The vector is located. Vectors on a plane and in space - basic definitions. Finding the angle between a straight line and a plane

DEFINITION

Vector(from lat. " vector" - "carrying") - a directed segment of a straight line in space or on a plane.

Graphically, a vector is depicted as a directed straight line segment of a certain length. A vector whose beginning is at point and end at point is denoted as (Fig. 1). A vector can also be denoted by one small letter, for example, .

If a coordinate system is specified in space, then the vector can be uniquely specified by a set of its coordinates. That is, a vector is understood as an object that has a magnitude (length), direction and point of application (the beginning of the vector).

The principles of vector calculus appeared in the works of the German mathematician, mechanician, physicist, astronomer and surveyor Johann Carl Friedrich Gauss (1777-1855) in 1831. Works on operations with vectors were published by the Irish mathematician, mechanic and theoretical physicist, Sir William Rowan Hamilton (1805-1865) as part of his quaternion calculus. The scientist proposed the term “vector” and described some operations on vectors. Vector calculus was further developed thanks to the work on electromagnetism by the British physicist, mathematician and mechanic James Clerk Maxwell (1831-1879). In the 1880s, the book “Elements of Vector Analysis” by the American physicist, physical chemist, mathematician and mechanic Josiah Willard Gibbs (1839-1903) was published. Modern vector analysis was described in 1903 in the works of the self-taught English scientist, engineer, mathematician and physicist Oliver Heaviside (1850-1925).

DEFINITION

Length or vector module is the length of the directed segment defining the vector. Denoted as .

Main types of vectors

Zero vector is called a vector whose starting point and ending point coincide. The length of the zero vector is zero.

Vectors parallel to one line or lying on one line are called collinear(Fig. 2).

co-directed, if their directions coincide.

In Figure 2 these are vectors and . The co-directionality of vectors is indicated as follows: .

Two collinear vectors are called oppositely directed, if their directions are opposite.

In Figure 3 these are vectors and . Designation: .

1. What is a vector?

2. Vector addition.

3. Equality of vectors.

4. The scalar product of two vectors and its properties.

5. Properties of operations on vectors.

6. Proofs and problem solving.

One of the fundamental concepts of modern mathematics is the vector and its generalization - the tensor. The evolution of the concept of a vector was carried out thanks to the widespread use of this concept in various fields of mathematics, mechanics, as well as in technology.

The end of the past and the beginning of the current century were marked by the widespread development of vector calculus and its applications. Vector algebra and vector analysis, and the general theory of vector space were created. These theories were used to construct the special and general theories of relativity, which play an extremely important role in modern physics.

In accordance with the requirements of the new mathematics program, the concept of a vector has become one of the leading concepts in the school mathematics course.

What is a vector? Oddly enough, the answer to this question presents certain difficulties. There are different approaches to defining the concept of a vector; Moreover, even if we limit ourselves only to the elementary geometric approach to the concept of a vector, which is most interesting for us here, then even then there will be different views on this concept. Of course, no matter what definition we take, a vector - from an elementary geometric point of view - is a geometric object characterized by direction (i.e., a given line with accuracy up to parallelism and the direction on it) and length. However, such a definition is too general, not evoking specific geometric ideas. According to this general definition, a parallel carry can be considered a vector. Indeed, one could accept the following definition: “Any parallel transfer is called a vector.” This definition is logically impeccable, and on its basis the entire theory of actions on vectors can be built and applications of this theory can be developed. However, this definition, despite its complete specificity, cannot satisfy us here either, since the idea of ​​a vector as a geometric transformation seems to us insufficiently clear and far from the physical concepts of vector quantities.

So, vector is the family of all segments parallel to each other, equally directed and having the same length (Fig. 1).

A vector is depicted in drawings by a segment with an arrow (i.e., not the entire family of segments representing a vector is depicted, but only one of these segments). Bold Latin letters are used to denote vectors in books and articles. a, b, c and so on, and in notebooks and on the blackboard - Latin letters with a dash on top , The same letter, but not bold, but light (and in the notebook and on the board, the same letter without a dash) denotes the length of the vector. Length is sometimes also indicated by vertical dashes - as the module (absolute value) of a number. Thus, the length of the vector A denoted by A or I A I, and in handwritten text the length of the vector A denoted by A or I A I. In connection with the depiction of vectors in the form of segments (Fig. 2), it should be remembered that the ends of the segment depicting the vector are unequal: one end of the segment to the other.

There is a distinction between the beginning and the end of a vector (more precisely, a segment representing a vector).

Quite often the concept of a vector is given a different definition: a vector is a directed segment. In this case, vectors (i.e. directed segments) having the same length and the same direction (Fig. 3) are agreed to be considered equal.

Vectors are called identically directed if their half-lines are identically directed.

Vector addition.

All that has been said does not yet make the concept of a vector sufficiently meaningful and useful. The concept of a vector gains greater meaning and richer application possibilities when we introduce a kind of “geometric arithmetic” - vector arithmetic, which allows us to add vectors, subtract them and perform a whole series of other operations on them. Let us note in this regard that the concept of number becomes interesting only with the introduction of arithmetic operations, and not in itself.

Vector sum A And V with coordinates a 1, a 2 and a 1, a 2 called a vector With with coordinates a 1 + in 1, and 2 + in 2, those. A(a 1; a 2) + V(in 1;in 2) = With(a 1 + in 1; a 2 + in 2).
Consequence:
To prove that addition of vectors on a plane is commutative, it is necessary to consider an example. A And V – vectors (Fig. 5).
Let

1. Construct a parallelogram OASV: AM II OV, VN II OA.

To prove associativity, we plot a vector from an arbitrary point O OA = a, from point A vector AB = in and from point in – vector BC = s. Then we have: AB + BC = AC.
whence the equality follows a + (in + c) = (a + b)+ s. Note that the above proof does not use a drawing at all. This is typical (with some skill) for solving problems using vectors. Let us now dwell on the case when the vectors A And V are directed in opposite directions and have equal lengths; such vectors are called opposite. Our vector addition rule results in the sum of two opposite vectors being a “vector” that has zero length and no direction; this “vector” is represented by a “segment of zero length”, i.e. dot. But this is also a vector, which is called zero and is denoted by the symbol 0.

Equality of vectors.

Two vectors are said to be equal if they are combined by parallel translation. This means that there is a parallel translation that takes the start and end of one vector to the start and end of another vector, respectively.

From this definition of equality of vectors it follows that different vectors are equally directed and equal in absolute value.

And vice versa: if the vectors are equally directed and equal in absolute value, then they are equal.

