Finding the coordinates of the midpoint of a segment: examples, solutions. Vectors for dummies

Finally, I got my hands on a vast and long-awaited topic analytic geometry... First, a little about this section of higher mathematics…. Surely you are now reminded of a school geometry course with numerous theorems, their proofs, drawings, etc. What to hide, an unloved and often obscure subject for a large proportion of students. Analytic geometry, oddly enough, may seem more interesting and accessible. What does the adjective analytic mean? Two stamped mathematical turns immediately come to mind: "graphical solution method" and "analytical solution method". Graphical method, of course, is associated with the construction of graphs, drawings. Analyticalthe same method involves solving problems predominantly through algebraic actions. 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, it will not do without drawings at all, besides, for a better understanding of the material, I will try to cite them beyond necessity.

The opened 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 more complete help on any subsection, I recommend the following readily available literature:

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

2) Geometry in 2 volumes... Authors L.S. Atanasyan, Bazylev V.T.... This is high school literature, you will need first volume... Rare tasks may fall out of my sight, and this tutorial will be of invaluable help.

Both books can be downloaded for free on the Internet. In addition, you can use my archive with ready-made solutions, which can be found on the page Download examples in higher mathematics.

As for the tools, I again suggest 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 shapes: point, line, plane, triangle, parallelogram, parallelepiped, cube, etc. It is advisable to remember some theorems, at least the Pythagorean theorem, hello to the repeaters)

And now we will sequentially consider: the concept of a vector, actions with vectors, vector coordinates. Further I recommend reading crucial article Dot product of vectorsand also Vector and mixed product of vectors... The local task - Division of a segment in this respect will not be superfluous either. Based on the above information, you can master equation of a straight line on a plane from the simplest examples of solutionswhich will allow learn 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 tasks on the line and 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 a point, the end of the segment is a 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 equate the concept of a vector with the motion of a physical body: you must 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 the plane, space as the so-called zero vector ... Such a vector has the same end and beginning.

!!! Note: Hereinafter, 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 true both for the plane and for space.

Legend: Many immediately noticed a wand without an arrow in the designation and said, there is also an arrow at the top! True, you can write with an arrow:, but also the record that I will use in the future... Why? Apparently, such a habit developed from practical considerations, my shooters turned out to be too variegated and shaggy at school and university. 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, but now about the ways of writing vectors:

1) Vectors can be written in two capital Latin letters:
and so on. Moreover, 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, for brevity, our vector can be redesignated with a small Latin letter.

Length or module a nonzero vector is the length of the segment. The length of the zero vector is zero. It is logical.

The vector length is indicated by the modulus sign:,

We will learn (or repeat, how to find the length of a vector) a little later.

Those were elementary information about the vector, familiar to all schoolchildren. In analytical geometry, the so-called free vector.

If it's quite simple - vector can be postponed 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 it is ONE AND 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 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 saying: Each lecturer in f ** k a vector. After all, it's not just a witty rhyme, everything is almost correct - a directed segment can be added there too. But do not rush to rejoice, students themselves suffer more often \u003d)

So, free vector - this is lots of identical directed segments. The school definition of a vector, given at the beginning of the paragraph: "A vector is called a directed segment ...", implies specific a directed segment taken from a given set, which is tied to a specific point in a 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 will suffice to develop my stupid example entails different consequences. However, not free vectors are also found in the course of high school (don't 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 vector difference, multiplication of a vector by a number, scalar product of vectors, etc. For the seed, let's repeat two rules that are especially relevant for solving problems of analytical geometry.

The rule for adding vectors according to the rule of triangles

Consider two arbitrary nonzero vectors and:

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

The sum of vectors is a vector. For a better understanding of the rule, it is advisable to put a physical meaning in it: let some body make a path along a vector, and then along a 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 the saying goes, the body can go its way strongly along the zigzag, and maybe on autopilot - according to the resulting sum vector.

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

First, about collinear vectors. The two vectors are called collinearif they lie on one straight 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 direction.

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

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

The rule of multiplying a vector by a number is easier to understand with the help of the figure:

Let's understand in more detail:

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

2) Length. If the factor is 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 is greater than one, then the vector length increases in time.

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

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

Which vectors are equal?

Two vectors are equal if they are codirectional and have the same length... Note that codirectionality implies collinear vectors. The definition will be inaccurate (redundant) if we say: "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 one and 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 postpone from the origin single vectors and:

Vectors and orthogonal... Orthogonal \u003d Perpendicular. I recommend to slowly get 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 perpendicularity symbol, for example:.

The vectors under consideration 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 basisIn 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 the basis of the plane: "ortho" - because the coordinate vectors are orthogonal, the adjective "normalized" means unit, i.e. the lengths of the vectors of the basis 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 can't rearrange.

Any vector plane unique way expressed as:
, where - numberswhich are called vector coordinates in this basis. And the expression itself called decomposition of the vector on the basis .

Dinner is served:

Let's start with the first letter of the alphabet:. The drawing clearly shows that when expanding the vector in terms of the basis, the ones just considered are used:
1) the rule for multiplying 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 decay will "follow him relentlessly." Here it is, the freedom of the vector - the vector "carries everything with itself." This property is, of course, true for any vector. It's funny that the basic (free) vectors themselves do not have to be postponed 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 do not need to do this, because the teacher will also show originality and draw you "credited" in an unexpected place.

Vectors, illustrate exactly the rule of multiplying a vector by a number, the vector is codirectional with the base vector, the vector is opposite to the base vector. These vectors have one of the coordinates equal to zero, it can be meticulously written as follows:


And the basis vectors, by the way, 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 anymore, 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 how the good old triangle addition of vectors works clearly in these situations.

The considered decomposition of the form sometimes called 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 equal 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: coordinates of vectors cannot be rearranged. Strictly in the 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 have figured out the coordinates on the plane. Now let's look at vectors in three-dimensional space, everything is almost the same here! Only one more coordinate will be added. It is difficult to carry out three-dimensional drawings, so I will limit myself to one vector, which I will postpone from the origin for simplicity:

Any vector of three-dimensional space can the only way expand on 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 rules work here. First, multiplying a vector by a number: (red arrow), (green arrow) and (crimson arrow). Secondly, here is an example of adding several, in this case three, vectors:. The sum vector starts at the starting point of departure (vector start) and ends at the final arrival point (vector end).

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 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 absent in the expansion, then zeros are put in their place. Examples:
vector (meticulously ) - write down;
vector (meticulously ) - write down;
vector (meticulously ) - write it down.

Basis vectors are written as follows:

Here, perhaps, is all the minimum theoretical knowledge required to solve problems in analytical geometry. Perhaps there are a lot of terms and definitions, so I recommend to 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 often be used in what follows. I note that the materials on 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 to your understanding of the subject. For a detailed theoretical background, please follow the 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 desirable to learn how to solve the tasks that will be considered fully automatic, and the formulas memorize, not even specifically memorizing, they themselves will be remembered \u003d) 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. There is no need to fasten the top buttons on the shirt, many things are familiar to you from school.

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

How to find a vector by two points?

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

If two points of 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.

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

Example 1

Two points of the plane and are given. Find vector coordinates

Decision: according to the corresponding formula:

Alternatively, the following entry could be used:

Aesthetes will decide this way:

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

Answer:

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

Must understand difference between point coordinates and vector coordinates:

Point coordinates Are the usual coordinates in a rectangular coordinate system. I think everyone knows how to put points on the coordinate plane since 5-6 grade. Each point has a strict place on the plane, and you cannot move them anywhere.

The coordinates of the vector Is its expansion in 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 (to avoid confusion, renaming it, for example, through). It is interesting that for vectors it is possible not to build axes at all, a rectangular coordinate system, only a basis is needed, in this case an orthonormal basis of the plane.

The records of the coordinates of points and coordinates of vectors seem to be similar:, and meaning of coordinates absolutely differentand you should understand this difference well. This difference, of course, is also true for space.

Ladies and gentlemen, we fill our hand:

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 an independent solution, 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 problems in analytical geometry? It is important to be EXTREMELY CAREFUL to avoid the “two plus two equals zero” workshop error. I apologize right away if I made a mistake \u003d)

How to find the length of a line segment?

The length, as already noted, is indicated by the module 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 of 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 rearranged: and, but the first option is more standard

Example 3

Decision: according to the corresponding formula:

Answer:

For clarity, I will make a drawing

Section - this is not a vector, and of course you cannot move it anywhere. In addition, if you complete a drawing to scale: 1 unit. \u003d 1 cm (two notebook cells), then the answer obtained can be checked with an ordinary ruler by directly measuring the length of the segment.

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

First, 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 “unit”.

Secondly, we will repeat the school material, which is useful not only for the problem under consideration:

pay attention to important technique – taking a factor out from under the root... As a result of the calculations, we got a result and good mathematical style involves taking the factor out from under the root (if possible). In more detail, the process looks like this: ... Of course, leaving the answer in the form will not be a mistake - but a defect, for sure, and a weighty argument for nagging on the part of the teacher.

Other common cases are:

Often a fairly large number is obtained under the root, for example. What to do in such cases? On the calculator, check if the number is divisible by 4:. Yes, it was split altogether, thus: ... Or maybe the number can be divided by 4 again? ... In this way: ... The last digit of the number is odd, so it is clearly not possible to divide by 4 a third time. We try to divide by nine:. As a result:
Done.

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

In the course of solving various problems, the roots are often encountered, always try to extract factors from under the root in order to avoid a lower score and unnecessary problems with revising your solutions as instructed by the teacher.

Let's also repeat the squaring and other powers:

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

Task for an independent solution with a segment in space:

Example 4

Points and are given. Find the length of the line segment.

Solution and answer at the end of the lesson.

How do I find the length of a vector?

