In a parallelogram, the point lies on side ,. Express the vector in terms of the vectors and .
The solution of the problem
This lesson shows how to use known vectors in the form of the sides of a parallelogram to express an arbitrary segment as a composition of the original vectors. This problem could not have a solution if we did not know in what ratio one of the sides of the parallelogram is divided by a point belonging to the required segment. Further actions come down to determining the beginning and end of the given vectors and the vectors into which the side is divided. All this is necessary to correctly use signs when combining vectors. After all, it is necessary to remember the rules for adding vectors: the sum of vectors gives a third vector, the beginning of which coincides with the beginning of the first vector, and the end with the end of the second; and the rule for subtracting vectors: the difference of two vectors is the third vector, the beginning of which coincides with the ends of the second vector, and the end with the end of the first vector. Based on these simple rules, we can get the combination we need.
There will also be problems for you to solve on your own, to which you can see the answers.
Vector concept
Before you learn everything about vectors and operations on them, get ready to solve a simple problem. There is a vector of your entrepreneurship and a vector of your innovative abilities. The vector of entrepreneurship leads you to Goal 1, and the vector of innovative abilities leads you to Goal 2. The rules of the game are such that you cannot move along the directions of these two vectors at once and achieve two goals at once. Vectors interact, or, speaking in mathematical language, some operation is performed on vectors. The result of this operation is the “Result” vector, which leads you to Goal 3.
Now tell me: the result of which operation on the vectors “Entrepreneurship” and “Innovative abilities” is the vector “Result”? If you can't tell right away, don't be discouraged. As you progress through this lesson, you will be able to answer this question.
As we have already seen above, the vector necessarily comes from a certain point A in a straight line to some point B. Consequently, each vector has not only a numerical value - length, but also a physical and geometric value - direction. From this comes the first, simplest definition of a vector. So, a vector is a directed segment coming from a point A to the point B. It is designated as follows: .
And to begin various operations with vectors , we need to get acquainted with one more definition of a vector.
A vector is a type of representation of a point that needs to be reached from some starting point. For example, a three-dimensional vector is usually written as (x, y, z) . In very simple terms, these numbers mean how far you need to walk in three different directions to get to a point.
Let a vector be given. Wherein x = 3 (right hand points to the right), y = 1 (left hand points forward) z = 5 (under the point there is a staircase leading up). Using this data, you will find a point by walking 3 meters in the direction indicated by your right hand, then 1 meter in the direction indicated by your left hand, and then a ladder awaits you and, rising 5 meters, you will finally find yourself at the end point.
All other terms are clarifications of the explanation presented above, necessary for various operations on vectors, that is, solving practical problems. Let's go through these more rigorous definitions, focusing on typical vector problems.
Physical examples vector quantities can be the displacement of a material point moving in space, the speed and acceleration of this point, as well as the force acting on it.
Geometric vector presented in two-dimensional and three-dimensional space in the form directional segment. This is a segment that has a beginning and an end.
If A- the beginning of the vector, and B- its end, then the vector is denoted by the symbol or one lowercase letter . In the figure, the end of the vector is indicated by an arrow (Fig. 1)
Length(or module) of a geometric vector is the length of the segment generating it
The two vectors are called equal , if they can be combined (if the directions coincide) by parallel transfer, i.e. if they are parallel, directed in the same direction and have equal lengths.
In physics it is often considered pinned vectors, specified by the point of application, length and direction. If the point of application of the vector does not matter, then it can be transferred, maintaining its length and direction, to any point in space. In this case, the vector is called free. We will agree to consider only free vectors.
Linear operations on geometric vectors
Multiplying a vector by a number
Product of a vector per number is a vector that is obtained from a vector by stretching (at ) or compressing (at ) by a factor, and the direction of the vector remains the same if , and changes to the opposite if . (Fig. 2)
From the definition it follows that the vectors and = are always located on one or parallel lines. Such vectors are called collinear. (We can also say that these vectors are parallel, but in vector algebra it is customary to say “collinear.”) The converse is also true: if the vectors are collinear, then they are related by the relation
Consequently, equality (1) expresses the condition of collinearity of two vectors.
Addition and subtraction of vectors
When adding vectors you need to know that amount vectors and is called a vector, the beginning of which coincides with the beginning of the vector, and the end with the end of the vector, provided that the beginning of the vector is attached to the end of the vector. (Fig. 3)
This definition can be distributed over any finite number of vectors. Let them be given in space n free vectors. When adding several vectors, their sum is taken to be the closing vector, the beginning of which coincides with the beginning of the first vector, and the end with the end of the last vector. That is, if you attach the beginning of the vector to the end of the vector, and the beginning of the vector to the end of the vector, etc. and, finally, to the end of the vector - the beginning of the vector, then the sum of these vectors is the closing vector 
, the beginning of which coincides with the beginning of the first vector, and the end - with the end of the last vector. (Fig. 4)
The terms are called components of the vector, and the formulated rule is polygon rule. This polygon may not be flat.