Indeed, let the vectors AB And WITH D – identically directed vectors, equal in absolute value (Fig. 6). A parallel translation that moves point C to point A combines the half-line CD with the half-line AB, since they are equally directed. And since the segments AB and CD are equal, then point D is combined with point B, that is, parallel translation translates the vector CD to vector AB. This means that the vectors AB And WITH D are equal, as required to prove.

Such a concept as a vector is considered in almost all natural sciences, and it can have completely different meanings, so it is impossible to give an unambiguous definition of a vector for all areas. But let's try to figure it out. So, what is a vector?

The concept of a vector in classical geometry

A vector in geometry is a segment for which it is indicated which of its points is the beginning and which is the end. That is, to put it simply, a directed segment is called a vector.

Accordingly, a vector is denoted (what it is - discussed above), as well as a segment, that is, by two capital letters of the Latin alphabet with the addition of a line or an arrow pointing to the right on top. It can also be signed with a lowercase (small) letter of the Latin alphabet with a line or arrow. The arrow always points to the right and does not change depending on the location of the vector.

Thus, a vector has a direction and a length.

The designation of a vector also contains its direction. This is expressed as in the figure below.

Changing the direction reverses the value of the vector.

The length of a vector is the length of the segment from which it is formed. It is denoted as the modulus of a vector. This is shown in the picture below.

Accordingly, a vector whose length is zero is zero. It follows from this that the zero vector is a point, and its beginning and end points coincide.

The length of a vector is always a non-negative quantity. In other words, if there is a segment, then it necessarily has a certain length or is a point, then its length is zero.

The very concept of a point is basic and has no definition.

Vector addition

There are special formulas and rules for vectors that can be used to perform addition.

Triangle rule. To add vectors according to this rule, it is enough to combine the end of the first vector and the beginning of the second, using parallel translation, and connect them. The resulting third vector will be equal to the addition of the other two.

Parallelogram rule. To add using this rule, you need to draw both vectors from one point, and then draw another vector from the end of each of them. That is, the second will be drawn from the first vector, and the first from the second. The result is a new intersection point and a parallelogram is formed. If you combine the point of intersection of the beginnings and ends of the vectors, then the resulting vector will be the result of addition.

Subtraction can be done in a similar way.

Vector difference

Similar to the addition of vectors, it is also possible to subtract them. It is based on the principle shown in the figure below.

That is, it is enough to represent the subtracted vector in the form of a vector opposite to it, and carry out the calculation using the principles of addition.

Also, absolutely any non-zero vector can be multiplied by any number k, this will change its length by k times.

In addition to these, there are other vector formulas (for example, for expressing the length of a vector through its coordinates).

Vector location

Surely many have come across such a concept as a collinear vector. What is collinearity?

Collinearity of vectors is the equivalent of parallelism of lines. If two vectors lie on lines that are parallel to each other, or on the same line, then such vectors are called collinear.

Direction. With respect to each other, collinear vectors can be co-directed or oppositely directed, this is determined by the direction of the vectors. Accordingly, if a vector is codirectional with another, then the vector opposite to it is oppositely directed.

The first figure shows two oppositely directed vectors and a third one that is not collinear with them.

After introducing the above properties, it is possible to define equal vectors - these are vectors that are directed in one direction and have the same length of the segments from which they are formed.

In many sciences, the concept of radius vector is also used. Such a vector describes the position of one point on the plane relative to another fixed point (often this is the origin).

Vectors in physics

Suppose that when solving a problem a condition arose: the body moves at a speed of 3 m/s. This means that the body moves with a specific direction along one straight line, so this variable will be a vector quantity. To solve, it is important to know both the value and the direction, since depending on the consideration, the speed can be either 3 m/s or -3 m/s.

In general, a vector in physics is used to indicate the direction of a force acting on a body and to determine the resultant.

When these forces are indicated in the figure, they are indicated by arrows with a vector label above it. Classically, the length of the arrow is just as important; it is used to indicate which force is stronger, but this is a secondary property and should not be relied upon.

Vector in linear algebra and calculus

Elements of linear spaces are also called vectors, but in this case they represent an ordered system of numbers that describe some of the elements. Therefore, the direction in this case no longer has any importance. The definition of a vector in classical geometry and in calculus are very different.

Projecting vectors

Projected vector - what is it?

Quite often, for a correct and convenient calculation, it is necessary to expand a vector located in two-dimensional or three-dimensional space along the coordinate axes. This operation is necessary, for example, in mechanics when calculating the forces acting on a body. A vector is used quite often in physics.

To perform a projection, it is enough to lower the perpendiculars from the beginning and end of the vector onto each of the coordinate axes; the segments obtained on them will be called the projection of the vector onto the axis.

To calculate the length of a projection, it is enough to multiply its original length by a certain trigonometric function, which is obtained by solving a mini-problem. Essentially, there is a right triangle in which the hypotenuse is the original vector, one of the legs is the projection, and the other leg is the dropped perpendicular.

In this article, we will begin to discuss one “magic wand” that will allow you to reduce many geometry problems to simple arithmetic. This “stick” can make your life much easier, especially when you feel unsure of constructing spatial figures, sections, etc. All this requires a certain imagination and practical skills. The method that we will begin to consider here will allow you to almost completely abstract 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 on the plane
3. Constructing a vector from two points
4. Vector length (distance between two points)​
5. Coordinates of the middle of the segment
6. Dot product of vectors
7. Angle between two vectors​

I think you've already guessed why the coordinate method is called that? That's right, it got this name because it operates not with geometric objects, but with their numerical characteristics (coordinates). And the transformation itself, which allows us to move 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 consider only 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 when solving problems on planimetry in Part B of the Unified State Exam). The next two sections on this topic are devoted to a 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 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 it that way. For example, if, then, if, then, etc. What did you get in the end? And you received points with coordinates: and. Next, 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 obtained on it, which you then connected with a straight line; the resulting line is the graph of the function.

There are a few 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 beautifully and compactly in the drawing.

2. It is accepted 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 writing the coordinates of a point, for example, on the left in parentheses there 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 specify any point on the coordinate axis, you need to indicate its coordinates (2 numbers)

6. For any point lying on the axis,

7. For any point lying on the axis,

8. The axis is called the x-axis

9. The axis is called the y-axis

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

Remember what another directional segment is called? That's right, it's called a vector!

So if we connect dot to dot, and the beginning will be point A, and the end will be point B, then we get a vector. You also did this construction in 8th grade, remember?