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

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

These formulas (as well as formulas for the length of a segment) can be easily derived using the well-known Pythagorean theorem.

The article below will highlight the issues of finding the coordinates of the midpoint of a segment if there are coordinates of its extreme points as the initial data. But, before starting to study the issue, we introduce a number of definitions.

Definition 1

Section - a straight line connecting two arbitrary points, called the ends of the line. As an example, let it be points A and B and, accordingly, segment A B.

If segment A B continues in both directions from points A and B, we get straight line A B. Then the segment A B is a part of the resulting line, bounded by points A and B. Segment A B joins points A and B, which are its ends, as well as a set of points lying between. If, for example, we take any arbitrary point K lying between points A and B, we can say that point K lies on the segment A B.

Definition 2

Segment length - the distance between the ends of the segment at a given scale (segment of unit length). The length of the segment A B is denoted as follows: A B.

Definition 3

Mid-point - a point lying on a segment and equidistant from its ends. If the midpoint of the segment A B is denoted by point C, then the equality will be true: A C \u003d C B

Initial data: coordinate line O x and non-coincident points on it: A and B. These points correspond to real numbers x A and x B. Point C - midpoint of segment A B: it is necessary to determine the coordinate x C.

Since point C is the midpoint of segment A B, the following equality will be true: | A C | \u003d | C B | ... The distance between points is determined by the module of the difference between their coordinates, i.e.

| A C | \u003d | C B | ⇔ x C - x A \u003d x B - x C

Then two equalities are possible: x C - x A \u003d x B - x C and x C - x A \u003d - (x B - x C)

From the first equality we derive the formula for the coordinates of the point C: x C \u003d x A + x B 2 (half the sum of the coordinates of the ends of the segment).

From the second equality we get: x A \u003d x B, which is impossible, since in the original data, mismatched points. In this way, formula for determining the coordinates of the midpoint of a segment A B with ends A (x A) and B (x B):

The resulting formula will be the basis for determining the coordinates of the midpoint of a segment on a plane or in space.

Initial data: rectangular coordinate system on the plane O x y, two arbitrary non-coinciding points with the given coordinates A x A, y A and B x B, y B. Point C is the midpoint of segment A B. It is necessary to determine the coordinates x C and y C for point C.

Let us take for analysis the case when points A and B do not coincide and do not lie on the same coordinate line or a straight line perpendicular to one of the axes. A x, A y; B x, B y and C x, C y - projections of points A, B and C on the coordinate axes (straight lines O x and O y).

According to the construction, the lines A A x, B B x, C C x are parallel; straight lines are also parallel to each other. Together with this, according to Thales' theorem, from the equality A C \u003d C B, the equalities follow: A x C x \u003d C x B x and A y C y \u003d C y In y, and they in turn indicate that the point C x is the middle of the segment A x B x, and C y is the midpoint of the segment A y B y. And then, based on the formula obtained earlier, we get:

x C \u003d x A + x B 2 and y C \u003d y A + y B 2

The same formulas can be used in the case when points A and B lie on the same coordinate line or a straight line perpendicular to one of the axes. We will not carry out a detailed analysis of this case, we will consider it only graphically:

Summarizing all of the above, coordinates of the midpoint of the segment A B on the plane with the coordinates of the ends A (x A, y A) and B (x B, y B) defined as:

(x A + x B 2, y A + y B 2)

Initial data: coordinate system О x y z and two arbitrary points with given coordinates A (x A, y A, z A) and B (x B, y B, z B). It is necessary to determine the coordinates of the point C, which is the midpoint of the segment A B.

A x, A y, A z; B x, B y, B z and C x, C y, C z - projections of all specified points on the axis of the coordinate system.

According to Thales' theorem, the following equalities are true: A x C x \u003d C x B x, A y C y \u003d C y B y, A z C z \u003d C z B z

Therefore, the points C x, C y, C z are the midpoints of the segments A x B x, A y B y, A z B z, respectively. Then, to determine the coordinates of the midpoint of a segment in space, the following formulas are valid:

x C \u003d x A + x B 2, y c \u003d y A + y B 2, z c \u003d z A + Z B 2

The formulas obtained are also applicable in cases where points A and B lie on one of the coordinate lines; on a straight line perpendicular to one of the axes; in one coordinate plane or a plane perpendicular to one of the coordinate planes.

Determining the coordinates of the midpoint of a segment through the coordinates of the radius vectors of its ends

The formula for finding the coordinates of the midpoint of a segment can also be derived according to the algebraic interpretation of vectors.

Initial data: rectangular Cartesian coordinate system O x y, points with given coordinates A (x A, y A) and B (x B, x B). Point C is the midpoint of segment A B.

According to the geometric definition of actions on vectors, the following equality will be true: O C → \u003d 1 2 · O A → + O B →. Point C in this case is the intersection point of the diagonals of the parallelogram built on the basis of the vectors O A → and O B →, i.e. midpoint of the diagonals. The coordinates of the radius vector of the point are equal to the coordinates of the point, then the equalities are true: O A → \u003d (x A, y A), O B → \u003d (x B, y B). Let's perform some operations on vectors in coordinates and get:

O C → \u003d 1 2 O A → + O B → \u003d x A + x B 2, y A + y B 2

Therefore, point C has coordinates:

x A + x B 2, y A + y B 2

By analogy, a formula is determined for finding the coordinates of the midpoint of a segment in space:

C (x A + x B 2, y A + y B 2, z A + z B 2)

Examples of solving problems for finding the coordinates of the midpoint of a segment

Among the tasks involving the use of the formulas obtained above, there are both those in which the question of calculating the coordinates of the midpoint of a segment is directly involved, and those that involve bringing the given conditions to this question: the term "median" is often used, the goal is to find from the ends of the segment, and also common problems on symmetry, the solution of which, in general, should also not cause difficulties after studying this topic. Let's consider typical examples.

Example 1

Initial data: on the plane - points with given coordinates A (- 7, 3) and B (2, 4). It is necessary to find the coordinates of the midpoint of the segment A B.

Decision

Denote the midpoint of the segment A B by point C. Its coordinates will be defined as the half-sum of the coordinates of the ends of the segment, i.e. points A and B.

x C \u003d x A + x B 2 \u003d - 7 + 2 2 \u003d - 5 2 y C \u003d y A + y B 2 \u003d 3 + 4 2 \u003d 7 2

Answer: coordinates of the middle of the segment A B - 5 2, 7 2.

Example 2

Initial data: the coordinates of the triangle A B C are known: A (- 1, 0), B (3, 2), C (9, - 8). It is necessary to find the length of the median A M.

Decision

1. By the hypothesis of the problem, M is the median, and therefore M is the midpoint of the segment B C. First of all, we find the coordinates of the midpoint of the segment B C, i.e. point M:

x M \u003d x B + x C 2 \u003d 3 + 9 2 \u003d 6 y M \u003d y B + y C 2 \u003d 2 + (- 8) 2 \u003d - 3

1. Since now we know the coordinates of both ends of the median (points A and M), we can use the formula to determine the distance between the points and calculate the length of the median A M:

A M \u003d (6 - (- 1)) 2 + (- 3 - 0) 2 \u003d 58

Answer: 58

Example 3

Initial data: in a rectangular coordinate system of three-dimensional space, a parallelepiped A B C D A 1 B 1 C 1 D 1 is given. The coordinates of the point C 1 (1, 1, 0) are given, and also the point M is defined, which is the middle of the diagonal B D 1 and has coordinates M (4, 2, - 4). It is necessary to calculate the coordinates of point A.

Decision

The diagonals of the parallelepiped have an intersection at one point, which is the midpoint of all the diagonals. Based on this statement, it can be borne in mind that the point M known from the conditions of the problem is the midpoint of the segment A C 1. Based on the formula for finding the coordinates of the midpoint of a segment in space, we find the coordinates of point A: x M \u003d x A + x C 1 2 ⇒ x A \u003d 2 x M - x C 1 \u003d 2 4 - 1 + 7 y M \u003d y A + y C 1 2 ⇒ y A \u003d 2 y M - y C 1 \u003d 2 2 - 1 \u003d 3 z M \u003d z A + z C 1 2 ⇒ z A \u003d 2 z M - z C 1 \u003d 2 (- 4) - 0 \u003d - 8

Answer: coordinates of point A (7, 3, - 8).

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In this article, you and I will begin a discussion of one "magic wand" that will allow you to reduce many geometry problems to simple arithmetic. This "stick" can make your life much easier, especially in the case when you feel insecure in the construction of spatial figures, sections, etc. All this requires a certain imagination and practical skills. The method, which we will begin to consider here, will allow you to abstract almost completely from all kinds of geometric constructions and reasoning. The method is called "Coordinate method"... In this article, we will consider the following questions:

1. Coordinate plane
2. Points and vectors in the plane
3. Constructing a vector from two points
4. Vector length (distance between two points)
5. Midpoint coordinates
6. Dot product of vectors
7. Angle between two vectors

I think you already guessed why the coordinate method is called that? It is true that he received such a name, since he operates not with geometric objects, but with their numerical characteristics (coordinates). And the transformation itself, which allows us to go from geometry to algebra, consists in introducing a coordinate system. If the original figure was flat, then the coordinates are two-dimensional, and if the figure is three-dimensional, then the coordinates are three-dimensional. In this article, we will only consider the two-dimensional case. And the main goal of the article is to teach you how to use some basic techniques of the coordinate method (they sometimes turn out to be useful in solving problems on planimetry in part B of the USE). The next two sections on this topic are devoted to the discussion of methods for solving problems C2 (the problem of stereometry).

