segment in a given ratio. Based on the above information, you can master equation of a line in a plane With simplest examples of solutions, which will allow learn to solve geometry problems. The following articles are also useful: Equation of a plane in space, Equations of a line in space, Basic problems on a straight line and a plane, other sections of analytical geometry. Naturally, standard tasks will be considered along the way.
Vector concept. 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 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 move the arrow to the other end of the segment, you get a vector, and this is already completely different vector. The concept of vector is conveniently identified with motion physical body: Agree, entering the doors of the institute or leaving the doors of the institute are completely different things.
It is convenient to consider individual points of a plane or space as the so-called zero vector. For such a vector, the end and beginning coincide.
!!! Note: Here and further, 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 noticed the stick without an arrow in the designation and said, there’s also an arrow at the top! True, you can write it with an arrow: , but it is also possible the entry that I will use in the future. Why? Apparently, this habit developed for practical reasons; my shooters at school and university turned out to be too different-sized and shaggy. IN educational literature sometimes they don’t bother with cuneiform writing at all, but highlight the letters in bold: , implying that it is a vector.
That was stylistics, and now about ways to write vectors:
1) Vectors can be written in two capital Latin letters: 
and so on. In this case, the first letter Necessarily denotes the beginning 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, for brevity our vector can be redesignated as small Latin letter.
Length or module a non-zero vector is called the length of the segment. The length of the zero vector is zero. Logical.
The length of the vector is indicated by the modulus sign: ,
We will learn how to find the length of a vector (or we will repeat it, depending on who) a little later.
They were basic information about vector, familiar to all schoolchildren. In analytical geometry, the so-called free vector.
To put it simply - the vector can be plotted from any point:
We are accustomed to calling such vectors equal (the definition of equal vectors will be given below), but from a purely mathematical point of view, they are the SAME VECTOR or free vector. Why free? Because in the course of solving problems, you can “attach” this or that vector to ANY point of the plane or space you need. This is a very cool feature! Imagine a vector of arbitrary length and direction - it can be “cloned” infinite number times and at any point in space, in fact, it exists EVERYWHERE. There is such a student saying: Every lecturer gives a damn about the vector. After all, it’s not just a witty rhyme, everything is mathematically correct - the vector can be attached there too. But don’t rush to rejoice, it’s the students themselves who often suffer =)
So, free vector- This a bunch of identical directed segments. School definition 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 tied to a specific point in the plane or space.
It should be noted that from the point of view of physics, the concept of a free vector in general case is incorrect, and the point of application of the vector matters. Indeed, a direct blow of the same force on the nose or forehead, enough to develop my stupid example, entails different consequences. However, unfree vectors are also found in the course of vyshmat (don’t go there :)).
Actions with vectors. Collinearity of vectors
IN school course geometry, a number of actions and rules with vectors are considered: addition according to the triangle rule, addition according to the parallelogram rule, vector difference rule, multiplication of a vector by a number, scalar product of vectors, etc. As a starting point, let us repeat two rules that are especially relevant for solving problems of analytical geometry.
The rule for adding vectors using the triangle rule
Consider two arbitrary non-zero vectors and : 
You need to find the sum of these vectors. Due to the fact that all vectors are considered free, we will set aside the vector from end vector: 
The sum of vectors is the vector. For a better understanding of the rule, it is advisable to include physical meaning: let some body travel along a vector, and then along a vector. Then the sum of vectors is the vector of the resulting path with the beginning at the departure point and the end at the arrival point. A similar rule is formulated for the sum of any number of vectors. As they say, the body can go its way very lean along a zigzag, or maybe on autopilot - along the resulting vector of the sum.
By the way, if the vector is postponed from started vector, then we get the equivalent parallelogram rule addition of vectors.
First, about 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-directed. If the arrows point towards different sides, then the vectors will be opposite directions.
Designations: collinearity of vectors is written with the usual parallelism symbol: , while detailing is possible: (vectors are co-directed) or (vectors are oppositely directed).
The work a non-zero vector on 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 the help of a picture: 
Let's look at it in more detail:
1) Direction. If the multiplier is negative, then the vector changes direction to the opposite.
2) Length. If the multiplier is contained within or , then the length of the vector decreases. So, the length of the vector is half the length of the vector. If the modulo multiplier more than one, then the vector length 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 through another, then such vectors are necessarily collinear. Thus: if we multiply a vector by a number, we get collinear(relative to the original) vector.
4) The vectors are co-directed. Vectors and are also co-directed. Any vector of the first group is oppositely directed with respect to any vector of the second group.