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

Thus, since in a vector the point is the beginning and the point 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 will be at the point. Then:

Look carefully, what is the difference between vectors and? Their only difference is the signs in the coordinates. They are opposites. This fact is usually written like this:

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

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

Examination:

Now solve a slightly more difficult problem:

A vector with a beginning at a point has a co-or-di-na-you. Find the abs-cis-su points.

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

I compiled the system based on the definition of what vector coordinates 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 can’t 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 non-zero 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:

A vector stretches or contracts or changes direction when multiplied or divided by a number:

However, here we will be interested in the question of what happens to the coordinates.

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

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

For example:

· Find the amount of co-or-di-nat century-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 equal.

Answer:

Now solve the following problem yourself:

· Find the sum of vector coordinates

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 us denote the distance between them by. Let's make the following drawing for clarity:

What I've done? Firstly, I connected the points 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, forming a remarkable figure? What's so special about her? Yes, you and I know almost everything about the right triangle. Well, the Pythagorean theorem for sure. The required segment is the hypotenuse of this triangle, and the segments are the legs. What are the coordinates of the point? Yes, they are easy to find from the picture: Since the segments are parallel to the axes and, respectively, their lengths are easy to find: if we denote the lengths of the segments by, respectively, 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 squared differences from the coordinates. Or - the distance between two points is the length of the segment connecting them. It is easy to see that the distance between points does not depend on the direction. Then:

From here we draw three conclusions:

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

For example, if, then the distance between and is equal to

Or let's go another way: find the coordinates of the vector

And find the length of the vector:

As you can see, it's the same thing!

Now practice a little yourself:

Task: find the distance between the indicated points:

We check:

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

1. Find the square of the length of the eyelid.

2. Find the square of the length of the eyelid

I think you dealt with them without difficulty? We check:

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

2. Find the coordinates of the vector

Then the square of its length is

Nothing complicated, right? Simple arithmetic, nothing more.

The following problems cannot be classified unambiguously; they are more about general erudition and the ability to draw simple pictures.

1. Find the sine of the angle from the cut, connecting the point, with the abscissa axis.

And

How are we going to proceed here? We need to find the sine of the angle between and the axis. Where can we look for sine? That's right, in a right triangle. So what do we need to do? Build this triangle!

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

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

Answer:

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

Task 2. From the point the per-pen-di-ku-lyar is lowered onto the ab-ciss axis. Nai-di-te abs-cis-su os-no-va-niya per-pen-di-ku-la-ra.

Let's make a drawing:

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

Answer: .

Task 3. In the conditions of the previous problem, find the sum of the distances from the 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 I will remind you:

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

Answer: .

Task 4. In the conditions of task 2, find the ordinate of a point symmetrical to the point relative to the abscissa axis.

I think it is intuitively clear to you what symmetry is? Many objects have it: many buildings, tables, airplanes, many geometric shapes: 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 symmetry. What then is an axis? This is exactly the line along which the figure can, relatively speaking, be “cut” into equal halves (in this picture the axis of symmetry is straight):

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

Did it work out the same way for you? Fine! We are interested in the ordinate of the found point. It is equal

Answer:

Now tell me, after thinking for a few seconds, what will be the abscissa of a point symmetrical 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 the coordinates:

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

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

Answer:

Now parallelogram problem:

Task 5: The points appear ver-shi-na-mi pa-ral-le-lo-gram-ma. Find or-di-on-that point.

You can solve this problem in two ways: logic and the coordinate method. I'll use the coordinate method first, and then I'll tell you how you can solve it differently.

It is quite clear that the abscissa of the point is equal. (it lies on the perpendicular drawn from the point to the abscissa axis). We need to find the ordinate. Let's take advantage of the fact that our figure is a parallelogram, this means that. Let's 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. I will denote the intersection point with a letter.

The length of the segment is equal. (find the problem yourself where we discussed this point), then we will find the length of the segment using the Pythagorean theorem:

The length of a segment coincides exactly with 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 length

3. Prove that.

Another one segment length problem:

The points appear on top of the triangle. Find the length of its midline, parallel.

Do you remember what the middle line of a triangle is? Then this task is elementary for you. If you don’t remember, I’ll 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 segment. We had to look for its length earlier, it is equal. Then the length of the middle line is half as large and equal.

Answer: .

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

In the meantime, here are a few problems for you, practice on them, they are very simple, but they help you get better at using the coordinate method!

1. The points are the top of the tra-pe-tions. Find the length of its midline.

2. Points and appearances ver-shi-na-mi pa-ral-le-lo-gram-ma. Find or-di-on-that point.

3. Find the length from the cut, connecting the point and

4. Find the area behind the colored figure on the co-ordi-nat plane.

5. A circle with a center in na-cha-le ko-or-di-nat passes through the point. Find her ra-di-us.

6. Find-di-te ra-di-us of the circle, describe-san-noy about the right-angle-no-ka, the tops of something have a co-or -di-na-you are so-responsible

Solutions:

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

Answer:

2. The easiest way to solve this problem is to note that (parallelogram rule). Calculating the coordinates of vectors is not difficult: . When adding vectors, the coordinates are added. Then has coordinates. The point also has these coordinates, since the origin of the vector is the point with the coordinates. We are interested in the ordinate. She is equal.

Answer:

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

Answer:

4. Look at the picture and tell me which two figures the shaded area is “sandwiched” between? It is sandwiched between two squares. Then the area of ​​the desired figure is equal to the area of ​​the large square minus the area of ​​the small one. The side of a small square is a segment connecting the points and Its length is

Then the area of ​​the 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 ​​the large square is

We find the area of ​​the desired figure using the formula:

Answer:

5. If a circle has the origin as its center and passes through a point, then its radius will be exactly equal to the length of the segment (make a drawing 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 its diagonal. Let's find the length of any of the two diagonals (after all, in a rectangle they are equal!)

Answer:

Well, did you cope with everything? It wasn't very difficult to figure it out, was it? There is only one rule here - 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 middle, then it has coordinates:

That is: coordinates of the middle of the segment = the arithmetic mean of the corresponding coordinates of the ends of the segment.

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

1. Find-di-te or-di-na-tu se-re-di-ny from-cut, connect-the-point and

2. The points appear to be the top of the world. Find-di-te or-di-na-tu points per-re-se-che-niya of his dia-go-na-ley.

3. Find-di-te abs-cis-su center of the circle, describe-san-noy about the rectangular-no-ka, the tops of something have co-or-di-na-you so-responsibly-but.