Where would it be logical to start discussing the coordinate method? Probably from the concept of a coordinate system. Remember when you first encountered her. It seems to me that in the 7th grade, when you learned about the existence of a linear function, for example. Let me remind you that you built it point by point. Do you remember? You chose an arbitrary number, substituted it into the formula and calculated that way. For example, if, then, if, then, etc. What did you get in the end? And you received points with coordinates: and. Then you drew a "cross" (coordinate system), chose a scale on it (how many cells you will have as a unit segment) and marked the points you received on it, which you then connected with a straight line, the resulting line is the graph of the function.

There are several points here that should be explained to you in a little more detail:

1. You choose a single segment for reasons of convenience, so that everything fits nicely and compactly in the picture

2. It is assumed that the axis goes from left to right, and the axis goes from bottom to top

3. They intersect at right angles, and the point of their intersection is called the origin. It is indicated by a letter.

4. In recording the coordinates of a point, for example, on the left in brackets is the coordinate of the point along the axis, and on the right, along the axis. In particular, it simply means that at the point

5. In order to set any point on the coordinate axis, you need to specify its coordinates (2 numbers)

6. For any point lying on the axis,

7. For any point on the axis,

8. The axis is called the abscissa axis

9. The axis is called the y-axis.

Now let's take the next step with you: mark two points. Let's connect these two points with a segment. And let's put the arrow as if we were drawing a segment from point to point: that is, we will make our segment directed!

Remember, what else is a directed segment called? That's right, it's called a vector!

Thus, if we connect a point with a point, where the beginning will be point A, and the end will be point B, then we get a vector. You also did this formation in the 8th grade, remember?

It turns out that vectors, like points, can be denoted by two numbers: these numbers are called the coordinates of the vector. The question is: do you think it is enough for us to know the coordinates of the beginning and end of the vector to find its coordinates? It turns out that yes! And this is done very simply:

Thus, since the point in the vector is the beginning and a is the end, the vector has the following coordinates:

For example, if, then the coordinates of the vector

Now let's do the opposite, find the coordinates of the vector. What do we need to change for this? Yes, you need to swap the beginning and end: now the beginning of the vector will be at the point, and the end - at the point. Then:

Look closely, how are vectors and? Their only difference is the signs in the coordinates. They are opposite. It is customary to write this fact like this:

Sometimes, if it is not specifically specified which point is the beginning of the vector and which is the end, then the vectors are denoted not by two capital letters, but by one lowercase, for example:, etc.

Now a little practice yourself and find the coordinates of the following vectors:

Verification:

Now solve the problem a little harder:

Vektor with na-cha-lom at the point has co-or-di-na-ty. Nay-di-those abs-cis-su points.

All the same is rather prosaic: Let be the coordinates of a point. Then

I made up the system by definition of what the coordinates of a vector are. Then the point has coordinates. We are interested in the abscissa. Then

Answer:

What else can you do with vectors? Yes, almost everything is the same as with ordinary numbers (except that you cannot divide, but you can multiply in two ways, one of which we will discuss here a little later)

1. Vectors can be added to each other
2. Vectors can be subtracted from each other
3. Vectors can be multiplied (or divided) by an arbitrary nonzero number
4. Vectors can be multiplied by each other

All these operations have a very clear geometric representation. For example, the triangle (or parallelogram) rule for addition and subtraction:

The vector expands or contracts or changes direction when multiplied or divided by a number:

However, here we will be interested in the question of what is happening with the coordinates.

1. When adding (subtracting) two vectors, we add (subtract) their coordinates element by element. I.e:

2. When multiplying (dividing) a vector by a number, all its coordinates are multiplied (divided) by this number:

For instance:

· Nay-di-te sum of co-or-di-nat vek-to-ra.

Let's first find the coordinates of each of the vectors. They both have the same origin - the origin point. Their ends are different. Then,. Now let's calculate the coordinates of the vector Then the sum of the coordinates of the resulting vector is.

Answer:

Now solve the following problem yourself:

Find the sum of the coordinates of a vector

We check:

Let's now consider the following problem: we have two points on the coordinate plane. How to find the distance between them? Let the first point be, and the second. Let's denote the distance between them through. Let's make the following drawing for clarity:

What I've done? I, firstly, connected the points and, and also from the point I drew a line parallel to the axis, and from the point I drew a line parallel to the axis. Did they intersect at a point, thus forming a wonderful figure? What is it remarkable for? Yes, you and I know almost everything about a right-angled triangle. Well, the Pythagorean theorem - for sure. The sought segment is the hypotenuse of this triangle, and the segments are the legs. What are the coordinates of a point? Yes, they are easy to find from the picture: Since the segments are parallel to the axes and, accordingly, their lengths are easy to find: if you denote the lengths of the segments, respectively, by, then

Now let's use the Pythagorean theorem. We know the lengths of the legs, we will find the hypotenuse:

Thus, the distance between two points is the root of the sum of the squares of the differences from the coordinates. Or, the distance between two points is the length of the line that connects them. It is easy to see that the distance between the points is independent of direction. Then:

From this we draw three conclusions:

Let's do a little practice calculating the distance between two points:

For example, if, then the distance between and is

Or let's go differently: find the coordinates of the vector

And find the length of the vector:

As you can see, the same thing!

Now do some practice yourself:

Task: find the distance between the specified points:

We check:

Here are a couple more problems for the same formula, although they sound a little different:

1. Nay-di-te square of the length of the century-to-ra.

2. Nay-di-te square-rat of the length of the century-to-ra

I think you did with them easily? We check:

1. And this is for attention) We have already found the coordinates of vectors and earlier:. Then the vector has coordinates. The square of its length will be:

2. Find the coordinates of the vector

Then the square of its length is

Nothing complicated, right? Simple arithmetic, nothing more.

The following tasks cannot be categorized unambiguously, they are more likely to general erudition and the ability to draw simple pictures.

1. Nay-di-te sine of an angle on-to-on from-cut, co-single-nya-yu-shch-th point, with the abscissa axis.

and

What are we going to do here? You need to find the sine of the angle between and the axis. And where do we know how to look for a sine? Right in a right-angled triangle. So what do we need to do? Build this triangle!

Since the coordinates of the point are and, the segment is equal, and the segment. We need to find the sine of the angle. Let me remind you that the sinus is the ratio of the opposite leg to the hypotenuse, then

What is left for us to do? Find the hypotenuse. You can do this in two ways: by the Pythagorean theorem (the legs are known!) Or by the formula for the distance between two points (in fact, the same thing as the first way!). I'll go the second way:

Answer:

The next task will seem even easier to you. She - on the coordinates of the point.

Objective 2. Per-pen-di-ku-lar is lowered from the point to the abs-ciss axis. Nay-di-te abs-cis-su os-no-va-nia per-pen-di-ku-la-ra.

Let's make a drawing:

The base of the perpendicular is the point at which it crosses the abscissa axis (axis), for me this is the point. The figure shows that it has coordinates:. We are interested in the abscissa - that is, the "x" component. It is equal.

Answer: .

Objective 3. Under the conditions of the previous problem, find the sum of the distances from a point to the coordinate axes.

The task is generally elementary if you know what the distance from a point to the axes is. You know? I hope, but still remind you:

So, in my picture, located a little higher, I have already drawn one such perpendicular? Which axis is it to? To the axis. And then what is its length equal to? It is equal. Now draw the perpendicular to the axis yourself and find its length. It will be equal, right? Then their sum is equal.

Answer: .

Problem 4. In the conditions of problem 2, find the ordinate of the point symmetrical to the point relative to the abscissa axis.

I think you intuitively understand what symmetry is? Many objects have it: many buildings, tables, airplanes, many geometric shapes: a ball, cylinder, square, rhombus, etc. Roughly speaking, symmetry can be understood as follows: a figure consists of two (or more) identical halves. This symmetry is called axial. What then is an axis? This is exactly the line along which a figure can, relatively speaking, be "cut" into identical halves (in this picture, the axis of symmetry is a straight line):

Now let's get back to our problem. We know that we are looking for a point that is symmetrical about the axis. Then this axis is the axis of symmetry. So, we need to mark a point so that the axis cuts the segment into two equal parts. Try to mark such a point yourself. Now compare with my solution:

Did you do the same? Good! At the found point, we are interested in the ordinate. She is equal

Answer:

Now tell me, after thinking about seconds, what will be the abscissa of a point symmetric to point A relative to the ordinate? What is your answer? Correct answer: .

In general, the rule can be written like this:

A point symmetrical to a point relative to the abscissa axis has coordinates:

A point symmetrical to a point about the ordinate axis has coordinates:

Well, now it's completely scary a task: find the coordinates of the point symmetric to the point, relative to the origin. You first think for yourself, and then look at my drawing!

Answer:

Now parallelogram problem:

Problem 5: The points are ver-shi-na-mi paral-le-lo-gram-ma. Nay-di-te or-di-na-tu points.

You can solve this problem in two ways: logic and the method of coordinates. I will first apply the coordinate method, and then I will tell you how you can decide otherwise.

It is quite clear that the abscissa of the point is. (it lies on the perpendicular drawn from a point to the abscissa). We need to find the ordinate. Let's use the fact that our figure is a parallelogram, which means that. Find the length of the segment using the formula for the distance between two points:

We lower the perpendicular connecting the point to the axis. The intersection point will be marked with a letter.

The segment length is. (find the problem itself, where we discussed this point), then we find the length of the segment by the Pythagorean theorem:

The length of the segment is exactly the same as its ordinate.

Answer: .

Another solution (I'll just give a picture that illustrates it)

Solution progress:

1. Conduct

2. Find the coordinates of the point and the length

3. Prove that.

Another one segment length puzzle:

The points are the top-shi-na-mi tre-coal-ni-ka. Nay-di-te is the length of its middle line, paral-lel-noy.

