In order for a single plane to be drawn through any three points in space, it is necessary that these points do not lie on the same straight line.

Consider the points M 1 (x 1, y 1, z 1), M 2 (x 2, y 2, z 2), M 3 (x 3, y 3, z 3) in general Cartesian system coordinates

In order for an arbitrary point M(x, y, z) to lie in the same plane with points M 1, M 2, M 3, it is necessary that the vectors be coplanar.

Definition 2.1.

Two lines in space are called parallel if they lie in the same plane and do not have common points.

If two lines a and b are parallel, then, as in planimetry, write a || b. In space, lines can be placed so that they do not intersect or are parallel. This case is special for stereometry.

Definition 2.2.

Lines that do not have common points and are not parallel are called intersecting.

Theorem 2.1.

Through a point outside a given line, one can draw a line parallel to the given one, and only one.

Sign of parallel lines

Two lines in space are called parallel if they lie in the same plane and do not intersect. Through a point outside a given line you can draw a straight line parallel to this straight line, and only one. This statement reduces to the axiom of parallels in a plane. Theorem. Two lines parallel to a third line are parallel. Let lines b and c be parallel to line a. Let us prove that b || With. The case when straight lines a, b and lie on the same plane is considered in planimetry; we omit it. Let us assume that a, b and c do not lie in the same plane. But since two parallel lines are located in the same plane, we can assume that a and b are located in the plane, and a b and c are in the plane (Fig. 61). On line c we mark a point (any) M and through line b and point M we draw a plane . She, , intersects in a straight line l. The straight line l does not intersect the plane, since if l intersected, then the point of their intersection must lie on a (a and l are in the same plane) and on b (b and l are in the same plane). Thus, one intersection point l and must lie on both line a and line b, which is impossible: a || b. Therefore, a || , l || a, l || b. Since a and l lie in the same plane, then l coincides with the line c (by the parallelism axiom), and therefore with || b. The theorem has been proven.

25.Sign of parallelism between a line and a plane

Theorem

If a line that does not belong to a plane is parallel to some line in this plane, then it is parallel to the plane itself.

Proof

Let α be a plane, a a line not lying in it, and a1 a line in the α plane parallel to line a. Let us draw the plane α1 through the lines a and a1. Planes α and α1 intersect along straight line a1. If line a intersected plane α, then the intersection point would belong to line a1. But this is impossible, since the lines a and a1 are parallel. Consequently, line a does not intersect the plane α, and therefore is parallel to the plane α. The theorem has been proven.

27.Existence of a plane parallel to a given plane

Theorem

Through a point outside a given plane it is possible to draw a plane parallel to the given one, and only one.

Proof

Let us draw in this plane α any two intersecting lines a and b. Through this point Let us draw lines a1 and b1 parallel to them. The plane β passing through the lines a1 and b1, according to the theorem on the parallelism of planes, is parallel to the plane α.

Suppose that another plane β1 passes through point A, also parallel to the planeα. Let us mark some point C on the β1 plane that does not lie in the β plane. Let us draw the plane γ through points A, C and some point B of the plane α. This plane will intersect planes α, β and β1 along straight lines b, a and c. Lines a and c do not intersect line b, since they do not intersect the plane α. Therefore, they are parallel to line b. But in the γ plane only one line parallel to line b can pass through point A. which contradicts the assumption. The theorem has been proven.

28.Properties of parallel planes th

29.

Perpendicular lines in space. Two lines in space are called perpendicular if the angle between them is 90 degrees. c. m. k. k. m. c. k. Intersecting. Crossbreeding.

Theorem 1 SIGN OF PERPENDICULARITY OF A LINE AND A PLANE. If a line intersecting a plane is perpendicular to two lines in this plane passing through the point of intersection of this line and the plane, then it is perpendicular to the plane.