Which vectors are equal?
Two vectors are equal if they are in the same direction and have same length . Note that codirectionality implies collinearity of vectors. The definition would be inaccurate (redundant) if we said: “Two vectors are equal if they are collinear, codirectional, and have the same length.”
From the point of view of the concept of a free vector, equal vectors– this is 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 the plane. Let us represent the Cartesian rectangular system coordinates and from the origin of coordinates we postpone single vectors and :
Vectors and orthogonal. Orthogonal = Perpendicular. I recommend that you slowly get used to the terms: instead of parallelism and perpendicularity, we use the words respectively collinearity And orthogonality.
Designation: The orthogonality of vectors is written with the usual perpendicularity symbol, for example: .
The vectors under consideration are called coordinate vectors or orts. These vectors form basis on surface. What a basis is, I think, is intuitively clear to many, more detailed information can be found in the article Linear (non) dependence of vectors. Basis of vectors In simple words, the basis and 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 sequence basis vectors are listed, for example: . Coordinate vectors it is forbidden rearrange.
Any plane vector the only way expressed as: 
, Where - numbers which are called vector coordinates V on this basis. And the expression itself 
called vector decompositionby basis .
Dinner served:
Let's start with the first letter of the alphabet: . The drawing clearly shows that when decomposing a vector into a basis, the ones just discussed are used:
1) the rule for multiplying a vector by a number: and ;
2) addition of vectors according to the triangle rule: .
Now mentally plot the vector from any other point on the plane. It is quite obvious that his decay will “follow him relentlessly.” Here it is, the freedom of the vector - the vector “carries everything with itself.” This property, of course, is true for any vector. It's funny that the basis (free) vectors themselves do not have to be plotted from the origin; one can be drawn, for example, at the bottom left, and the other at the top right, and nothing will change! True, you don’t need to do this, since the teacher will also show originality and draw you a “credit” in an unexpected place.
Vectors illustrate exactly the rule for multiplying a vector by a number, the vector is co-directed with the base vector, the vector is directed opposite to the base vector. For these vectors, one of the coordinates is equal to zero; you can meticulously write it like this:
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 talk about the subtraction rule? Somewhere in linear algebra, I don’t remember where, I noted that subtraction is special case addition. Thus, the expansions of the vectors “de” and “e” are easily written as a sum: , 
. Rearrange the terms and see in the drawing how well the good old addition of vectors according to the triangle rule works in these situations.
The considered decomposition of the form 
sometimes called vector decomposition in the ort system(i.e. in a system of unit vectors). But this is not the only way to write a vector, it is common next option:
Or with an equal sign:
The basis vectors themselves are written as follows: and
That is, the coordinates of the vector are indicated in parentheses. IN practical problems All three recording options are used.
I doubted whether to speak, but I’ll say it anyway: vector coordinates cannot be rearranged. Strictly in first place we write down the coordinate that corresponds to the unit vector, strictly in second place we 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 let's look at vectors in three-dimensional space, almost everything is the same here! It will just add one more coordinate. It’s hard to make three-dimensional drawings, so I’ll limit myself to one vector, which for simplicity I’ll set aside from the origin: 
Any vector three-dimensional space Can the only way
expand over an orthonormal basis: 
, where are the coordinates of the vector (number) in this basis.
Example from the picture: 
. Let's see how the vector rules work here. First, multiplying the vector by a number: (red arrow), (green arrow) and (raspberry arrow). Secondly, here is an example of adding several, in this case three, vectors: . The sum vector starts at starting point departure (beginning of the vector) and ends up at the final point of arrival (end of the vector).
All vectors of three-dimensional space, naturally, are also free; try to mentally set aside the vector from any other point, and you will understand that its decomposition “will remain with it.”
Similar to the flat case, in addition to writing 
versions with brackets are widely used: either .
If the expansion is missing one (or two) coordinate vectors, then zeros are put in their place. Examples:
vector (meticulously 
) – let’s write ;
vector (meticulously 
) – let’s write ;
vector (meticulously 
) – let’s write .
The basis vectors are written in the following way:
That's probably all the minimum theoretical knowledge, necessary for solving problems of analytical geometry. There may be a lot of terms and definitions, so I recommend that dummies re-read and comprehend this information again. And it will be useful for any reader to refer to basic lesson For better absorption material. Collinearity, orthogonality, orthonormal basis, vector decomposition - these and other concepts will be often used in the future. I would like to note that the site materials are not enough to pass a theoretical test or a colloquium in geometry, since I carefully encrypt all theorems (and without proofs) - to the detriment of scientific style presentation, but a plus to your understanding of the subject. To receive detailed theoretical information, please bow to Professor Atanasyan.