Do you remember what the middle line of a triangle is? Then this task is elementary for you. If you don't remember, then I will remind you: the middle line of a triangle is the line that connects the midpoints of opposite sides. It is parallel to the base and equal to half of it.

The base is a line segment. We had to search for its length earlier, it is equal. Then the length of the middle line is half and equal.

Answer: .

Commentary: this problem can be solved in another way, which we will turn to a little later.

In the meantime - here are a few tasks for you, practice them, they are quite simple, but they help you to "get your hand" using the method of coordinates!

1. The points are the ver-shi-na-mi tra-petsii. Nay-di-te is the length of its middle line.

2. Dots and are-la-is-Xia ver-shi-na-mi pa-ra-le-lo-gram-ma. Nay-di-te or-di-na-tu points.

3. Nay-di-te length from-cut, co-single-nya-yu-shch-go point and

4. Nay-di-te area of \u200b\u200bthe beautiful fi-gu-ry on the co-or-di-nat-noy plane.

5. The circle with the center at na-cha-le ko-or-di-nat passes through the point. Nay-di-te her ra-di-us.

6. Nai-di-te ra-di-us of the circle, described-san-noy around the rectangular-coal-nik-ka, the vertices of the ko-to-ro-go have co-op -di-na-you co-vet-but

Solutions:

1. It is known that the middle line of a trapezoid is equal to the half-sum of its bases. The base is equal, and the base is. Then

Answer:

2. The easiest way to solve this problem is to notice that (parallelogram rule). Calculate the coordinates of vectors and is not difficult:. When adding vectors, the coordinates are added. Then has coordinates. The point has the same coordinates, since the origin of the vector is the point with coordinates. We are interested in the ordinate. It is equal.

Answer:

3. We act immediately according to the formula for the distance between two points:

Answer:

4. Look at the picture and tell me, between which two shapes is the shaded area "sandwiched"? It is sandwiched between two squares. Then the area of \u200b\u200bthe required figure is equal to the area of \u200b\u200bthe large square minus the area of \u200b\u200bthe small one. The side of the small square is a line segment connecting the points and Its length is

Then the area of \u200b\u200bthe small square is

We do the same with a large square: its side is a segment connecting the points and its length is

Then the area of \u200b\u200bthe big square is

The area of \u200b\u200bthe required figure is found by the formula:

Answer:

5. If the circle has the origin of coordinates as its center and passes through a point, then its radius will be exactly equal to the length of the segment (draw a picture and you will understand why this is obvious). Let's find the length of this segment:

Answer:

6. It is known that the radius of a circle circumscribed about a rectangle is equal to half of its diagonal. Let's find the length of any of the two diagonals (after all, in a rectangle they are equal!)

Answer:

Well, have you dealt with everything? It wasn't very hard to figure it out, was it? The rule here is one - to be able to make a visual picture and simply "read" all the data from it.

We have very little left. There are literally two more points that I would like to discuss.

Let's try to solve this simple problem. Let two points and be given. Find the coordinates of the midpoint of the segment. The solution to this problem is as follows: let the point be the desired midpoint, then it has coordinates:

I.e: midpoint coordinates \u003d arithmetic mean of the corresponding coordinates of the segment ends.

This rule is very simple and usually does not cause difficulties for students. Let's see what tasks and how it is used:

1. Nay-di-te or-di-na-tu-re-di-us from-cut, co-uni-nya-yu-shch-go point

2. The points are-la-yut-sya ver-shi-na-mi-you-rekh-coal-no-ka. Nay-di-te or-di-na-tu points of pe-re-se-ch-niya his dia-go-na-lei.

3. Nay-di-those abs-cis-su center-tra of the circle, described-san-noy around the coal-no-ka, the vertices of the ko-ro-go co-op-di-na-you co-vet-but.

Solutions:

1. The first problem is just a classic. We act immediately to determine the middle of the segment. It has coordinates. The ordinate is.

Answer:

2. It is easy to see that the given quadrangle is a parallelogram (even a rhombus!). You yourself can prove this by calculating the lengths of the sides and comparing them with each other. What do I know about a parallelogram? Its diagonals are halved by the intersection point! Aha! So the point of intersection of the diagonals is what? This is the middle of any of the diagonals! I will choose, in particular, the diagonal. Then the point has coordinates The ordinate of the point is.

Answer:

3.What does the center of the circumscribed circle about the rectangle coincide with? It coincides with the intersection point of its diagonals. What do you know about the diagonals of a rectangle? They are equal and the intersection point is halved. The task was reduced to the previous one. Take the diagonal, for example. Then if is the center of the circumscribed circle, then is the middle. Looking for coordinates: Abscissa is equal.

Answer:

Now practice a little yourself, I will just give the answers to each problem so that you can test yourself.

1. Nai-di-te ra-di-us of the circle, described-san-noy around the triangle, the vertices of the co-to-ro-go have co-or-di -no misters

2. Nay-di-te or-di-na-tu center-tra of the circle, described-san-noy around the triangle-nik, the vertices of ko-to-ro-go have coordinates

3. How-to-ra-di-u-sa should there be a circle with a center at a point so that it would line the absciss axis?

4. Nay-di-te or-di-na-tu points of pe-re-seeding of the axis and from-cut, co-uni-nya-yu-shch-go point

Answers:

Did you succeed? I really hope for it! Now - the last push. Be especially careful now. The material that I will now explain is directly related not only to simple problems on the coordinate method from part B, but also occurs everywhere in the C2 problem.

Which of my promises have I not kept yet? Remember what operations on vectors I promised to introduce and what ones I eventually introduced? Have I forgotten anything? I forgot! Forgot to explain what vector multiplication means.

There are two ways to multiply a vector by a vector. Depending on the chosen method, we will get objects of a different nature:

The vector product is pretty tricky. How to do it and what it is for, we will discuss with you in the next article. And in this one we will focus on the dot product.

There are two ways to calculate it:

As you guessed, the result should be the same! So let's look at the first way first:

Dot product in terms of coordinates

Find: - common dot product notation

The formula for the calculation is as follows:

That is, the dot product \u003d the sum of the products of the coordinates of the vectors!

Example:

Nai di te

Decision:

Let's find the coordinates of each of the vectors:

We calculate the dot product by the formula:

Answer:

See, absolutely nothing complicated!

Well, now try it yourself:

Nay-di-te scalar-noe pro-iz-ve-de-vek-to-moat and

Did you manage? Maybe you noticed a small catch? Let's check:

The coordinates of the vectors are the same as in the previous task! Answer:.

In addition to the coordinate, there is another way to calculate the dot product, namely, through the lengths of the vectors and the cosine of the angle between them:

Indicates the angle between vectors and.

That is, the dot product is equal to the product of the vector lengths by the cosine of the angle between them.

Why do we need this second formula, if we have the first, which is much simpler, at least there are no cosines in it. And it is needed so that we can derive from the first and second formulas how to find the angle between vectors!

Let Then remember the formula for the length of the vector!

Then if I substitute this data into the dot product formula, then I get:

But on the other side:

So what did you and I get? We now have a formula to calculate the angle between two vectors! Sometimes it is also written like this for brevity:

That is, the algorithm for calculating the angle between vectors is as follows:

1. Calculate the dot product in terms of coordinates
2. Find the lengths of the vectors and multiply them
3. Divide the result of point 1 by the result of point 2

Let's practice with examples:

1. Nay-di-those angle between the century-to-ra-mi and. Give the answer in gra-du-sakh.

2. Under the conditions of the previous problem, find the cosine between the vectors

Let's do this: I'll help you solve the first problem, and try to do the second yourself! I agree? Then let's get started!

1. These vectors are our old acquaintances. We have already counted their dot product and it was equal. Their coordinates are:,. Then we find their lengths:

Then we are looking for the cosine between the vectors:

What is the cosine of the angle? This is the corner.

Answer:

Now solve the second problem yourself, and then we will compare! I will only give a very short solution:

2. has coordinates, has coordinates.

Let be the angle between the vectors and, then

Answer:

It should be noted that problems directly on vectors and the method of coordinates in part B of the examination work are quite rare. However, the vast majority of C2 problems can be easily solved by introducing a coordinate system. So you can consider this article as the foundation, on the basis of which we will make quite cunning constructions that we will need to solve complex problems.

COORDINATES AND VECTORS. MEDIUM ROVEN

You and I continue to study the method of coordinates. In the last part, we derived a number of important formulas that allow:

1. Find vector coordinates
2. Find the length of a vector (alternatively: the distance between two points)
3. Add, subtract vectors. Multiply them by a real number
4. Find the midpoint of a line segment
5. Calculate dot product of vectors
6. Find the angle between vectors

Of course, the whole coordinate method does not fit into these 6 points. It lies at the heart of such a science as analytical geometry, which you will get to know at the university. I just want to build a foundation that will allow you to solve problems in a single state. exam. We figured out the tasks of Part B in Now it's time to move to a qualitatively new level! This article will be devoted to the method for solving those problems C2, in which it would be reasonable to switch to the method of coordinates. This rationality is determined by what is required to find in the problem, and what figure is given. So, I would use the coordinate method if the questions are:

1. Find the angle between two planes
2. Find the angle between a line and a plane
3. Find the angle between two straight lines
4. Find the distance from a point to a plane
5. Find the distance from point to line
6. Find the distance from a straight line to a plane
7. Find the distance between two straight lines

If the figure given in the problem statement is a body of revolution (ball, cylinder, cone ...)

Suitable shapes for the coordinate method are:

1. Rectangular parallelepiped
2. Pyramid (triangular, quadrangular, hexagonal)

Also in my experience it is inappropriate to use the coordinate method for:

1. Finding the cross-sectional areas
2. Calculating body volumes

However, it should be noted right away that three situations “unfavorable” for the coordinate method are quite rare in practice. In most tasks, he can become your savior, especially if you are not very strong in three-dimensional constructions (which are sometimes quite intricate).