Proof: Let a be a line perpendicular to lines b and c in the plane. Then line a passes through the point A of the intersection of lines b and c. Let us prove that straight line a is perpendicular to the plane. Let us draw an arbitrary line x through point A in the plane and show that it is perpendicular to line a. Let us draw an arbitrary line in the plane that does not pass through point A and intersects the lines b, c and x. Let the points of intersection be B, C and X. Let us plot a on the straight line from point A to different sides equal segments AA 1 and AA 2. Triangle A 1 CA 2 is isosceles, since segment AC is the height according to the theorem and the median by construction (AA 1 = AA 2). For the same reason, triangle A 1 BA 2 is also isosceles. Therefore, triangles A 1 BC and A 2 BC are equal on three sides. From the equality of triangles A 1 BC and A 2 BC, it follows that the angles A 1 BC and A 2 BC are equal and, therefore, the triangles A 1 BC and A 2 BC are equal on two sides and the angle between them. From the equality of the sides A 1 X and A 2 X of these triangles, we conclude that the triangle A 1 XA 2 is isosceles. Therefore its median XA is also its height. And this means that line x is perpendicular to a. By definition, a straight line is perpendicular to a plane. The theorem has been proven.

Theorem 2 1st PROPERTY OF PERPENDICULAR LINES AND PLANES. If a plane is perpendicular to one of two parallel lines, then it is also perpendicular to the other.
Proof: Let a 1 and a 2 - 2 be parallel lines and a plane perpendicular to the line a 1. Let us prove that this plane is perpendicular to the line a 2. Let us draw an arbitrary straight line x 2 in the plane through the point A 2 of the intersection of the line a 2 with the plane. Let us draw in the plane through point A 1 the intersection of line a 1 with line x 1 parallel to line x 2. Since line a 1 is perpendicular to the plane, then lines a 1 and x 1 are perpendicular. And by Theorem 1, the intersecting lines parallel to them, a 2 and x 2, are also perpendicular. Thus, line a 2 is perpendicular to any line x 2 in the plane. And this (by definition) means that straight line a 2 is perpendicular to the plane. The theorem has been proven. See also reference problem №2.
Theorem 3 2nd PROPERTY OF PERPENDICULAR LINES AND PLANES. Two lines perpendicular to the same plane are parallel.
Proof: Let a and b be 2 straight lines perpendicular to the plane. Let us assume that lines a and b are not parallel. Let us choose a point C on line b that does not lie in the plane. Let us draw a line b 1 through point C, parallel to line a. Line b 1 is perpendicular to the plane according to Theorem 2. Let B and B 1 be the points of intersection of lines b and b 1 with the plane. Then straight line BB 1 is perpendicular to the intersecting lines b and b 1. And this is impossible. We have arrived at a contradiction. The theorem has been proven.

33.Perpendicular, dropped from a given point given plane, is called a segment connecting a given point with a point in the plane and lying on a straight line, perpendicular to the plane. The end of this segment lying in a plane is called base of the perpendicular.
Inclined drawn from a given point to a given plane is any segment connecting a given point with a point on the plane that is not perpendicular to the plane. The end of a segment lying in a plane is called inclined base. A segment connecting the bases of a perpendicular to an inclined one drawn from the same point is called oblique projection.

AB is perpendicular to the α plane.
AC – oblique, CB – projection.

Statement of the theorem

If a straight line drawn on a plane through the base of an inclined line is perpendicular to its projection, then it is perpendicular to the inclined one.

Proof

Let AB- perpendicular to plane α, A.C.- inclined and c- a straight line in the α plane passing through the point C And perpendicular to the projection B.C.. Let's make a direct CK parallel to the line AB. Straight CK is perpendicular to the plane α (since it is parallel AB), and therefore any straight line of this plane, therefore, CK perpendicular to a straight line c. Let's draw through parallel lines AB And CK plane β (parallel lines define a plane, and only one). Straight c perpendicular to two intersecting lines lying in the β plane, this is B.C. according to the condition and CK by construction, it means that it is perpendicular to any line belonging to this plane, which means it is perpendicular to the line A.C..

Equation of a plane. How to write an equation of a plane?
Mutual arrangement of planes. Tasks

Spatial geometry is not much more complicated than “flat” geometry, and our flights in space begin with this article. To master the topic, you need to have a good understanding of vectors, in addition, it is advisable to be familiar with the geometry of the plane - there will be many similarities, many analogies, so the information will be digested much better. In a series of my lessons, the 2D world opens with an article Equation of a straight line on a plane. But now Batman has left the flat TV screen and is launching from the Baikonur Cosmodrome.