When a vector is multiplied by the number -1, the opposite vector is obtained. The vectors and have the same lengths and opposite directions. Their sum gives zero vector, whose length is zero. The direction of the zero vector is not defined.
In vector algebra, there is no need to consider the subtraction operation separately: subtracting a vector from a vector means adding the opposite vector to the vector, i.e. 
Example 1. Simplify the expression:
.
,
that is, vectors can be added and multiplied by numbers in the same way as polynomials (in particular, also problems on simplifying expressions). Typically, the need to simplify linearly similar expressions with vectors arises before calculating the products of vectors.
Example 2. Vectors and serve as diagonals of the parallelogram ABCD (Fig. 4a). Express through and the vectors , , and , which are the sides of this parallelogram.
Solution. The point of intersection of the diagonals of a parallelogram bisects each diagonal. We find the lengths of the vectors required in the problem statement either as half the sums of the vectors that form a triangle with the required ones, or as half the differences (depending on the direction of the vector serving as the diagonal), or, as in the latter case, half the sum taken with a minus sign. The result is the vectors required in the problem statement:
There is every reason to believe that you have now correctly answered the question about the vectors “Entrepreneurship” and “Innovative abilities” at the beginning of this lesson. Correct answer: an addition operation is performed on these vectors.
Solve vector problems yourself and then look at the solutions
How to find the length of the sum of vectors?
This problem occupies a special place in operations with vectors, since it involves the use of trigonometric properties. Let's say you come across a task like the following:
The vector lengths are given 
and the length of the sum of these vectors. Find the length of the difference between these vectors.
Solutions to this and other similar problems and explanations of how to solve them are in the lesson " Vector addition: length of the sum of vectors and the cosine theorem ".
And you can check the solution to such problems at Online calculator "Unknown side of a triangle (vector addition and cosine theorem)" .
Where are the products of vectors?
Vector-vector products are not linear operations and are considered separately. And we have lessons "Scalar product of vectors" and "Vector and mixed products of vectors".
Projection of a vector onto an axis
The projection of a vector onto an axis is equal to the product of the length of the projected vector and the cosine of the angle between the vector and the axis:
As is known, the projection of a point A on the straight line (plane) is the base of the perpendicular dropped from this point onto the straight line (plane).
Let be an arbitrary vector (Fig. 5), and and be the projections of its origin (points A) and end (points B) per axis l. (To construct a projection of a point A) draw a straight line through the point A a plane perpendicular to a straight line. The intersection of the line and the plane will determine the required projection.
Vector component on the l axis is called such a vector lying on this axis, the beginning of which coincides with the projection of the beginning, and the end with the projection of the end of the vector.
Projection of the vector onto the axis l called number
,
equal to the length of the component vector on this axis, taken with a plus sign if the direction of the components coincides with the direction of the axis l, and with a minus sign if these directions are opposite.
Basic properties of vector projections onto an axis:
1. Projections of equal vectors onto the same axis are equal to each other.
2. When a vector is multiplied by a number, its projection is multiplied by the same number.
3. The projection of the sum of vectors onto any axis is equal to the sum of the projections of the summands of the vectors onto the same axis.
4. The projection of the vector onto the axis is equal to the product of the length of the projected vector and the cosine of the angle between the vector and the axis:
.
Solution. Let's project vectors onto the axis l as defined in the theoretical background above. From Fig. 5a it is obvious that the projection of the sum of vectors is equal to the sum of the projections of vectors. We calculate these projections:
We find the final projection of the sum of vectors:
Relationship between a vector and a rectangular Cartesian coordinate system in space
Getting to know rectangular Cartesian coordinate system in space took place in the corresponding lesson, it is advisable to open it in a new window.
In an ordered system of coordinate axes 0xyz axis Ox called x-axis, axis 0y – y-axis, and axis 0z – axis applicate.
With an arbitrary point M space connect vector
called radius vector points M and project it onto each of the coordinate axes. Let us denote the magnitudes of the corresponding projections:
Numbers x, y, z are called coordinates of point M, respectively abscissa, ordinate And applicate, and are written as an ordered point of numbers: M(x;y;z)(Fig. 6).
A vector of unit length whose direction coincides with the direction of the axis is called unit vector(or ortom) axes. Let us denote by
Accordingly, the unit vectors of the coordinate axes Ox, Oy, Oz
Theorem. Any vector can be expanded into unit vectors of coordinate axes:
(2)
Equality (2) is called the expansion of the vector along the coordinate axes. The coefficients of this expansion are the projections of the vector onto the coordinate axes. Thus, the coefficients of expansion (2) of the vector along the coordinate axes are the coordinates of the vector.