And we move on to the practical part:
The simplest problems of analytical geometry.
Actions with vectors in coordinates
It is highly advisable to learn how to solve the tasks that will be considered fully automatically, and the formulas memorize, don’t even specifically remember, they will remember themselves =) This is very important, because in the simplest elementary examples other problems of analytical geometry are based, and it will be annoying to spend Extra time for eating pawns. There is no 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 from 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:
That is, from the coordinates of the end of the vector you need to subtract the corresponding coordinates beginning of the vector.
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 of the plane and . Find vector coordinates
Solution: according to the appropriate formula:
Alternatively, the following entry could be used:
Aesthetes will decide this:
Personally, I'm used to the first version of the recording.
Answer:
According to the condition, it was not necessary to construct a drawing (which is typical for problems of analytical geometry), but in order to clarify some points for dummies, I will not be lazy: 
You definitely need to understand difference between point coordinates and vector coordinates:
Point coordinates– these are ordinary coordinates in a rectangular coordinate system. I think everyone knows how to plot points on a coordinate plane from the 5th-6th grade. Each point has a strict place on the plane, and they cannot be moved anywhere.
The coordinates of the vector– this is its expansion according to the basis, in this case. Any vector is free, so if necessary, we can easily move it away from some other point in the plane. It is interesting that for vectors you don’t have to build axes or a rectangular coordinate system at all; you only need a basis, in this case an orthonormal basis of the plane.
The records of coordinates of points and coordinates of vectors seem to be similar: , and meaning of coordinates absolutely different, and you should be well aware of this difference. This difference, of course, also applies to space.
Ladies and gentlemen, let's fill our hands:
Example 2
a) Points and are given. Find vectors and .
b) Points are given 
And . Find vectors and .
c) Points and are given. Find vectors and .
d) Points are given. Find vectors 
.
Perhaps that's enough. These are examples for independent decision, try not to neglect them, it will pay off ;-). There is no need to make drawings. Solutions and answers at the end of the lesson.
What is important when solving analytical geometry problems? It is important to be EXTREMELY CAREFUL to avoid making the masterful “two plus two equals zero” mistake. I apologize right away if I made a mistake somewhere =)
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 are given and , then the length of the segment can be calculated using the formula
If two points in space and are given, then the length of the segment can be calculated using the formula
Note: The formulas will remain correct if the corresponding coordinates are swapped: and , but the first option is more standard
Example 3
Solution: according to the appropriate formula:
Answer: 
For clarity, I will make a drawing 
Line segment - this is not a vector, and, of course, you cannot move it anywhere. In addition, if you draw to scale: 1 unit. = 1 cm (two notebook cells), then the resulting 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 more in it important points that I would like to clarify:
Firstly, in the answer we put the dimension: “units”. The condition does not say WHAT it is, millimeters, centimeters, meters or kilometers. Therefore, a mathematically correct solution would be the general formulation: “units” - abbreviated as “units.”
Secondly, let's repeat school material, which is useful not only for the problem considered:
pay attention to important technical technique
– removing the multiplier from under the root. As a result of the calculations, we have a result and good mathematical style involves removing the factor from under the root (if possible). In more detail the process looks like this: 
. Of course, leaving the answer as is would not be a mistake - but it would certainly be a shortcoming and a weighty argument for quibbling on the part of the teacher.
Here are other common cases: 
Often there is enough at the root big number, For example . What to do in such cases? Using the calculator, we check whether the number is divisible by 4: . Yes, it was completely divided, 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 will obviously not work. Let's try to divide by nine: . As a result:
Ready.
Conclusion: if under the root we get a number that cannot be extracted as a whole, then we try to remove the factor from under the root - using a calculator we check whether the number is divisible by: 4, 9, 16, 25, 36, 49, etc.
During the decision various tasks roots are common, always try to extract factors from under the root in order to avoid a lower grade and unnecessary problems with finalizing your solutions based on the teacher’s comments.
Let's also repeat squaring roots and other powers: 
Rules for actions with degrees in general view can be found in school textbook in algebra, but I think from the examples given, everything or almost everything is already clear.
Task for independent solution with a segment in space:
Example 4
Points and are given. Find the length of the segment.
The solution and answer are 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 
.