Let's start with drawings and symbols. Schematically, the plane can be drawn in the form of a parallelogram, which creates the impression of space:

The plane is infinite, but we have the opportunity to depict only a piece of it. In practice, in addition to the parallelogram, an oval or even a cloud is also drawn. For technical reasons, it is more convenient for me to depict the plane in exactly this way and in exactly this position. Real planes that we will consider in practical examples, can be positioned in any way - mentally take the drawing in your hands and rotate it in space, giving the plane any inclination, any angle.

Designations: planes are usually denoted in small Greek letters, apparently so as not to confuse them with straight line on a plane or with straight line in space. I'm used to using the letter . In the drawing it is the letter “sigma”, and not a hole at all. Although, the holey plane is certainly quite funny.

In some cases, it is convenient to use the same symbols to designate planes. greek letters with subscripts, for example, .

It is obvious that the plane is uniquely defined by three different points that do not lie on the same line. Therefore, three-letter designations of planes are quite popular - by the points belonging to them, for example, etc. Often letters are enclosed in parentheses: , so as not to confuse the plane with another geometric figure.

For experienced readers I will give quick access menu:

• How to create an equation of a plane using a point and two vectors?
• How to create an equation of a plane using a point and a normal vector?

and we will not languish in long waits:

General plane equation

The general equation of the plane has the form , where the coefficients are not equal to zero at the same time.

A number of theoretical calculations and practical problems valid both for the usual orthonormal basis and for affine basis space (if the oil is oil, return to the lesson Linear (non) dependence of vectors. Basis of vectors). For simplicity, we will assume that all events occur in an orthonormal basis and a Cartesian rectangular system coordinates

Now let's practice a little spatial imagination. It’s okay if yours is bad, now we’ll develop it a little. Even playing on nerves requires training.

In the very general case, when the numbers are not zero, the plane intersects all three coordinate axes. For example, like this:

I repeat once again that the plane continues indefinitely in all directions, and we have the opportunity to depict only part of it.

Let's consider the simplest equations of planes:

How to understand given equation? Think about it: “Z” is ALWAYS equal to zero, for any values ​​of “X” and “Y”. This equation is "native" coordinate plane. Indeed, formally the equation can be rewritten as follows: , from where you can clearly see that we don’t care what values ​​“x” and “y” take, it is important that “z” is equal to zero.

Likewise:
– equation of the coordinate plane;
– equation of the coordinate plane.

Let's complicate the problem a little, consider a plane (here and further in the paragraph we assume that the numerical coefficients are not equal to zero). Let's rewrite the equation in the form: . How to understand it? “X” is ALWAYS, for any values ​​of “Y” and “Z”, equal to a certain number. This plane is parallel to the coordinate plane. For example, a plane is parallel to a plane and passes through a point.

Likewise:
– equation of a plane that is parallel to the coordinate plane;
– equation of a plane that is parallel to the coordinate plane.

Let's add members: . The equation can be rewritten as follows: , that is, “zet” can be anything. What does it mean? “X” and “Y” are connected by the relation, which draws a certain straight line in the plane (you will find out equation of a line in a plane?). Since “z” can be anything, this straight line is “replicated” at any height. Thus the equation defines a plane parallel to coordinate axis

Likewise:
– equation of a plane that is parallel to the coordinate axis;
– equation of a plane that is parallel to the coordinate axis.

If the free terms are zero, then the planes will directly pass through the corresponding axes. For example, the classic “direct proportionality”: . Draw a straight line in the plane and mentally multiply it up and down (since “Z” is any). Conclusion: plane, given by the equation, passes through the coordinate axis.

We complete the review: the equation of the plane passes through the origin. Well, here it is quite obvious that the point satisfies this equation.

And finally, the case shown in the drawing: – the plane is friendly with all coordinate axes, while it always “cuts off” a triangle, which can be located in any of the eight octants.

Linear inequalities in space

To understand the information you need to study well linear inequalities in the plane, because many things will be similar. The paragraph will be of a brief overview nature with several examples, since the material is quite rare in practice.

If the equation defines a plane, then the inequalities
ask half-spaces. If the inequality is not strict (the last two in the list), then the solution of the inequality, in addition to the half-space, also includes the plane itself.