Solutions:

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

Answer:

2. It is easy to see that this quadrilateral is a parallelogram (even a rhombus!). You can prove this yourself by calculating the lengths of the sides and comparing them with each other. What do I know about parallelograms? Its diagonals are divided in half by the point of intersection! Yeah! So what is the point of intersection of the diagonals? 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 equal to.

Answer:

3. What does the center of the circle circumscribed 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 point of intersection divides them in half. The task was reduced to the previous one. Let's take, for example, the diagonal. Then if is the center of the circumcircle, then is the midpoint. I'm looking for coordinates: The abscissa is equal.

Answer:

Now practice a little on your own, I’ll just give the answers to each problem so you can test yourself.

1. Find-di-te ra-di-us of the circle, describe-san-noy about the tri-angle-no-ka, the tops of something have a co-or-di -no misters

2. Find-di-te or-di-on-that center of the circle, describe-san-noy about the triangle-no-ka, the tops of which have coordinates

3. What kind of ra-di-u-sa should there be a circle with a center at a point so that it touches the ab-ciss axis?

4. Find-di-those or-di-on-that point of re-se-ce-tion of the axis and from-cut, connect-the-point and

Answers:

Was everything successful? I really hope for it! Now - the last push. Now be especially careful. The material that I will now explain is directly related not only to simple problems on the coordinate method from Part B, but is also found everywhere in Problem C2.

Which of my promises have I not yet kept? Remember what operations on vectors I promised to introduce and which ones I ultimately introduced? Are you sure I haven't forgotten anything? Forgot! I 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 different natures:

The cross product is done quite cleverly. We will discuss how to do it and why it is needed in the next article. And in this one we will focus on the scalar product.

There are two ways that allow us to calculate it:

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

Dot product via coordinates

Find: - generally accepted notation for scalar product

The formula for calculation is as follows:

That is, the scalar product = the sum of the products of vector coordinates!

Example:

Find-di-te

Solution:

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

We calculate the scalar product using the formula:

Answer:

See, absolutely nothing complicated!

Well, now try it yourself:

· Find a scalar pro-iz-ve-de-nie of centuries and

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

Vector coordinates, as in the previous problem! Answer: .

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

Denotes the angle between the vectors and.

That is, the scalar product is equal to the product of the lengths of the vectors and the cosine of the angle between them.

Why do we need this second formula, if we have the first one, which is much simpler, at least there are no cosines in it. And it is needed so that from the first and second formulas you and I can deduce 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 scalar product formula, I get:

But in other way:

So what did you and I get? We now have a formula that allows us 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 scalar product through 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. Find the angle between the eyelids and. Give the answer in grad-du-sah.

2. In 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! Agree? Then let's begin!

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

Then we look for the cosine between the vectors:

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

Answer:

Well, now solve the second problem yourself, and then compare! I will give just 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 coordinate method in Part B of the exam paper 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 the foundation on the basis of which we will make quite clever constructions that we will need to solve complex problems.

COORDINATES AND VECTORS. AVERAGE LEVEL

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

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

Of course, the entire coordinate method does not fit into these 6 points. It underlies such a science as analytical geometry, which you will become familiar with at university. I just want to build a foundation that will allow you to solve problems in a single state. exam. We have dealt with the tasks of Part B. Now it’s time to move to a whole new level! This article will be devoted to a method for solving those C2 problems in which it would be reasonable to switch to the coordinate method. This reasonableness is determined by what is required to be found 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 straight 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 a point to a line
6. Find the distance from a straight line to a plane
7. Find the distance between two lines

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

Suitable figures for the coordinate method are:

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

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

1. Finding cross-sectional areas
2. Calculation of volumes of bodies

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

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

All of them are mutually perpendicular and intersect at one point, which we will call the origin of coordinates. As before, we will denote the abscissa axis, the ordinate axis - , and the introduced applicate axis - .

If previously 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, the ordinate, and the applicate. For example:

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

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

called the projection of a point onto a plane

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 have the same appearance. For a small detail. I think you've already guessed which one it is. In all formulas we will have to add one more term responsible for the applicate axis. Namely.

1. If two points are given: , then:

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

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

• Their scalar product is equal to:
• The cosine of the angle between the vectors is equal to:

However, space is not so simple. As you understand, adding one more coordinate introduces significant diversity into the spectrum of figures “living” in this space. And for further narration I will need to introduce some, roughly speaking, “generalization” of the straight line. This “generalization” will be a plane. What do you know about 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 “sheet” stuck into space. “Infinity” should be understood that the plane extends in all directions, that is, its area is equal to infinity. However, this “hands-on” explanation does not give the slightest idea about the structure of the plane. And it is she who will be interested in us.

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

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

Or its analogue in space:

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

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

For example, a line passes through points:

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

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

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

Once again, I won’t be very interested in the equation of a straight line, but I really need you to remember what a direction vector is! Again: this is ANY non-zero vector lying on a line or parallel to it.

Withdraw equation of a plane based on three given points is no longer so trivial, and the issue is not usually addressed in high school courses. But in vain! This technique is vital when we resort to 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 to use a technique that is usually studied in an analytical geometry course. 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 a plane is not very different from the equation of a straight line (linear function). However, remember what you and I argued? We said that if we have three points that do not lie on the same line, then the equation of the plane can be uniquely reconstructed from them. But how? I'll try to explain it 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 obtain the correct identity:

Thus, there is a need to solve three equations with unknowns! Dilemma! However, you can always assume that (to do this you need to divide by). Thus, we get three equations with three unknowns:

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

Equation of a plane passing through three given points

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

Stop! What is this? Some very unusual module! However, the object that 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 encounter these same determinants. What is a third order determinant? Oddly enough, it's 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 row number, and by the index we mean the column number. For example, it means that this number is at the intersection of the second row and third column. Let's pose the following question: how exactly will we calculate such a determinant? That is, what specific number will we compare to it? For the third-order determinant there is a heuristic (visual) triangle rule, 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 the 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 the 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 ​​obtained at the step and

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

However, you don’t need to remember the method of calculation in this form; it’s enough to just keep in your head the triangles and the very idea of ​​what 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 equal to

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

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

Add up three numbers:

Terms that come with a minus

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

The first triangle, “perpendicular to the secondary diagonal: the product of the elements is equal to

The second triangle, “perpendicular to the secondary diagonal: the product of the elements is equal to

Add up three numbers:

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

Thus,

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

We check:

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

Here are a couple more determinants, calculate their values ​​yourself and compare them with the answers:

Answers:

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

Now let's go back 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 triangle method) and set the result to zero. Naturally, since these 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 the same straight line!