What are all the figures I have listed above? They are no longer flat, like, for example, a square, triangle, circle, but three-dimensional! Accordingly, we need to consider not a two-dimensional, but a three-dimensional coordinate system. It is easy to build: just in addition to the abscissa and ordinate axes, we will introduce one more axis, the applicate axis. The figure schematically shows their relative position:

All of them are mutually perpendicular, intersect at one point, which we will call the origin. The abscissa axis, as before, will be denoted, the ordinate axis -, and the entered applicate axis -.

If earlier each point on the plane was characterized by two numbers - the abscissa and the ordinate, then each point in space is already described by three numbers - the abscissa, ordinate, applicate. For instance:

Accordingly, the abscissa of the point is equal, the ordinate is, and the applicate is.

Sometimes the abscissa of a point is also called the projection of the point onto the abscissa axis, the ordinate is the projection of the point onto the ordinate axis, and the applicate is the projection of the point onto the applicate axis. Accordingly, if a point is specified, then a point with coordinates:

is called the projection of a point onto a plane

is called the projection of a point onto a plane

A natural question arises: are all the formulas derived for the two-dimensional case valid in space? The answer is yes, they are fair and look the same. For a little detail. I think you already guessed for which one. In all formulas, we will have to add one more term responsible for the axis of the applicate. Namely.

1. If two points are given:, then:

• Vector coordinates:
• Distance between two points (or vector length)
• The middle of the segment has coordinates

2. If two vectors are given: and, then:

• Their dot product is:
• The cosine of the angle between vectors is:

However, space is not so simple. As you can imagine, the addition of one more coordinate introduces a significant variety in the spectrum of figures "living" in this space. And for further narration I need to introduce some, roughly speaking, "generalization" of the straight line. This "generalization" is the plane. What do you know about a plane? Try to answer the question, what is a plane? It's very difficult to say. However, we all intuitively imagine what it looks like:

Roughly speaking, this is a kind of endless "leaf" tucked into space. "Infinity" should be understood that the plane extends in all directions, that is, its area is equal to infinity. However, this explanation "on the fingers" does not give the slightest idea of \u200b\u200bthe structure of the plane. And we will be interested in it.

Let's remember one of the basic axioms of geometry:

• a straight line passes through two different points on the plane, moreover, only one:

Or its counterpart in space:

Of course, you remember how to derive the equation of a straight line from two given points, it is not at all difficult: if the first point has coordinates: and the second, then the equation of the straight line will be as follows:

You went through this in 7th grade. In space, the equation of a straight line looks like this: let us have two points with coordinates:, then the equation of a straight line passing through them has the form:

For example, a straight line passes through the points:

How should this be understood? It should be understood as follows: a point lies on a straight line if its coordinates satisfy the following system:

We will not be very interested in the equation of the line, but we need to pay attention to the very important concept of the direction vector of a line. - any nonzero vector lying on the given line or parallel to it.

For example, both vectors are direction vectors of a straight line. Let be a point lying on a straight line, and be its direction vector. Then the equation of the straight line can be written in the following form:

Once again, I will not be very interested in the equation of a straight line, but I really need you to remember what a direction vector is! Again: it is ANY nonzero vector lying on a line or parallel to it.

Withdraw equation of a plane at three given points is no longer so trivial, and usually this issue is not addressed in a high school course. But in vain! This technique is vital when we use the coordinate method to solve complex problems. However, I assume that you are eager to learn something new? Moreover, you will be able to impress your teacher at the university when it turns out that you already know how with the methodology that is usually studied in the course of analytical geometry. So let's get started.

The equation of a plane is not too different from the equation of a straight line on a plane, namely, it has the form:

some numbers (not all equal to zero), but variables, for example: etc. As you can see, the equation of the plane is not very different from the equation of a straight line (linear function). However, remember what you and I said? We said that if we have three points that do not lie on one straight line, then the equation of the plane is uniquely reconstructed from them. But how? I'll try to explain to you.

Since the equation of the plane is:

And the points belong to this plane, then when substituting the coordinates of each point into the equation of the plane, we should get the correct identity:

Thus, it becomes necessary to solve three equations even with unknowns! Dilemma! However, you can always assume that (for this you need to divide by). Thus, we get three equations with three unknowns:

However, we will not solve such a system, but write out a mysterious expression that follows from it:

Equation of a plane passing through three given points

\\ [\\ left | (\\ begin (array) (* (20) (c)) (x - (x_0)) & ((x_1) - (x_0)) & ((x_2) - (x_0)) \\\\ (y - (y_0) ) & ((y_1) - (y_0)) & ((y_2) - (y_0)) \\\\ (z - (z_0)) & ((z_1) - (z_0)) & ((z_2) - (z_0)) \\ end (array)) \\ right | \u003d 0 \\]

Stop! What is this? Some very unusual module! However, the object you see in front of you has nothing to do with the module. This object is called a third-order determinant. From now on, when you deal with the method of coordinates on a plane, you will very often come across these same determinants. What is a third-order determinant? Oddly enough, this is just a number. It remains to understand what specific number we will compare with the determinant.

Let's first write the third-order determinant in a more general form:

Where are some numbers. Moreover, by the first index we mean the line number, and by the index - the column number. For example, it means that this number is at the intersection of the second row and the third column. Let's pose the next question: how exactly are we going to calculate such a determinant? That is, what specific number will we match to it? For the determinant of the third order there is a heuristic (visual) rule of the triangle, it looks like this:

1. The product of the elements of the main diagonal (from the upper left corner to the lower right) the product of the elements forming the first triangle "perpendicular" to the main diagonal product of the elements forming the second triangle "perpendicular" to the main diagonal
2. The product of the elements of the secondary diagonal (from the upper right corner to the lower left) the product of the elements forming the first triangle "perpendicular" to the secondary diagonal product of the elements forming the second triangle "perpendicular" to the secondary diagonal
3. Then the determinant is equal to the difference between the values \u200b\u200bobtained at step and

If we write all this in numbers, then we get the following expression:

Nevertheless, you do not need to memorize the calculation method in this form, it is enough to just keep the triangles in your head and the very idea of \u200b\u200bwhat adds up to what and what is then subtracted from what).

Let's illustrate the triangle method with an example:

1. Calculate the determinant:

Let's figure out what we add and what we subtract:

Terms that come with a "plus":

This is the main diagonal: the product of the elements is

The first triangle, "perpendicular to the main diagonal: the product of the elements is

The second triangle, "perpendicular to the main diagonal: the product of the elements is

Add three numbers:

Terms that come with a "minus"

This is a side diagonal: the product of the elements is

The first triangle, "perpendicular to the side diagonal: the product of the elements is

Second triangle, "perpendicular to the side diagonal: the product of the elements is

Add three numbers:

All that remains to be done is to subtract from the sum of the plus terms the sum of the minus terms:

In this way,

As you can see, there is nothing complicated and supernatural in calculating third-order determinants. It's just important to remember about triangles and not make arithmetic errors. Now try to calculate yourself:

We check:

1. First triangle perpendicular to the main diagonal:
2. Second triangle perpendicular to the main diagonal:
3. Sum of terms with plus:
4. First triangle perpendicular to the side diagonal:
5. Second triangle perpendicular to the side diagonal:
6. Sum of terms with minus:
7. The sum of terms with a plus minus the sum of terms with a minus:

Here's a couple more determinants, calculate their values \u200b\u200byourself and compare them with the answers:

Answers:

Well, did it all coincide? Great, then you can move on! If there are difficulties, then my advice is this: on the Internet there are a bunch of programs for calculating the determinant on-line. All you need is to come up with your own determinant, calculate it yourself, and then compare it with what the program will calculate. And so on until the results begin to coincide. I am sure this moment will not be long in coming!

Now let's return to the determinant that I wrote out when I talked about the equation of a plane passing through three given points:

All you need is to calculate its value directly (using the triangles method) and set the result to zero. Naturally, since they are variables, you will get some expression that depends on them. It is this expression that will be the equation of a plane passing through three given points that do not lie on one straight line!

Let's illustrate this with a simple example:

1. Construct the equation of the plane passing through the points

We compose the determinant for these three points:

Let's simplify:

Now we calculate it directly by the rule of triangles:

\\ [(\\ left | (\\ begin (array) (* (20) (c)) (x + 3) & 2 & 6 \\\\ (y - 2) & 0 & 1 \\\\ (z + 1) & 5 & 0 \\ end (array)) \\ \\ cdot 5 \\ cdot 6 -) \\]

Thus, the equation of the plane passing through the points has the form:

Now try to solve one problem yourself, and then we will discuss it:

2. Find the equation of the plane passing through the points

Well, let's now discuss the solution:

We compose the determinant:

And we calculate its value:

Then the equation of the plane has the form:

Or, by reducing by, we get:

Now two tasks for self-control:

1. Construct the equation of a plane passing through three points:

Answers:

Did it all coincide? Again, if there are certain difficulties, then my advice is this: you take three points from your head (with a high degree of probability they will not lie on one straight line), you build a plane along them. And then you check yourself online. For example, on the site:

However, with the help of determinants, we will construct not only the equation of the plane. Remember, I told you that not only dot product is defined for vectors. There is also a vector product as well as a mixed product. And if the dot product of two vectors is a number, then the vector product of two vectors will be a vector, and this vector will be perpendicular to the given ones:

Moreover, its module will be equal to the area of \u200b\u200bthe parallelogram built on the vectors and. We will need this vector to calculate the distance from a point to a straight line. How do we calculate the cross product of vectors and, if their coordinates are given? The determinant of the third order comes to our aid again. However, before I turn to the algorithm for calculating the vector product, I have to make a small lyrical digression.