Example 5

Find the unit normal vector of the plane .

Solution: A unit vector is a vector whose length is one. Let us denote this vector by . It is absolutely clear that the vectors are collinear:

First, we remove the normal vector from the equation of the plane: .

How to find unit vector? In order to find the unit vector, you need every divide the vector coordinate by the vector length.

Let's rewrite the normal vector in the form and find its length:

According to the above:

Answer:

Verification: what was required to be verified.

Readers who carefully studied the last paragraph of the lesson probably noticed that the coordinates of the unit vector are exactly the direction cosines of the vector:

Let's take a break from the problem at hand: when you are given an arbitrary non-zero vector, and according to the condition it is required to find its direction cosines (see the last problems of the lesson Dot product of vectors), then you, in fact, find a unit vector collinear to this one. Actually two tasks in one bottle.

The need to find the unit normal vector arises in some problems of mathematical analysis.

We’ve figured out how to fish out a normal vector, now let’s answer the opposite question:

How to create an equation of a plane using a point and a normal vector?

This rigid construction of a normal vector and a point is well known to the dartboard. Please stretch your hand forward and mentally select an arbitrary point in space, for example, a small cat in the sideboard. Obviously, through this point you can draw a single plane perpendicular to your hand.

The equation of a plane passing through a point perpendicular to the vector is expressed by the formula:

First level

Coordinates and vectors. Comprehensive guide (2019)

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 a lot easier, especially when you feel unsure about building 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 that name because it doesn't operate with geometric objects, and with them 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 coordinate method (they sometimes turn out to be useful when solving problems on planimetry in Part B of the Unified State Examination). 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 linear function, For example. Let me remind you that you built it point by point. Do you remember? You chose arbitrary number, substituted it into the formula and calculated it this 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 do it with you next step: Let's 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 by more than two in capital letters, and one lowercase, 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? First of all, I connected dots and,a also drew a line from the point, 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 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 a ratio 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 the point, symmetrical 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 figures: 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 it is middle line triangle? Then this task is elementary for you. If you don’t remember, then I’ll remind you: the middle line of a triangle is the line that connects the midpoints 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. Side small square is a segment connecting points and Its length is

Then the area of ​​the small square is

We do exactly the same with big 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 length 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 equal to half its diagonals. 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 designation dot 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 last task! 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 the problems directly on vectors and the coordinate method in part B exam paper 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 tasks.

COORDINATES AND VECTORS. AVERAGE LEVEL

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

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 on to high-quality 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:

• in two various points there is a straight line on the 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 the line, but we need to pay attention to the very important concept directing vector straight 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 usually this issue is not addressed in the course high school. 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 can already use the technique that is usually studied in the course analytical geometry. So let's get started.

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

some numbers (not all equal to zero), and 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 more general view:

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 given number stands at the intersection of the second row and third column. Let's put it on next 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 in the following way:

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 allow 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 to a large extent chances are they will not lie on the same straight line), you 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 area parallelogram constructed on vectors and. This vector We will need it to calculate the distance from a point to a line. How can we count? vector product 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 decision:

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’s the choice relative position coordinate systems and shapes in space 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 remember how best to position the cube or cuboid- desirable.

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, didn’t you and I decide? similar examples earlier? 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 you to accurately calculate the angle, 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 in brief, and for the last two problems I will only give answers; you must carry out all the calculations for them yourself.

Tasks:

1. In the 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, then 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 heights equilateral triangle the intersection point is 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's draw the correct one hexagonal pyramid together 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:

Sum of angles regular n-gon equal to .

Thus, the sum of the angles regular hexagon 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 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 side rib. 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 count them equal to one. Thus, since ALL the 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 side faces- 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:

Cube - 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 on the side of the pi-ra-mi-dy’s edge.

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've already had to deal with the triangular and quadrangular pyramids, but with prisms - not yet.