Let's illustrate this with a simple example:

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

We compile a determinant for these three points:

Let's simplify:

Now we calculate it directly using the triangle rule:

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

Thus, the equation of the plane passing through the points is:

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:

Let's create a determinant:

And calculate its value:

Then the equation of the plane has the form:

Or, reducing by, we get:

Now two tasks for self-control:

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

Answers:

Did everything coincide? Again, if there are certain difficulties, then my advice is this: take three points from your head (with a high degree of probability they will not lie on the same straight line), build a plane based on 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 scalar 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 ​​a parallelogram built on the vectors and. We will need this vector to calculate the distance from a point to a line. How can we calculate the vector product of vectors and, if their coordinates are given? The third-order determinant comes to our aid again. However, before I move on to the algorithm for calculating the vector product, I have to make a small 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 artwork

Now I can start introducing the cross product:

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

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

Example 1: Find the cross product of vectors:

Solution: I make up a determinant:

And I calculate it:

Now from writing through basis vectors, I will return to the usual vector notation:

Thus:

Now try it.

Ready? We check:

And traditionally two tasks for control:

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

Answers:

Mixed product of three vectors

The last construction I'll need is the 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 be given three vectors:

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

1. - that is, the mixed product is the scalar product of a vector and the vector product of two other vectors

For example, the mixed product of three vectors is:

Try to calculate it yourself using the vector product and make sure that the results match!

And again, two examples for independent solutions:

Answers:

Selecting a coordinate system

Well, now we have all the necessary foundation of knowledge to solve complex stereometric geometry problems. However, before proceeding directly to examples and algorithms for solving them, I believe that it will be useful to dwell on the following 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 consider the following figures:

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

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

That is, I will place the figure “in the corner”. The cube and parallelepiped are very good figures. 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 position a cube or rectangular parallelepiped is advisable.

Straight prism

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

Triangular prism:

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

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:

The situation is similar to a cube: we align two sides of the base with the coordinate axes, and align one of the vertices with the origin of coordinates. The only slight 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 to find 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 starting to solve 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 look at the problems of finding an angle. They are in turn divided into the following categories (as complexity increases):

Problems for finding angles

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

Let's look at these problems sequentially: let's start by finding the angle between two straight lines. Well, remember, haven’t you and I solved similar examples before? Do you remember, we already had something similar... We were looking for the angle between two vectors. Let me remind you, if two vectors are given: and, then the angle between them is found from the relation:

Now our goal is to find the angle between two straight lines. Let's look at the “flat picture”:

How many angles did we get when two straight lines intersected? Just a few things. True, only two of them are not equal, while the others are vertical to them (and therefore coincide with them). So which 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 two straight lines is equal. In order not to bother each time with finding the smallest of two angles, cunning mathematicians suggested using a modulus. Thus, the angle between two straight lines is determined by the formula:

You, as an attentive reader, should have had a question: where, exactly, do we get these very numbers that we need to calculate the cosine of an angle? Answer: we will take them from the direction vectors of the 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 scalar product
4. We are looking for the length of the first vector
5. We are looking for the length of the second vector
6. Multiply the results of point 4 by the results of point 5
7. We 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 us to accurately calculate the angle, we look for it
9. Otherwise we write through arc cosine

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

Tasks:

1. In the right tet-ra-ed-re, find the angle between the height of the tet-ra-ed-ra and the middle side.

2. In the right-hand six-corner pi-ra-mi-de, the hundred os-no-va-niyas are equal, and the side edges are equal, find the angle between the lines and.

3. The lengths of all the edges of the right four-coal pi-ra-mi-dy are equal to each other. Find the angle between the straight lines and if from the cut - you are with the given pi-ra-mi-dy, the point is se-re-di-on its bo-co- second ribs

4. On the edge of the cube there is a point so that Find the angle between the straight lines and

5. Point - on the edges of the cube Find the angle between the straight lines and.

It is no coincidence that I arranged the tasks in this order. While you have not yet begun to navigate the coordinate method, I will analyze the most “problematic” figures myself, and I will leave you to deal with the simplest cube! Gradually you will have to learn how to work with all the figures; I will increase the complexity of the tasks 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 its faces (including the base) are regular triangles. Since we are not given the length of the side, I can take it to be equal. I think you understand that the angle will not actually 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 need to find the coordinates of the points. Now we think: a point is the point of intersection of the altitudes (or bisectors or medians) of the triangle. And a point is a raised point. The point is the middle of the segment. Then we finally need to find: the coordinates of the points: .

Let's start with the simplest thing: the coordinates of a point. Look at the figure: It is clear that the applicate of a point is equal to zero (the point lies on the plane). Its ordinate is equal (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 its 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 the point, that is. Let's find its abscissa. This is done quite trivially if you remember that the heights of an equilateral triangle by the point of intersection are divided in proportion, counting 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:

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 sought for reasons that I have highlighted in bold:

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

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

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

Thus,

Answer:

You should not be scared by such “scary” answers: for C2 tasks this is common practice. I would rather be surprised by the “beautiful” answer in this part. Also, as you noticed, I practically did not resort to anything other than the Pythagorean theorem and the property of altitudes 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 us depict a regular hexagonal pyramid along with the coordinate system, as well as its base:

We need to find the angle between the lines and. Thus, our task comes down to finding the coordinates of the points: . We will find the coordinates of the last three using a small drawing, and we will find the coordinate of the vertex through the coordinate of the point. There's a lot of work to do, but we need to get started!

a) Coordinate: it is clear that its applicate and ordinate are equal to zero. Let's find the abscissa. To do this, consider a right triangle. Alas, in it we only know the hypotenuse, which is equal. We will try to find the leg (for it is clear that double the length of the leg will give us the abscissa of the point). How can we look for it? 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 all sides and all angles are equal. We need to find one such angle. 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:

Let's 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 from.

Thus, 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. Finding the ordinate is also not very difficult: if we connect the dots and designate the point of intersection of the line as, say, . (do it yourself simple 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 let's find the coordinates of the point. Consider the 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 applica. Since, then. Consider a right triangle. According to the conditions of the problem, a side edge. This is the hypotenuse of my triangle. Then the height of the pyramid is a leg.

Then the point has coordinates:

Well, that's it, I have the coordinates of all the points that interest me. I am looking for the coordinates of the directing vectors of straight lines:

We are looking for the angle between these vectors:

Answer:

Again, in solving this problem I did not use any sophisticated techniques other than the formula for the sum of the angles of a regular n-gon, as well as the definition of the cosine and sine of a right triangle.