This digression concerns basis vectors.

They are shown schematically in the figure:

Why do you think they are called basic? The fact is that :

Or in the picture:

The validity of this formula is obvious, because:

Vector product

Now I can start introducing the cross product:

The vector product of two vectors is a vector that is calculated according to the following rule:

Now let's give some examples of calculating a cross product:

Example 1: Find the cross product of vectors:

Solution: I compose a determinant:

And I calculate it:

Now, from writing in terms of basis vectors, I'll return to the usual notation of a vector:

In this way:

Now try it.

Are you ready? We check:

And traditionally two tasks for control:

1. Find the cross product of the following vectors:
2. Find the cross product of the following vectors:

Answers:

Mixed product of three vectors

The last construction I need is a mixed product of three vectors. It, like a scalar, is a number. There are two ways to calculate it. - through a determinant, - through a mixed product.

Namely, let us have three vectors:

Then the mixed product of three vectors, denoted by, can be calculated as:

1. - that is, the mixed product is the dot product of a vector by the cross product of two other vectors

For example, the mixed product of three vectors is:

Try to calculate it yourself through the cross product and make sure that the results match!

And again - two examples for an independent solution:

Answers:

Coordinate system selection

Well, now we have all the necessary knowledge base to solve complex stereometric problems in geometry. However, before proceeding directly to the examples and algorithms for their solution, I believe that it will be useful to dwell on another question: how exactly choose a coordinate system for a particular figure. After all, it is the choice of the relative position of the coordinate system and the figure in space that will ultimately determine how cumbersome the calculations will be.

Let me remind you that in this section we are looking at the following shapes:

1. Rectangular parallelepiped
2. Straight prism (triangular, hexagonal ...)
3. Pyramid (triangular, quadrangular)
4. Tetrahedron (same as triangular pyramid)

For a rectangular box or cube, I recommend you the following construction:

That is, I will place the figure "in the corner". The cube and the parallelepiped are very nice shapes. For them, you can always easily find the coordinates of its vertices. For example, if (as shown in the picture)

then the coordinates of the vertices are as follows:

Of course, you don't need to remember this, but remembering how best to place a cube or rectangular parallelepiped is desirable.

Straight prism

The prism is a more harmful figure. You can place it in space in different ways. However, the following option seems to me the most acceptable:

Triangular prism:

That is, we put one of the sides of the triangle entirely on the axis, and one of the vertices coincides with the origin.

Hexagonal prism:

That is, one of the vertices coincides with the origin, and one of the sides lies on the axis.

Quadrangular and hexagonal pyramid:

A situation similar to a cube: align the two sides of the base with the coordinate axes, align one of the vertices with the origin. The only small difficulty will be to calculate the coordinates of the point.

For a hexagonal pyramid - the same as for a hexagonal prism. The main task will again be in finding the coordinates of the vertex.

Tetrahedron (triangular pyramid)

The situation is very similar to the one I gave for a triangular prism: one vertex coincides with the origin, one side lies on the coordinate axis.

Well, now you and I are finally close to getting down to solving problems. From what I said at the very beginning of the article, you could draw the following conclusion: most C2 problems are divided into 2 categories: angle problems and distance problems. First, we will consider the problem of finding an angle. They, in turn, are divided into the following categories (as the difficulty increases):

Finding corners

1. Finding the angle between two straight lines
2. Finding the angle between two planes

Let's consider these tasks in sequence: start by finding the angle between two straight lines. Well, remember, did you and I solve similar examples before? Remember, we already had something similar ... We were looking for an angle between two vectors. I will remind you, if two vectors are given: and, then the angle between them is found from the ratio:

Now we have a goal - to find the angle between two straight lines. Let's turn to the "flat picture":

How many corners did we get when two straight lines intersect? As many things. True, only two of them are unequal, while others are vertical to them (and therefore coincide with them). So what angle should we consider the angle between two straight lines: or? Here the rule is: the angle between two straight lines is always no more than degrees... That is, from two angles we will always choose the angle with the smallest degree measure. That is, in this picture, the angle between the two straight lines is equal. In order not to bother with finding the smallest of two angles every time, cunning mathematicians suggested using the module. Thus, the angle between two straight lines is determined by the formula:

As an attentive reader, you should have had a question: where, in fact, will we take these very numbers that we need to calculate the cosine of an angle? Answer: we will take them from the direction vectors of the straight lines! Thus, the algorithm for finding the angle between two straight lines is as follows:

1. We apply formula 1.

Or in more detail:

1. We are looking for the coordinates of the direction vector of the first straight line
2. We are looking for the coordinates of the direction vector of the second straight line
3. We calculate the modulus of their dot product
4. We are looking for the length of the first vector
5. We are looking for the length of the second vector
6. Multiplying the results from point 4 by the results from point 5
7. Divide the result of point 3 by the result of point 6. We get the cosine of the angle between the lines
8. If this result allows you to calculate the angle exactly, look for it
9. Otherwise, we write through the inverse cosine

Well, now is the time to move on to the problems: I will demonstrate the solution of the first two in detail, I will present the solution of another one in a short form, and for the last two problems I will only give answers, you must carry out all the calculations for them yourself.

Tasks:

1. In the correct tet-ra-ed-re, nay-di-those angle between you-so-that tet-ra-ed-ra and the med-di-a-noy bo-kov face.

2. In the right-handed six-coal-noy pi-ra-mi-de, the sides of the os-no-va-nia are equal, and the ribs are equal, find the angle between the straight lines and.

3. The lengths of all the ribs of the correct four-you-rekh-coal pi-ra-mi-dy are equal to each other. Nay-di-those angle between the straight lines and if from-cut is you-co-that given pi-ra-mi-dy, the point is se-re-di-na her bo-ko- second rib

4. On the edge of the cube from-me-che-na point so that Nay-di-te is the angle between straight lines and

5. Point - se-re-di-on the edges of the cube Nay-di-te angle between straight lines and.

It is no coincidence that I have arranged the tasks in this order. While you have not yet had time to start navigating in the method of coordinates, I myself will analyze the most "problematic" figures, and I will leave you to deal with the simplest cube! Gradually, you will have to learn how to work with all the figures, the complexity of the tasks I will increase from topic to topic.

Let's start solving problems:

1. Draw a tetrahedron, place it in the coordinate system as I suggested earlier. Since the tetrahedron is regular, all of its faces (including the base) are regular triangles. Since we are not given the length of the side, I can take it equal. I think you understand that the angle will not really depend on how much our tetrahedron is "stretched"? I will also draw the height and median in the tetrahedron. Along the way, I will draw its base (it will also be useful to us).

I need to find the angle between and. What do we know? We only know the coordinate of the point. This means that we also need to find the coordinates of the points. Now we think: a point is the intersection point of the heights (or bisectors or medians) of the triangle. A point is a raised point. The point is the middle of the segment. Then finally we need to find: coordinates of points:.

Let's start with the simplest: point coordinates. Look at the picture: It is clear that the applicate of a point is zero (the point lies on the plane). Its ordinate is (since it is the median). It is more difficult to find its abscissa. However, this is easily done based on the Pythagorean theorem: Consider a triangle. Its hypotenuse is equal, and one of the legs is equal Then:

Finally we have:.

Now let's find the coordinates of the point. It is clear that its applicate is again equal to zero, and its ordinate is the same as that of a point, that is. Let's find its abscissa. This is done quite trivially if you remember that the heights of an equilateral triangle are divided by the point of intersection in proportioncounting from the top. Since:, then the required abscissa of the point, equal to the length of the segment, is equal to:. Thus, the coordinates of the point are equal:

Let's find the coordinates of the point. It is clear that its abscissa and ordinate coincide with the abscissa and ordinate of the point. And the applicate is equal to the length of the segment. - this is one of the legs of the triangle. The hypotenuse of a triangle is a segment - a leg. It is searched for from the considerations that I have highlighted in bold:

The point is the midpoint of the line. Then we need to remember the formula for the coordinates of the midpoint of the segment:

That's it, now we can search for the coordinates of the direction vectors:

Well, everything is ready: we substitute all the data into the formula:

In this way,

Answer:

You should not be intimidated by such "scary" answers: for C2 problems this is a common practice. I would rather be surprised at the "nice" answer in this part. Also, as you noticed, I practically did not resort to anything other than the Pythagorean theorem and the property of heights of an equilateral triangle. That is, to solve the stereometric problem, I used the very minimum of stereometry. The gain in this is partially "extinguished" by rather cumbersome calculations. But they are quite algorithmic!

2. Let's draw a regular hexagonal pyramid together with a coordinate system, as well as its base:

We need to find the angle between straight lines and. Thus, our task is reduced to finding the coordinates of points:. We will find the coordinates of the last three from the small picture, and we will find the coordinate of the vertex through the coordinate of the point. Work in bulk, but you have to start it!

a) Coordinate: it is clear that its applicate and ordinate are zero. Let's find the abscissa. To do this, consider a right-angled triangle. Alas, in it we know only the hypotenuse, which is equal to. We will try to find the leg (for it is clear that the doubled leg length will give us the abscissa of the point). How can we find her? Let's remember what kind of figure we have at the base of the pyramid? This is a regular hexagon. What does it mean? This means that it has all sides and all angles. One such corner should be found. Any ideas? There are a lot of ideas, but there is a formula:

The sum of the angles of a regular n-gon is .

Thus, the sum of the angles of a regular hexagon is equal to degrees. Then each of the angles is equal to:

We look at the picture again. It is clear that the segment is the bisector of the angle. Then the angle is equal to degrees. Then:

Then where.