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, we 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 last coordinate you can easily find it 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 is it? rectangular prism, How do you think? 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 previous tasks, which 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: the solution to this problem is traditional methods(via construction), it 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 coordinate method. I demonstrated this solution only to show you universal method, which 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 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 directing 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, in fact, is 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 the right direction 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:

Suppose we need to find the equation of a plane passing through three given points that do not lie on the same line. Denoting their radius vectors by and the current radius vector by , we can easily obtain the required equation in vector form. In fact, the vectors must be coplanar (they all lie in the desired plane). Therefore, the vector-scalar product of these vectors must be equal to zero:

This is the equation of a plane passing through three given points, in vector form.

Moving on to the coordinates, we get the equation in coordinates:

If three given points lay on the same line, then the vectors would be collinear. Therefore, the corresponding elements of the two last lines determinant in equation (18) would be proportional and the determinant would be identical equal to zero. Consequently, equation (18) would become identical for any values ​​of x, y and z. Geometrically, this means that through each point in space there passes a plane in which the three given points lie.

Remark 1. The same problem can be solved without using vectors.

Denoting the coordinates of the three given points, respectively, we will write the equation of any plane passing through the first point:

To obtain the equation of the desired plane, it is necessary to require that equation (17) be satisfied by the coordinates of two other points:

From equations (19), it is necessary to determine the ratio of two coefficients to the third and enter the found values ​​into equation (17).

Example 1. Write an equation for a plane passing through the points.

The equation of the plane passing through the first of these points will be:

The conditions for the plane (17) to pass through two other points and the first point are:

Adding the second equation to the first, we find:

Substituting into the second equation, we get:

Substituting into equation (17) instead of A, B, C, respectively, 1, 5, -4 (numbers proportional to them), we obtain:

Example 2. Write an equation for a plane passing through the points (0, 0, 0), (1, 1, 1), (2, 2, 2).

The equation of any plane passing through the point (0, 0, 0) will be]

The conditions for the passage of this plane through points (1, 1, 1) and (2, 2, 2) are:

Reducing the second equation by 2, we see that to determine two unknowns, there is one equation with

From here we get . Now substituting the value of the plane into the equation, we find:

This is the equation of the desired plane; it depends on arbitrary

quantities B, C (namely, from the relation i.e. there are an infinite number of planes passing through three given points (three given points lie on the same straight line).

Remark 2. The problem of drawing a plane through three given points that do not lie on the same line can be easily solved in general form if we use determinants. Indeed, since in equations (17) and (19) the coefficients A, B, C cannot be simultaneously equal to zero, then, considering these equations as homogeneous system with three unknowns A, B, C, write the necessary and sufficient condition existence of a solution to this system other than zero (Part 1, Chapter VI, § 6):

Having expanded this determinant into the elements of the first row, we obtain an equation of the first degree with respect to the current coordinates, which will be satisfied, in particular, by the coordinates of the three given points.

You can also verify this latter directly by substituting the coordinates of any of these points instead of . On the left side we get a determinant in which either the elements of the first row are zeros or there are two identical rows. Thus, the equation constructed represents a plane passing through the three given points.

In order for a single plane to be drawn through any three points in space, it is necessary that these points do not lie on the same straight line.

Consider the points M 1 (x 1, y 1, z 1), M 2 (x 2, y 2, z 2), M 3 (x 3, y 3, z 3) in the general Cartesian coordinate system.

In order for an arbitrary point M(x, y, z) to lie in the same plane with points M 1, M 2, M 3, it is necessary that the vectors be coplanar.

(
) = 0

Thus,

Equation of a plane passing through three points:

Equation of a plane given two points and a vector collinear to the plane.

Let the points M 1 (x 1,y 1,z 1),M 2 (x 2,y 2,z 2) and the vector be given
.

Let's create an equation for a plane passing through the given points M 1 and M 2 and an arbitrary point M (x, y, z) parallel to the vector .

Vectors
and vector
must be coplanar, i.e.

(
) = 0

Plane equation:

Equation of a plane using one point and two vectors,

collinear to the plane.

Let two vectors be given
And
, collinear planes. Then for arbitrary point M(x, y, z), belonging to the plane, vectors
must be coplanar.

Plane equation:

Equation of a plane by point and normal vector .

Theorem. If a point M is given in space 0 (X 0 , y 0 , z 0 ), then the equation of the plane passing through the point M 0 perpendicular to the normal vector (A, B, C) has the form:

A(x – x 0 ) + B(y – y 0 ) + C(z – z 0 ) = 0.