After choosing a certain coordinate system in space, the vector and the triplet of its coordinates uniquely determine each other, so the vector can be written in the form
Representations of the vector in the form (2) and (3) are identical.
Condition for collinearity of vectors in coordinates
As we have already noted, vectors are called collinear if they are related by the relation
Let the vectors be given 
. These vectors are collinear if the coordinates of the vectors are related by the relation
,
that is, the coordinates of the vectors are proportional.
Example 6. Vectors are given 
. Are these vectors collinear?
Solution. Let's find out the relationship between the coordinates of these vectors:
.
The coordinates of the vectors are proportional, therefore, the vectors are collinear, or, what is the same, parallel.
Vector length and direction cosines
Due to the mutual perpendicularity of the coordinate axes, the length of the vector
equal to the length of the diagonal of a rectangular parallelepiped built on vectors
and is expressed by the equality
(4)
A vector is completely defined by specifying two points (start and end), so the coordinates of the vector can be expressed in terms of the coordinates of these points.
Let, in a given coordinate system, the origin of the vector be at the point
and the end is at the point
From equality
Follows that
or in coordinate form
Hence, vector coordinates are equal to the differences between the same coordinates of the end and beginning of the vector . Formula (4) in this case will take the form
The direction of the vector is determined direction cosines . These are the cosines of the angles that the vector makes with the axes Ox, Oy And Oz. Let us denote these angles accordingly α , β And γ . Then the cosines of these angles can be found using the formulas
The direction cosines of a vector are also the coordinates of the vector of that vector and thus the vector of the vector
.
Considering that the length of the unit vector is equal to one unit, that is
,
we obtain the following equality for the direction cosines:
Example 7. Find the length of the vector x = (3; 0; 4).
Solution. The length of the vector is
Example 8. Points given:
Find out whether the triangle constructed on these points is isosceles.
Solution. Using the vector length formula (6), we find the lengths of the sides and determine whether there are two equal ones among them:
Two equal sides have been found, therefore there is no need to look for the length of the third side, and the given triangle is isosceles.
Example 9. Find the length of the vector and its direction cosines if 
.
Solution. The vector coordinates are given:
.
The length of the vector is equal to the square root of the sum of the squares of the vector coordinates:
.
Finding direction cosines:
Solve the vector problem yourself, and then look at the solution
Operations on vectors given in coordinate form
Let two vectors and be given, defined by their projections:
Let us indicate actions on these vectors.
Finally, I got my hands on this extensive and long-awaited topic. analytical geometry. First, a little about this section of 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 solution method.” Graphical method, of course, is associated with the construction of graphs and drawings. Analytical or 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; often it is enough to carefully apply the 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 the tutorial will be of invaluable help.
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 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. A local task - Division of a segment in this respect - will also not be superfluous. 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 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. It is convenient to identify the concept of a vector with the movement of a physical body: you must agree, 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 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: , thereby implying that this 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, our vector can be redesignated for brevity by a 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.
This was basic information about vectors, 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 “school” vector to ANY point of the plane or space you need. This is a very cool feature! 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 saying: Every lecturer gives a damn about the vector. After all, it’s not just a witty rhyme, everything is almost correct - a directed segment can be added 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. 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 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 is generally incorrect, and the point of application 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
A school geometry course covers a number of actions and rules with vectors: 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 put a physical meaning into it: let some body travel along the vector , and then along the 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 in different directions, 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 modulus of the 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 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 the 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 are the same vector, as 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 depict a Cartesian rectangular coordinate system and plot it from the origin of coordinates 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 in 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 codirectional 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 a special case of addition. Thus, the expansions of the vectors “de” and “e” are easily written as a sum: , 
. Follow the drawing to see how clearly 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; the following option is common:
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 notation 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 3D space vector 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 begins at the initial point of departure (beginning of the vector) and ends 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 one (or two) coordinate vectors are missing in the expansion, then zeros are put in their place. Examples:
vector (meticulously 
) – let’s write ;
vector (meticulously) – write down;
vector (meticulously 
) – let’s write .
The basis vectors are written as follows:
This, perhaps, is all the minimum theoretical knowledge necessary to solve problems of analytical geometry. There may be a lot of terms and definitions, so I recommend that teapots 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 to better assimilate the material. Collinearity, orthogonality, orthonormal basis, vector decomposition - these and other concepts will be often used in the future. I note that the materials on the site are not enough to pass the theoretical test or colloquium on geometry, since I carefully encrypt all theorems (and without proofs) - to the detriment of the scientific style of 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, you don’t even have to 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 additional time 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 desired or necessary, we can easily move it away from some other point on 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 you to decide on your own, 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 important points in it 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 us repeat the school material, which is useful not only for the task considered:
pay attention to important 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 the root produces a fairly large 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.
When solving various problems, roots are often encountered; 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: 
The rules for operating with powers in general form can be found in a school algebra textbook, 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 
.