Thus, it has coordinates

b) Now we can easily find the coordinate of the point:.

c) Find the coordinates of the point. Since its abscissa coincides with the length of the segment, it is equal to. Finding the ordinate is also not very difficult: if we connect the points and denote the intersection point of the straight line, say by. (DIY easy construction). Then Thus, the ordinate of point B is equal to the sum of the lengths of the segments. Let's look at the triangle again. Then

Then since Then the point has coordinates

d) Now find the coordinates of the point. Consider a rectangle and prove that Thus, the coordinates of the point are:

e) It remains to find the coordinates of the vertex. It is clear that its abscissa and ordinate coincide with the abscissa and ordinate of the point. Let's find the applicator. Since, then. Consider a right-angled triangle. By the condition of the problem, the side edge. This is the hypotenuse of my triangle. Then the height of the pyramid is the leg.

Then the point has coordinates:

All right, I have the coordinates of all the points of interest to me. Looking for the coordinates of the direction vectors of straight lines:

We are looking for the angle between these vectors:

Answer:

Again, when solving this problem, I did not use any sophisticated tricks, except for the formula for the sum of the angles of a regular n-gon, as well as determining the cosine and sine of a right triangle.

3. Since we are again not given the lengths of the ribs in the pyramid, I will consider them equal to one. Thus, since ALL edges, and not only the side ones, are equal to each other, then at the base of the pyramid and me lies a square, and the side faces are regular triangles. Let's draw such a pyramid, as well as its base on a plane, marking all the data given in the text of the problem:

We are looking for the angle between and. I will be doing very brief calculations when looking for the coordinates of points. You will need to "decipher" them:

b) - the middle of the segment. Its coordinates:

c) I will find the length of the segment by the Pythagorean theorem in a triangle. I will find it in a triangle by the Pythagorean theorem.

Coordinates:

d) - the middle of the segment. Its coordinates are equal

e) Vector coordinates

f) Vector coordinates

g) Looking for an angle:

The cube is the simplest figure. I'm sure you can deal with her yourself. The answers to problems 4 and 5 are as follows:

Finding the angle between a straight line and a plane

Well, the time for simple tasks is over! Now the examples will be even more complicated. To find the angle between a straight line and a plane, we will proceed as follows:

1. From three points we construct the equation of the plane
   ,
   using a third-order determinant.
2. Looking for the coordinates of the directing vector of the straight line by two points:
3. We apply the formula to calculate the angle between a straight line and a plane:

As you can see, this formula is very similar to the one we used to find the angles between two lines. The structure of the right side is just the same, and on the left we are now looking for the sine, not the cosine as before. Well, one nasty action was added - the search for the equation of the plane.

Let's not postpone solution of examples:

1. The main-but-va-no-em direct prize-we are-la-is-equal-but-poor-worn triangles You-so-that prize-we are equal. Nai di te angle between straight line and plane

2. In a rectangular pa-ra-le-le-pi-pe-de from the West Nay-di-te angle between straight line and plane

3. In the correct six-coal prism, all edges are equal. Nay-di-those angle between a straight line and a plane.

4. In the right-handed triangular pi-ra-mi-de with os-no-va-ni-it is the ribs Nay-di-te angle, ob-ra-zo-van flat-to-bone os-no-va-nia and straight, pro-ho-dya-shi through se-re-di-us of the ribs and

5. The lengths of all the ribs of the right four-corner pyramid with apex are equal to each other. Nay-di-te is the angle between a straight line and a plane, if the point is se-re-di-na bo-ko-th ribs pi-ra-mi-dy.

Again I will solve the first two problems in detail, the third - briefly, and I leave the last two to you for your own decision. Besides, you have already dealt with triangular and quadrangular pyramids, but not with prisms yet.

Solutions:

1. Let's depict the prism, as well as its base. Let's combine it with the coordinate system and mark all the data given in the problem statement:

I apologize for some non-observance of proportions, but for solving the problem, this, in fact, is not so important. The plane is just the "back wall" of my prism. It is easy enough to guess that the equation of such a plane has the form:

However, this can be shown directly:

Let's choose arbitrary three points on this plane: for example,.

Let's compose the equation of the plane:

Exercise for you: calculate this determinant yourself. Did you do it? Then the equation of the plane has the form:

Or simply

In this way,

To solve the example, I need to find the coordinates of the direction vector of a straight line. Since the point coincides with the origin, the coordinates of the vector will simply coincide with the coordinates of the point. To do this, we first find the coordinates of the point.

To do this, consider a triangle. Let's draw the height (it is the median and the bisector) from the vertex. Since, then the ordinate of the point is. In order to find the abscissa of this point, we need to calculate the length of the segment. By the Pythagorean theorem we have:

Then the point has coordinates:

A point is a “raised” point:

Then the coordinates of the vector:

Answer:

As you can see, there is nothing fundamentally difficult in solving such problems. In fact, the process further simplifies the "straightness" of a shape such as a prism. Now let's move on to the next example:

2. Draw a parallelepiped, draw a plane and a straight line in it, and also separately draw its lower base:

First, we find the equation of the plane: Coordinates of three points lying in it:

(The first two coordinates were obtained in an obvious way, and you can easily find the last coordinate from the picture from the point). Then we compose the equation of the plane:

We calculate:

We are looking for the coordinates of the direction vector: It is clear that its coordinates coincide with the coordinates of the point, isn't it? How do I find the coordinates? These are the coordinates of the point, raised along the axis of the application by one! ... Then we are looking for the required angle:

Answer:

3. Draw a regular hexagonal pyramid, and then draw a plane and a line in it.

Here, even drawing a plane is problematic, not to mention solving this problem, but the coordinate method does not care! It is in its versatility that its main advantage lies!

The plane passes through three points:. We are looking for their coordinates:

1) . Draw the coordinates for the last two points yourself. The solution to the problem with a hexagonal pyramid will be useful for you!

2) We build the equation of the plane:

We are looking for the coordinates of the vector:. (see the triangular pyramid problem again!)

3) Looking for an angle:

Answer:

As you can see, there is nothing supernaturally difficult in these tasks. You just need to be very careful with your roots. For the last two problems, I will only give answers:

As you can see, the technique for solving problems is the same everywhere: the main task is to find the coordinates of the vertices and substitute them in some formulas. It remains for us to consider one more class of problems for calculating angles, namely:

Calculating angles between two planes

The solution algorithm will be as follows:

1. We look for the equation of the first plane by three points:
2. For the other three points, we look for the equation of the second plane:
3. We apply the formula:

As you can see, the formula is very similar to the two previous ones, with the help of which we looked for the angles between straight lines and between a straight line and a plane. So remembering this one will not be difficult for you. Let's go straight to the analysis of tasks:

1. One hundred-ro-na of the os-no-va-nia of the right-handed triangular prism is equal, and the dia-go-nal of the big face is equal. Nay-di-those are the angle between the plane and the plane of the prism.

2. In the correct four-you-rekh-coal-noy pi-ra-mi-de, all the edges of the swarm are equal, find the sine of the angle between the plane and the plane to-stu, pro-ho-dya-shchey through the point per-pen-di-ku-lar-but straight.

3. In the correct four-you-rekh-coal prism, the sides of the axis are equal, and the sides are equal. On the edge from-me-che-to point so that. Find the angle between the plane-to-sti-mi and

4. In the right four-corner prism, the sides of the os-no-va-nia are equal, and the side edges are equal. On the edge from-me-che-to point so that Nay-di-te is the angle between plane-to-st-mi and.

5. In the cube nay-di-te ko-si-nus of the angle between the plane

Problem solutions:

1. I draw a regular (at the base is an equilateral triangle) triangular prism and mark on it the planes that appear in the problem statement:

We need to find the equations of two planes: The equation of the base is trivial: you can compose the corresponding determinant by three points, but I will compose the equation at once:

Now we find the equation Point has coordinates Point - Since is the median and the height of the triangle, it is easy to find in a triangle using the Pythagorean theorem. Then the point has coordinates: Find the applicate of the point To do this, consider a right-angled triangle

Then we get the following coordinates: We make the equation of the plane.

We calculate the angle between the planes:

Answer:

2. Making a drawing:

The most difficult thing is to understand what this mysterious plane is, passing through a point perpendicularly. Well, the main thing is what is this? The main thing is attentiveness! Indeed, the line is perpendicular. The straight line is also perpendicular. Then the plane passing through these two lines will be perpendicular to the line, and, by the way, pass through the point. This plane also goes through the top of the pyramid. Then the desired plane - And the plane has already been given to us. We are looking for the coordinates of the points.

Find the coordinate of the point through the point. From the small figure it is easy to deduce that the coordinates of the point will be as follows: What is now left to find to find the coordinates of the top of the pyramid? You also need to calculate its height. This is done using the same Pythagorean theorem: first, prove that (trivially from small triangles forming a square at the base). Since by condition, we have:

Now everything is ready: vertex coordinates:

We compose the equation of the plane:

You are already special in calculating determinants. You can easily get:

Or else (if we multiply both parts by the root of two)

Now we find the equation of the plane:

(You have not forgotten how we get the equation of the plane, right? If you do not understand where this minus one came from, then go back to the definition of the equation of the plane! It's just that before that it turned out that the origin of coordinates belonged to my plane!)

We calculate the determinant:

(You can see that the equation of the plane coincides with the equation of the straight line passing through the points and! Think why!)

Now we calculate the angle:

We need to find the sine:

Answer:

3. Tricky question: what do you think is a rectangular prism? It's just a parallelepiped you know well! Make a drawing immediately! It is even possible not to depict the base separately, there is little benefit from it here:

The plane, as we noted earlier, is written in the form of an equation:

Now we make a plane

We immediately compose the equation of the plane:

Looking for an angle:

Now the answers to the last two problems:

Well, now is the time to take a break, because you and I are great and have done a great job!