First of all, we need to understand the concept of a vector itself. In order to introduce the definition of a geometric vector, let us remember what a segment is. Let us introduce the following definition.
Definition 1
A segment is a part of a line that has two boundaries in the form of points.
A segment can have 2 directions. To denote the direction, we will call one of the boundaries of the segment its beginning, and the other boundary its end. The direction is indicated from its beginning to the end of the segment.
Definition 2
A vector or directed segment will be a segment for which it is known which of the boundaries of the segment is considered the beginning and which is its end.
Designation: In two letters: $\overline(AB)$ – (where $A$ is its beginning, and $B$ is its end).
In one small letter: $\overline(a)$ (Fig. 1).
Let us now introduce directly the concept of vector lengths.
Definition 3
The length of the vector $\overline(a)$ will be the length of the segment $a$.
Notation: $|\overline(a)|$
The concept of vector length is associated, for example, with such a concept as the equality of two vectors.
Definition 4
We will call two vectors equal if they satisfy two conditions: 1. They are codirectional; 1. Their lengths are equal (Fig. 2).
In order to define vectors, enter a coordinate system and determine the coordinates for the vector in the entered system. As we know, any vector can be decomposed in the form $\overline(c)=m\overline(i)+n\overline(j)$, where $m$ and $n$ are real numbers, and $\overline(i )$ and $\overline(j)$ are unit vectors on the $Ox$ and $Oy$ axis, respectively.
Definition 5
We will call the expansion coefficients of the vector $\overline(c)=m\overline(i)+n\overline(j)$ the coordinates of this vector in the introduced coordinate system. Mathematically:
$\overline(c)=(m,n)$
How to find the length of a vector?
In order to derive a formula for calculating the length of an arbitrary vector given its coordinates, consider the following problem:
Example 1
Given: vector $\overline(α)$ with coordinates $(x,y)$. Find: the length of this vector.
Let us introduce a Cartesian coordinate system $xOy$ on the plane. Let us set aside $\overline(OA)=\overline(a)$ from the origins of the introduced coordinate system. Let us construct projections $OA_1$ and $OA_2$ of the constructed vector on the $Ox$ and $Oy$ axes, respectively (Fig. 3).
The vector $\overline(OA)$ we have constructed will be the radius vector for point $A$, therefore, it will have coordinates $(x,y)$, which means
$=x$, $[OA_2]=y$
Now we can easily find the required length using the Pythagorean theorem, we get
$|\overline(α)|^2=^2+^2$
$|\overline(α)|^2=x^2+y^2$
$|\overline(α)|=\sqrt(x^2+y^2)$
Answer: $\sqrt(x^2+y^2)$.
Conclusion: To find the length of a vector whose coordinates are given, it is necessary to find the root of the square of the sum of these coordinates.
Sample tasks
Example 2
Find the distance between points $X$ and $Y$, which have the following coordinates: $(-1.5)$ and $(7.3)$, respectively.
Any two points can be easily associated with the concept of a vector. Consider, for example, the vector $\overline(XY)$. As we already know, the coordinates of such a vector can be found by subtracting the corresponding coordinates of the starting point ($X$) from the coordinates of the end point ($Y$). We get that
Sum of vectors. Vector length. Dear friends, as part of the back exam types there is a group of problems with vectors. The tasks are quite wide range(It is important to know theoretical basis). Most are resolved orally. The questions are related to finding the length of a vector, the sum (difference) of vectors, and the scalar product. There are also many tasks in which it is necessary to perform actions with vector coordinates.
The theory surrounding the topic of vectors is not complicated, and it must be well understood. In this article we will analyze problems related to finding the length of a vector, as well as the sum (difference) of vectors. Some theoretical points:
Vector concept
A vector is a directed segment.
All vectors that have the same direction and are equal in length are equal.
*All four vectors presented above are equal!
That is, if we use parallel transfer move the vector given to us, we will always get a vector equal to the original one. Thus, there can be an infinite number of equal vectors.
Vector notation
The vector can be denoted by Latin in capital letters, For example:
With this form of notation, first the letter denoting the beginning of the vector is written, then the letter denoting the end of the vector.
Another vector is denoted by one letter Latin alphabet(capital):
Designation without arrows is also possible:
The sum of two vectors AB and BC will be the vector AC.
It is written as AB + BC = AC.
This rule is called - triangle rule.
That is, if we have two vectors – let’s call them conventionally (1) and (2), and the end of vector (1) coincides with the beginning of vector (2), then the sum of these vectors will be a vector whose beginning coincides with the beginning of vector (1) , and the end coincides with the end of vector (2).