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

We are looking for the angle between and. I will make very brief calculations when I search for the coordinates of the points. You will need to “decipher” them:

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

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

Coordinates:

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

e) Vector coordinates

f) Vector coordinates

g) Looking for the angle:

A cube is the simplest figure. I'm sure you'll figure it out on your own. 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 puzzles 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. Using three points we construct an equation of the plane
   ,
   using a third order determinant.
2. Using two points, we look for the coordinates of the directing vector of the straight line:
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 angles between two straight lines. The structure on the right side is simply the same, and on the left we are now looking for the sine, not the cosine as before. Well, one nasty action was added - searching for the equation of the plane.

Let's not procrastinate solution examples:

1. The main-but-va-ni-em direct prism-we are an equal-to-poor triangle. Find the angle between the straight line and the plane

2. In a rectangular par-ral-le-le-pi-pe-de from the West Find the angle between the straight line and the plane

3. In a right six-corner prism, all edges are equal. Find the angle between the straight line and the plane.

4. In the right triangular pi-ra-mi-de with the os-no-va-ni-em of the known ribs Find a corner, ob-ra-zo-van -flat in base and straight, passing through the gray ribs and

5. The lengths of all the edges of a right quadrangular pi-ra-mi-dy with a vertex are equal to each other. Find the angle between the straight line and the plane, if the point is in the middle of the edge of the pi-ra-mi-dy.

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

Solutions:

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

I apologize for some non-compliance with the proportions, but for solving the problem this is, in fact, not so important. The plane is simply the "back wall" of my prism. It is enough to simply 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 create the equation of the plane:

Exercise for you: calculate this determinant yourself. Did you succeed? Then the equation of the plane looks like:

Or simply

Thus,

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

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

Then the point has coordinates:

A dot is a "raised" dot:

Then the vector coordinates are:

Answer:

As you can see, there is nothing fundamentally difficult when solving such problems. In fact, the process is simplified a little more by the “straightness” of a figure 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: The coordinates of the three points lying in it:

(the first two coordinates are 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 guiding vector: It is clear that its coordinates coincide with the coordinates of the point, isn’t it? How to find coordinates? These are the coordinates of the point, raised along the applicate axis by one! . Then we look for the desired angle:

Answer:

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

Here it’s even problematic to draw a plane, not to mention solving this problem, but the coordinate method doesn’t care! Its versatility is its main advantage!

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

1) . Find out the coordinates for the last two points yourself. You'll need to solve the hexagonal pyramid problem for this!

2) We construct 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 the roots. I will only give answers to the last two problems:

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 into certain formulas. We still have to consider one more class of problems for calculating angles, namely:

Calculating angles between two planes

The solution algorithm will be as follows:

1. Using three points we look for the equation of the first plane:
2. Using 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 angles between straight lines and between a straight line and a plane. So it won’t be difficult for you to remember this one. Let's move on to the analysis of the tasks:

1. The side of the base of the right triangular prism is equal, and the dia-go-nal of the side face is equal. Find the angle between the plane and the plane of the axis of the prism.

2. In the right four-corner pi-ra-mi-de, all the edges of which are equal, find the sine of the angle between the plane and the plane bone, passing through the point per-pen-di-ku-lyar-but straight.

3. In a regular four-corner prism, the sides of the base are equal, and the side edges are equal. There is a point on the edge from-me-che-on so that. Find the angle between the planes and

4. In a right quadrangular prism, the sides of the base are equal, and the side edges are equal. There is a point on the edge from the point so that Find the angle between the planes and.

5. In a cube, find the co-si-nus of the angle between the planes and

Problem solutions:

1. I draw a regular (an equilateral triangle at the base) 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 using three points, but I will compose the equation right away:

Now let’s find the equation Point has coordinates Point - Since is the median and altitude of the triangle, it is easily found using the Pythagorean theorem in the triangle. Then the point has coordinates: Let's find the applicate of the point. To do this, consider a right triangle

Then we get the following coordinates: We compose 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 kind of mysterious plane this is, passing perpendicularly through the point. Well, the main thing is, what is it? The main thing is attentiveness! In fact, 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 passes 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.

We find the coordinate of the point through the point. From the small picture it is easy to deduce that the coordinates of the point will be as follows: What now remains to be found 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 an expert in calculating determinants. Without difficulty you will receive:

Or otherwise (if we multiply both sides by the root of two)

Now let's find the equation of the plane:

(You haven’t forgotten how we get the equation of a plane, right? If you don’t understand where this minus one came from, then go back to the definition of the equation of a plane! It just always turned out before that my plane belonged to the origin of coordinates!)

We calculate the determinant:

(You may notice that the equation of the plane coincides with the equation of the line passing through the points and! Think about why!)

Now let's calculate the angle:

We need to find the sine:

Answer:

3. Tricky question: what do you think a rectangular prism is? This is just a parallelepiped that you know well! Let's make a drawing right away! You don’t even have to depict the base separately; it’s of little use here:

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

Now let's create a plane

We immediately create 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 little 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 calculation problems. Namely, we will consider the following cases:

1. Calculation of the distance between intersecting lines.

I have ordered these assignments in order of increasing difficulty. It turns out to be 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 proceed to consider 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 receive all the necessary data, we apply the formula:

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

Tasks:

1. Given a cube. The length of the edge of the cube is equal. Find the distance from the se-re-di-na from the cut to the plane

2. Given the right four-coal pi-ra-mi-yes, the side of the side is equal to the base. Find the distance from the point to the plane where - se-re-di-on the edges.

3. In the right triangular pi-ra-mi-de with the os-no-va-ni-em, the side edge is equal, and the hundred-ro-on the os-no-va- nia is equal. Find the distance from the top to the plane.

4. In a right hexagonal prism, all edges are equal. Find the distance from a point to a plane.

Solutions:

1. Draw a cube with single edges, construct a segment and a plane, denote the middle of the segment with a letter

.

First, let's start with the easy one: find the coordinates of the point. Since then (remember the coordinates of the middle of the segment!)

Now we compose the equation of the plane using three points

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

Now I can start finding the distance:

2. We start again with a drawing on which we mark all the data!

For a pyramid, it would be useful to draw its base separately.

Even the fact that I draw like a chicken with its paw will not prevent 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 middle of the segment, then

Without any problems, we can find the coordinates of two more points on the plane. We create an equation for the plane and simplify it:

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

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

Answer (very rare!):

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

Calculating the distance from a straight line to a plane

In fact, there is nothing new here. How can a straight line and a plane be positioned relative to each other? They have only one possibility: to 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. Not an interesting case.