Coordinates and vectors. Advanced level

In this article we will discuss with you another class of problems that can be solved using the coordinate method: distance problems. Namely, we will consider the following cases:

1. Calculation of the distance between crossed lines.

I have ordered these tasks as their complexity increases. It turns out to be the easiest to find distance from point to plane, and the most difficult thing is to find distance between crossing lines... Although, of course, nothing is impossible! Let's not procrastinate and immediately start considering the first class of problems:

Calculating the distance from a point to a plane

What do we need to solve this problem?

1. Point coordinates

So, as soon as we get all the necessary data, then we apply the formula:

You should already know how we construct the equation of the plane from the previous problems that I discussed in the last part. Let's get down to the tasks right away. The scheme is as follows: 1, 2, I help you solve, and in some detail, 3, 4 - only the answer, you make the decision yourself and compare. Let's start!

Tasks:

1. Given a cube. The length of the edge of the cube is. Nay-di-te distance-i-ni from se-re-di-us from-cut to flat-to-sti

2. Given the right-vil-naya four-you-rekh-coal-naya pi-ra-mi-da Bo-ko-th edge of the side-ro-na os-no-va-nia is equal. Nay-di-those distance from point to plane-to-sti where - se-re-di-on ribs.

3. In the right-handed triangular pi-ra-mi-de with os-no-va-ni, the bo-kov edge is equal, and the side-ro-na is-no-va- is equal to. Nay-di-te distance-i-nye from the top to the plane.

4. In the correct six-coal prism, all edges are equal. Nay-di-te distance-i-nye from point to plane.

Solutions:

1. Draw a cube with unit edges, build a segment and a plane, denote the middle of the segment by the letter

.

Let's start with an easy one: find the coordinates of a point. Since then (remember the coordinates of the midpoint of the segment!)

Now we compose the equation of the plane by three points

\\ [\\ left | (\\ begin (array) (* (20) (c)) x & 0 & 1 \\\\ y & 1 & 0 \\\\ z & 1 & 1 \\ end (array)) \\ right | \u003d 0 \\]

Now I can start looking for distance:

2. Start again with the drawing, on which we mark all the data!

For the pyramid, it would be helpful to draw its base separately.

Even the fact that I draw like a chicken with its paw won't stop us from solving this problem with ease!

Now it's easy to find the coordinates of a point

Since the coordinates of the point, then

2. Since the coordinates of point a are the midpoint of the segment, then

We can also find the coordinates of two more points on the plane without any problems.Create the equation of the plane and simplify it:

\\ [\\ left | (\\ left | (\\ begin (array) (* (20) (c)) x & 1 & (\\ frac (3) (2)) \\\\ y & 0 & (\\ frac (3) (2)) \\\\ z & 0 & (\\ frac ( (\\ sqrt 3)) (2)) \\ end (array)) \\ right |) \\ right | \u003d 0 \\]

Since the point has coordinates:, then we calculate the distance:

Answer (very rare!):

Well, figured it out? It seems to me that everything here is as technical as in the examples that we considered with you in the previous part. So I'm sure that if you have mastered that material, then it will not be difficult for you to solve the remaining two problems. I will just give the answers:

Calculating the distance from a straight line to a plane

In fact, there is nothing new here. How can a line and a plane be located relative to each other? They have all the possibilities: intersect, or a straight line is parallel to the plane. What do you think is the distance from a straight line to the plane with which this straight line intersects? It seems to me that it is clear here that such a distance is equal to zero. An uninteresting case.

The second case is trickier: here the distance is already nonzero. However, since the line is parallel to the plane, then each point of the line is equidistant from this plane:

In this way:

This means that my task has been reduced to the previous one: we are looking for the coordinates of any point on a straight line, we are looking for the equation of the plane, and we calculate the distance from a point to a plane. In fact, such tasks are extremely rare in the exam. I managed to find only one problem, and the data in it was such that the coordinate method was not very applicable to it!

Now let's move on to another, much more important class of problems:

Calculating the distance of a point to a line

What do we need?

1. The coordinates of the point from which we are looking for the distance:

2. Coordinates of any point lying on a straight line

3. Coordinates of the direction vector of the straight line

What formula do we use?

What does the denominator of this fraction mean to you and so it should be clear: this is the length of the directing vector of a straight line. There is a very tricky numerator here! The expression means the modulus (length) of the vector product of vectors and How to calculate the cross product, we studied in the previous part of the work. Refresh your knowledge, they will be very useful to us now!

Thus, the algorithm for solving problems will be as follows:

1. We are looking for the coordinates of the point from which we are looking for the distance:

2. We are looking for the coordinates of any point on the straight line to which we are looking for the distance:

3. Build a vector

4. Build the direction vector of the straight line

5. Calculate the cross product

6. We are looking for the length of the resulting vector:

7. Calculate the distance:

We have a lot of work, and the examples will be quite complex! So now focus all your attention!

1. Dana is a right-vil-naya triangular pi-ra-mi-da with a top. One hundred-ro-na os-no-va-nia pi-ra-mi-dy is equal, you-so-that is equal. Nay-di-those distance from the se-re-di-ny of the bo-ko-th rib to the straight line, where the points and are the se-re-di-ny of the ribs and co- vet-but.

2. The lengths of the ribs and the right-angled pa-ral-le-le-pi-pe-da are equal co-vet-n-but and Nay-di-te distance from top to top to straight

3. In the right-handed six-coal prism, all the edges of a swarm are equal find-di-those distance from a point to a straight line

Solutions:

1. We make a neat drawing on which we mark all the data:

We have a lot of work with you! First, I would like to describe in words what we will be looking for and in what order:

1. Coordinates of points and

2. Point coordinates

3. Coordinates of points and

4. Coordinates of vectors and

5. Their cross product

6. Vector length

7. The length of the vector product

8. Distance from to

Well, we have a lot of work to do! We get down to it, rolling up our sleeves!

1. To find the coordinates of the height of the pyramid, we need to know the coordinates of the point. Its applicate is zero, and the ordinate is equal to the Abscissa, it is equal to the length of the segment. Since is the height of an equilateral triangle, it is divided in relation, counting from the top, henceforth. Finally, we got the coordinates:

Point coordinates

2. - the middle of the segment

3. - the middle of the segment

Mid-point

4.Coordinates

Vector coordinates

5. We calculate the cross product:

6. Vector length: the easiest way to change is that the segment is the middle line of the triangle, which means that it is equal to half the base. So that.

7. We consider the length of the vector product:

8. Finally, we find the distance:

Phew, that's it! Honestly, the solution to this problem using traditional methods (through constructions) would be much faster. But here I have reduced everything to a ready-made algorithm! I think that the solution algorithm is clear to you? Therefore, I will ask you to solve the remaining two problems yourself. Let's compare the answers?

Again, I repeat: it is easier (faster) to solve these problems through constructions, and not resorting to the coordinate method. I have demonstrated this solution only to show you a universal method that allows you to "complete nothing".

Finally, consider the last class of problems:

Calculating the distance between crossed lines

Here the problem solving algorithm will be similar to the previous one. What we have:

3. Any vector connecting points of the first and second straight lines:

How do we find the distance between straight lines?

The formula is as follows:

The numerator is the modulus of the mixed product (we introduced it in the previous part), and the denominator is the same as in the previous formula (the modulus of the vector product of the direction vectors of the straight lines, the distance between which we are looking for).

I will remind you that

then the formula for the distance can be rewritten as:

A sort of determinant divided by a determinant! Although, to be honest, I have no time for jokes here! This formula, in fact, is very cumbersome and leads to rather complicated calculations. If I were you, I would only use it as a last resort!

Let's try to solve several problems using the above method:

1. In the correct triangular prism, all the edges of which are equal, find the distance between the straight lines and.

2. Given the right-vil-naya triangular prism, all the edges of the os-no-va-nia of the co-swarm are equal Se-c, passing through the god rib and se-re-di-well ribs yav-la-et-sya square-ra-tom. Nai di te distance between straight we and

I decide the first, and based on it, you decide the second!

1. Draw a prism and mark the straight lines and

Point C coordinates: then

Point coordinates

Vector coordinates

Point coordinates

Vector coordinates

Vector coordinates

\\ [\\ left ((B, \\ overrightarrow (A (A_1)) \\ overrightarrow (B (C_1))) \\ right) \u003d \\ left | (\\ begin (array) (* (20) (l)) (\\ begin (array) (* (20) (c)) 0 & 1 & 0 \\ end (array)) \\\\ (\\ begin (array) (* (20) (c)) 0 & 0 & 1 \\ end (array)) \\\\ (\\ begin (array) (* (20) (c)) (\\ frac ((\\ sqrt 3)) (2)) & (- \\ frac (1) (2)) & 1 \\ end (array)) \\ end (array)) \\ right | \u003d \\ frac ((\\ sqrt 3)) (2) \\]

We consider the cross product between vectors and

\\ [\\ overrightarrow (A (A_1)) \\ cdot \\ overrightarrow (B (C_1)) \u003d \\ left | \\ begin (array) (l) \\ begin (array) (* (20) (c)) (\\ overrightarrow i) & (\\ overrightarrow j) & (\\ overrightarrow k) \\ end (array) \\\\\\ begin (array ) (* (20) (c)) 0 & 0 & 1 \\ end (array) \\\\\\ begin (array) (* (20) (c)) (\\ frac ((\\ sqrt 3)) (2)) & (- \\ - \\ frac ((\\ sqrt 3)) (2) \\ overrightarrow k + \\ frac (1) (2) \\ overrightarrow i \\]

Now we calculate its length:

Answer:

Now try to complete the second task carefully. The answer to it will be:.

Coordinates and vectors. Brief description and basic formulas

Vector is a directed line segment. - the beginning of the vector, - the end of the vector.
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