Conclusion: if we have two vectors on a plane, we can always find their sum. Using parallel translation, you can move any of these vectors and connect its beginning to the end of another. For example:
Let's move the vector b, or in other words, let’s construct an equal one:
How is the sum of several vectors found? By the same principle:
* * *
Parallelogram rule
This rule is a consequence of the above.
For vectors with common beginning their sum is represented by the diagonal of a parallelogram constructed on these vectors.
Let's construct a vector equal to the vector b so that its beginning coincides with the end of the vector a, and we can build a vector that will be their sum:
A bit more important information necessary to solve problems.
A vector equal in length to the original one, but oppositely directed, is also denoted but has the opposite sign:
This information is extremely useful for solving problems that involve finding the difference between vectors. As you can see, the vector difference is the same sum in a modified form.
Let two vectors be given, find their difference:
We built a vector opposite vector b, and found the difference.
Vector coordinates
To find the coordinates of a vector, you need to subtract the corresponding coordinates of the beginning from the end coordinates:
That is, the vector coordinates are a pair of numbers.
If
And the coordinates of the vectors look like:
Then c 1 = a 1 + b 1 c 2 = a 2 + b 2
If
Then c 1 = a 1 – b 1 c 2 = a 2 – b 2
Vector module
The modulus of a vector is its length, determined by the formula:
Formula for determining the length of a vector if the coordinates of its beginning and end are known:
Let's consider the tasks:
The two sides of rectangle ABCD are equal to 6 and 8. The diagonals intersect at point O. Find the length of the difference between the vectors AO and BO.
Let’s find the vector that will be the result of AO–VO:
AO –VO =AO +(–VO )=AB
That is, the difference between the vectors AO and VO will be a vector AB. And its length is eight.
Diagonals of a rhombus ABCD are equal to 12 and 16. Find the length of the vector AB + AD.
Let's find a vector that will be the sum of vectors AD and AB BC equal to the vector A.D. So AB +AD =AB +BC =AC
AC is the length of the diagonal of the rhombus AC, it is equal to 16.The diagonals of rhombus ABCD intersect at the point O and are equal to 12 and 16. Find the length of the vector AO + BO.
Let's find a vector that will be the sum of the vectors AO and VO VO is equal to the vector OD, which means
AD is the length of the side of the rhombus. The problem comes down to finding the hypotenuse in right triangle AOD. Let's calculate the legs:
According to the Pythagorean theorem:
The diagonals of the rhombus ABCD intersect at point O and are equal to 12 and 16. Find the length of the vector AO – BO.
Let’s find the vector that will be the result of AO–VO:
AB is the length of a side of a rhombus. The problem comes down to finding the hypotenuse AB in the right triangle AOB. Let's calculate the legs:
According to the Pythagorean theorem:
Sides correct triangle ABC are equal to 3.
Find the length of the vector AB –AC.
Let's find the result of the vector difference:
CB is equal to three, since the condition says that the triangle is equilateral and its sides are equal to 3.27663. Find the length of the vector a (6;8).
27664. Find the square of the length of the vector AB.
The abscissa and ordinate axis are called coordinates vector. Vector coordinates are usually indicated in the form (x, y), and the vector itself as: =(x, y).
Formula for determining vector coordinates for two-dimensional problems.
When two-dimensional problem vector with famous coordinates of points A(x 1;y 1) And B(x 2 ; y 2 ) can be calculated:
= (x 2 - x 1; y 2 - y 1).
Formula for determining vector coordinates for spatial problems.
When spatial problem vector with famous coordinates of points A (x 1;y 1;z 1 ) and B (x 2 ; y 2 ; z 2 ) can be calculated using the formula:
= (x 2 - x 1 ; y 2 - y 1 ; z 2 - z 1 ).
The coordinates are given comprehensive description vector, since it is possible to construct the vector itself using the coordinates. Knowing the coordinates, it is easy to calculate and vector length. (Property 3 below).
Properties of vector coordinates.
1. Any equal vectors V unified system coordinates have equal coordinates.
2. Coordinates collinear vectors proportional. Provided that none of the vectors is zero.
3. Square of the length of any vector equal to the sum square it coordinates.
4.During surgery vector multiplication on real number each of its coordinates is multiplied by this number.
5. When adding vectors, we calculate the sum of the corresponding vector coordinates.
6. Scalar product two vectors is equal to the sum of the products of their corresponding coordinates.