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

Thus:

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, looking for the equation of the plane, and calculating the distance from the point to the plane. In fact, such tasks are extremely rare in the Unified State Examination. I managed to find only one problem, and the data in it were 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. Coordinates of the point from which we are looking for the distance:

2. Coordinates of any point lying on a line

3. Coordinates of the directing vector of the straight line

What formula do we use?

What the denominator of this fraction means should be clear to you: this is the length of the directing vector of the straight line. This is a very tricky numerator! The expression means the modulus (length) of the vector product of vectors and How to calculate the vector product, we studied in the previous part of the work. Refresh your knowledge, we will need it very much 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 line to which we are looking for the distance:

3. Construct a vector

4. Construct a directing vector of a straight line

5. Calculate the vector product

6. We look for the length of the resulting vector:

7. Calculate the distance:

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

1. Given a right triangular pi-ra-mi-da with a top. The hundred-ro-on the basis of the pi-ra-mi-dy is equal, you are equal. Find the distance from the gray edge to the straight line, where the points and are the gray edges and from veterinary.

2. The lengths of the ribs and the straight-angle-no-go par-ral-le-le-pi-pe-da are equal accordingly and Find the distance from the top to the straight line

3. In a right hexagonal prism, all edges are equal, find the 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 to do! First, I would like to describe in words what we will look 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. Length of the vector product

8. Distance from to

Well, we have a lot of work ahead of us! Let's get to it with our sleeves rolled up!

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 its ordinate is equal to its abscissa is equal to the length of the segment. Since is the height of an equilateral triangle, it is divided in the ratio, counting from the vertex, from here. Finally, we got the coordinates:

Point coordinates

2. - middle of the segment

3. - middle of the segment

Midpoint of the segment

4.Coordinates

Vector coordinates

5. Calculate the vector product:

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

7. Calculate the length of the vector product:

8. Finally, we find the distance:

Ugh, that's it! I’ll tell you honestly: solving this problem using traditional methods (through construction) would be much faster. But here I reduced everything to a ready-made algorithm! I think 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, rather than resorting to the coordinate method. I demonstrated this method of solution only to show you a universal method that allows you to “not finish building anything.”

Finally, consider the last class of problems:

Calculating the distance between intersecting lines

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

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

How do we find the distance between 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, 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'll remind you that

Then the formula for the distance can be rewritten as:

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

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

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

2. Given a right triangular prism, all the edges of the base are equal to the section passing through the body rib and se-re-di-well ribs are a square. Find the distance between the straight lines and

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

1. I draw a prism and mark straight lines and

Coordinates of point C: then

Point coordinates

Vector coordinates

Point coordinates

Vector coordinates

Vector coordinates

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

We calculate the vector product between vectors and

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

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

A vector is a directed segment. - the beginning of the vector, - the end of the vector.
A vector is denoted by or.

Absolute value vector - the length of the segment representing the vector. Denoted as.

Vector coordinates:

,
where are the ends of the vector \displaystyle a .

Sum of vectors: .

Product of vectors:

Dot product of vectors:

The scalar product of vectors is equal to the product of their absolute values ​​and the cosine of the angle between them:

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Finally, I got my hands on this extensive and long-awaited topic. analytical geometry. First, a little about this section of higher mathematics... Surely you now remember a school geometry course with numerous theorems, their proofs, drawings, etc. What to hide, an unloved and often obscure subject for a significant proportion of students. Analytical geometry, oddly enough, may seem more interesting and accessible. What does the adjective “analytical” mean? Two cliched mathematical phrases immediately come to mind: “graphical solution method” and “analytical solution method.” Graphical method, of course, is associated with the construction of graphs and drawings. Analytical or method involves solving problems mainly through algebraic operations. In this regard, the algorithm for solving almost all problems of analytical geometry is simple and transparent; often it is enough to carefully apply the necessary formulas - and the answer is ready! No, of course, we won’t be able to do this without drawings at all, and besides, for a better understanding of the material, I will try to cite them beyond necessity.

The newly opened course of lessons on geometry does not pretend to be theoretically complete; it is focused on solving practical problems. I will include in my lectures only what, from my point of view, is important in practical terms. If you need more complete help on any subsection, I recommend the following quite accessible literature:

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

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

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

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

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

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

Vector concept. Free vector

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

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

It is convenient to consider individual points of a plane or space as the so-called zero vector. For such a vector, the end and beginning coincide.

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

Designations: Many immediately noticed the stick without an arrow in the designation and said, there’s also an arrow at the top! True, you can write it with an arrow: , but it is also possible the entry that I will use in the future. Why? Apparently, this habit developed for practical reasons; my shooters at school and university turned out to be too different-sized and shaggy. In educational literature, sometimes they don’t bother with cuneiform writing at all, but highlight the letters in bold: , thereby implying that this is a vector.

That was stylistics, and now about ways to write vectors:

1) Vectors can be written in two capital Latin letters:
and so on. In this case, the first letter Necessarily denotes the beginning point of the vector, and the second letter denotes the end point of the vector.

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

Length or module a non-zero vector is called the length of the segment. The length of the zero vector is zero. Logical.

The length of the vector is indicated by the modulus sign: ,

We will learn how to find the length of a vector (or we will repeat it, depending on who) a little later.

This was basic information about vectors, familiar to all schoolchildren. In analytical geometry, the so-called free vector.

To put it simply - the vector can be plotted from any point:

We are accustomed to calling such vectors equal (the definition of equal vectors will be given below), but from a purely mathematical point of view, they are the SAME VECTOR or free vector. Why free? Because in the course of solving problems, you can “attach” this or that “school” vector to ANY point of the plane or space you need. This is a very cool feature! Imagine a directed segment of arbitrary length and direction - it can be “cloned” an infinite number of times and at any point in space, in fact, it exists EVERYWHERE. There is such a student saying: Every lecturer gives a damn about the vector. After all, it’s not just a witty rhyme, everything is almost correct - a directed segment can be added there too. But don’t rush to rejoice, it’s the students themselves who often suffer =)

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

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

Actions with vectors. Collinearity of vectors

A school geometry course covers a number of actions and rules with vectors: addition according to the triangle rule, addition according to the parallelogram rule, vector difference rule, multiplication of a vector by a number, scalar product of vectors, etc. As a starting point, let us repeat two rules that are especially relevant for solving problems of analytical geometry.

The rule for adding vectors using the triangle rule

Consider two arbitrary non-zero vectors and :

You need to find the sum of these vectors. Due to the fact that all vectors are considered free, we will set aside the vector from end vector:

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

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

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

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

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

The work a non-zero vector on a number is a vector whose length is equal to , and the vectors and are co-directed at and oppositely directed at .