Proof. For an arbitrary point M(x, y, z) belonging to the plane, we compose a vector. Because vector is the normal vector, then it is perpendicular to the plane, and, therefore, perpendicular to the vector
. Then the scalar product

= 0

Thus, we obtain the equation of the plane

The theorem has been proven.

Equation of a plane in segments.

If in general equation Ax + Bu + Cz + D = 0 divide both sides by (-D)

,

replacing
, we obtain the equation of the plane in segments:

The numbers a, b, c are the intersection points of the plane with the x, y, z axes, respectively.

Equation of a plane in vector form.

Where

- radius vector of the current point M(x, y, z),

A unit vector having the direction of a perpendicular dropped onto a plane from the origin.

,  and  are the angles formed by this vector with the x, y, z axes.

p is the length of this perpendicular.

In coordinates, this equation looks like:

xcos + ycos + zcos - p = 0.

Distance from a point to a plane.

The distance from an arbitrary point M 0 (x 0, y 0, z 0) to the plane Ax+By+Cz+D=0 is:

Example. Find the equation of the plane, knowing that point P(4; -3; 12) is the base of the perpendicular dropped from the origin to this plane.

So A = 4/13; B = -3/13; C = 12/13, we use the formula:

A(x – x 0 ) + B(y – y 0 ) + C(z – z 0 ) = 0.

Example. Find the equation of a plane passing through two points P(2; 0; -1) and

Q(1; -1; 3) perpendicular to the plane 3x + 2y – z + 5 = 0.

Normal vector to the plane 3x + 2y – z + 5 = 0
parallel to the desired plane.

We get:

Example. Find the equation of the plane passing through points A(2, -1, 4) and

B(3, 2, -1) perpendicular to the plane X + at + 2z – 3 = 0.

The required equation of the plane has the form: A x+B y+C z+ D = 0, normal vector to this plane (A, B, C). Vector
(1, 3, -5) belongs to the plane. The plane given to us, perpendicular to the desired one, has a normal vector (1, 1, 2). Because points A and B belong to both planes, and the planes are mutually perpendicular, then

So the normal vector (11, -7, -2). Because point A belongs to the desired plane, then its coordinates must satisfy the equation of this plane, i.e. 112 + 71 - 24 +D= 0;D= -21.

In total, we get the equation of the plane: 11 x - 7y – 2z – 21 = 0.

Example. Find the equation of the plane, knowing that point P(4, -3, 12) is the base of the perpendicular dropped from the origin to this plane.

Finding the coordinates of the normal vector
= (4, -3, 12). The required equation of the plane has the form: 4 x – 3y + 12z+ D = 0. To find the coefficient D, we substitute the coordinates of point P into the equation:

16 + 9 + 144 + D = 0

In total, we get the required equation: 4 x – 3y + 12z – 169 = 0

Example. Given are the coordinates of the vertices of the pyramid A 1 (1; 0; 3), A 2 (2; -1; 3), A 3 (2; 1; 1),

Find the length of edge A 1 A 2.

Find the angle between edges A 1 A 2 and A 1 A 4.

Find the angle between edge A 1 A 4 and face A 1 A 2 A 3.

First we find the normal vector to the face A 1 A 2 A 3 as a cross product of vectors
And
.

= (2-1; 1-0; 1-3) = (1; 1; -2);

Let's find the angle between the normal vector and the vector
.

-4 – 4 = -8.

The desired angle  between the vector and the plane will be equal to  = 90 0 - .

Find the area of ​​face A 1 A 2 A 3.

Find the volume of the pyramid.

Find the equation of the plane A 1 A 2 A 3.

Let's use the formula for the equation of a plane passing through three points.

2x + 2y + 2z – 8 = 0

x + y + z – 4 = 0;

When using the computer version “ Higher mathematics course” you can run a program that will solve the above example for any coordinates of the vertices of the pyramid.

To start the program, double-click on the icon:

In the program window that opens, enter the coordinates of the vertices of the pyramid and press Enter. In this way, all decision points can be obtained one by one.

Note: To run the program, the Maple program ( Waterloo Maple Inc.) of any version, starting with MapleV Release 4, must be installed on your computer.