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

Let's look at it in more detail:

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

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

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

4) The vectors are co-directed. Vectors and are also co-directed. Any vector of the first group is oppositely directed with respect to any vector of the second group.

Which vectors are equal?

Two vectors are equal if they are in the same direction and have the same length. Note that codirectionality implies collinearity of vectors. The definition would be inaccurate (redundant) if we said: “Two vectors are equal if they are collinear, codirectional, and have the same length.”

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

Vector coordinates on the plane and in space

The first point is to consider vectors on the plane. Let us depict a Cartesian rectangular coordinate system and plot it from the origin of coordinates single vectors and :

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

Designation: The orthogonality of vectors is written with the usual perpendicularity symbol, for example: .

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

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

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

Any plane vector the only way expressed as:
, Where - numbers which are called vector coordinates in this basis. And the expression itself called vector decompositionby basis .

Dinner served:

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

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

Vectors illustrate exactly the rule for multiplying a vector by a number, the vector is codirectional with the base vector, the vector is directed opposite to the base vector. For these vectors, one of the coordinates is equal to zero; you can meticulously write it like this:


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

And finally: , . By the way, what is vector subtraction, and why didn’t I talk about the subtraction rule? Somewhere in linear algebra, I don’t remember where, I noted that subtraction is a special case of addition. Thus, the expansions of the vectors “de” and “e” are easily written as a sum: , . Follow the drawing to see how clearly the good old addition of vectors according to the triangle rule works in these situations.

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

Or with an equal sign:

The basis vectors themselves are written as follows: and

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

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

We figured out the coordinates on the plane. Now let's look at vectors in three-dimensional space, almost everything is the same here! It will just add one more coordinate. It’s hard to make three-dimensional drawings, so I’ll limit myself to one vector, which for simplicity I’ll set aside from the origin:

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

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

All vectors of three-dimensional space, naturally, are also free; try to mentally set aside the vector from any other point, and you will understand that its decomposition “will remain with it.”

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

If one (or two) coordinate vectors are missing in the expansion, then zeros are put in their place. Examples:
vector (meticulously ) – let’s write ;
vector (meticulously ) – let’s write ;
vector (meticulously ) – let’s write .

The basis vectors are written as follows:

This, perhaps, is all the minimum theoretical knowledge necessary to solve problems of analytical geometry. There may be a lot of terms and definitions, so I recommend that teapots re-read and comprehend this information again. And it will be useful for any reader to refer to the basic lesson from time to time to better assimilate the material. Collinearity, orthogonality, orthonormal basis, vector decomposition - these and other concepts will be often used in the future. I note that the materials on the site are not enough to pass the theoretical test or colloquium on geometry, since I carefully encrypt all theorems (and without proofs) - to the detriment of the scientific style of presentation, but a plus to your understanding of the subject. To receive detailed theoretical information, please bow to Professor Atanasyan.

And we move on to the practical part:

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

It is highly advisable to learn how to solve the tasks that will be considered fully automatically, and the formulas memorize, you don’t even have to remember it on purpose, they will remember it themselves =) This is very important, since other problems of analytical geometry are based on the simplest elementary examples, and it will be annoying to spend additional time eating pawns. There is no need to fasten the top buttons on your shirt; many things are familiar to you from school.

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

How to find a vector from two points?

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

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

That is, from the coordinates of the end of the vector you need to subtract the corresponding coordinates beginning of the vector.

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

Example 1

Given two points of the plane and . Find vector coordinates

Solution: according to the appropriate formula:

Alternatively, the following entry could be used:

Aesthetes will decide this:

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

Answer:

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

You definitely need to understand difference between point coordinates and vector coordinates:

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

The coordinates of the vector– this is its expansion according to the basis, in this case. Any vector is free, so if desired or necessary, we can easily move it away from some other point on the plane (to avoid confusion, redesignating it, for example, by ). It is interesting that for vectors you don’t have to build axes or a rectangular coordinate system at all; you only need a basis, in this case an orthonormal basis of the plane.

The records of coordinates of points and coordinates of vectors seem to be similar: , and meaning of coordinates absolutely different, and you should be well aware of this difference. This difference, of course, also applies to space.

Ladies and gentlemen, let's fill our hands:

Example 2

a) Points and are given. Find vectors and .
b) Points are given And . Find vectors and .
c) Points and are given. Find vectors and .
d) Points are given. Find vectors .

Perhaps that's enough. These are examples for you to decide on your own, try not to neglect them, it will pay off ;-). There is no need to make drawings. Solutions and answers at the end of the lesson.

What is important when solving analytical geometry problems? It is important to be EXTREMELY CAREFUL to avoid making the masterful “two plus two equals zero” mistake. I apologize right away if I made a mistake somewhere =)

How to find the length of a segment?

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

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

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

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

Example 3

Solution: according to the appropriate formula:

Answer:

For clarity, I will make a drawing

Line segment - this is not a vector, and, of course, you cannot move it anywhere. In addition, if you draw to scale: 1 unit. = 1 cm (two notebook cells), then the resulting answer can be checked with a regular ruler by directly measuring the length of the segment.

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

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

Secondly, let us repeat the school material, which is useful not only for the task considered:

pay attention to important technique – removing the multiplier from under the root. As a result of the calculations, we have a result and good mathematical style involves removing the factor from under the root (if possible). In more detail the process looks like this: . Of course, leaving the answer as is would not be a mistake - but it would certainly be a shortcoming and a weighty argument for quibbling on the part of the teacher.

Here are other common cases:

Often the root produces a fairly large number, for example . What to do in such cases? Using the calculator, we check whether the number is divisible by 4: . Yes, it was completely divided, thus: . Or maybe the number can be divided by 4 again? . Thus: . The last digit of the number is odd, so dividing by 4 for the third time will obviously not work. Let's try to divide by nine: . As a result:
Ready.

Conclusion: if under the root we get a number that cannot be extracted as a whole, then we try to remove the factor from under the root - using a calculator we check whether the number is divisible by: 4, 9, 16, 25, 36, 49, etc.

When solving various problems, roots are often encountered; always try to extract factors from under the root in order to avoid a lower grade and unnecessary problems with finalizing your solutions based on the teacher’s comments.

Let's also repeat squaring roots and other powers:

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

Task for independent solution with a segment in space:

Example 4

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

The solution and answer are at the end of the lesson.

How to find the length of a vector?

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

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

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