How to find the coordinates of a vector. Finding the length of a vector, examples and solutions How to find the length of a vector in space formula
12. Length of a vector, length of a segment, angle between vectors, condition of perpendicularity of vectors.
Vector – it is a directed segment connecting two points in space or in a plane. Vectors are usually denoted either by small letters or by starting and ending points. There is usually a dash at the top.
For example, a vector directed from the point A to the point B, can be designated a ,
Zero vector 0 or 0 - This is a vector whose starting and ending points coincide, i.e. A = B. From here, 0 = – 0 .
Vector length (modulus)a is the length of the segment representing it AB, denoted by |a | . In particular, | 0 | = 0.
The vectors are called collinear, if their directed segments lie on parallel lines. Collinear vectors a And b are designated a || b .
Three or more vectors are called coplanar, if they lie in the same plane.
Vector addition. Since vectors are directed segments, then their addition can be performed geometrically. (Algebraic addition of vectors is described below, in the paragraph “Unit orthogonal vectors”). Let's pretend that
a = AB and b = CD,
then the vector __ __
a + b = AB+ CD
is the result of two operations:
a)parallel transfer one of the vectors so that its starting point coincides with the end point of the second vector;
b)geometric addition, i.e. constructing a resulting vector going from the starting point of the fixed vector to the ending point of the transferred vector.
Subtraction of vectors. This operation is reduced to the previous one by replacing the subtrahend vector with its opposite one: a – b =a + (– b ) .
Laws of addition.
I. a + b = b + a (Transitional law).
II. (a + b ) + c = a + (b + c ) (Combinative law).
III. a + 0 = a .
IV. a + (– a ) = 0 .
Laws for multiplying a vector by a number.
I. 1 · a = a , 0 · a = 0 , m· 0 = 0 , (– 1) · a = – a .
II. ma = a m,| ma | = | m | · | a | .
III. m(na ) = (mn)a . (C o m b e t a l
law of multiplication by number).
IV. (m+n) a = ma +na , (DISTRIBUTION
m(a + b ) = ma +mb . law of multiplication by number).
Dot product of vectors. __ __
Angle between non-zero vectors AB And CD– this is the angle formed by the vectors when they are transferred in parallel until the points are aligned A And C. Dot product of vectorsa And b is called a number equal to the product of their lengths and the cosine of the angle between them:
If one of the vectors is zero, then their scalar product, in accordance with the definition, is equal to zero:
(a, 0 ) = ( 0 , b ) = 0 .
If both vectors are non-zero, then the cosine of the angle between them is calculated by the formula:
Scalar product ( a , a ), equal to | a | 2, called scalar square. a Vector length
and its scalar square are related by the relation:
- Dot product of two vectors: positively , if the angle between the vectors;
- spicy negative, if the angle between the vectors.
blunt The scalar product of two non-zero vectors is equal to zero then
and only when the angle between them is straight, i.e. when these vectors are perpendicular (orthogonal): Properties of the scalar product. For any vectors a, b,c m and any number
I. (For any vectors b ) = (the following relations are valid: ) . b,a
II. (mFor any vectors b ) = m(a, b ) .
III.((Transitional law) ) = (For any vectors c ) + (a+b,c c ). b,
(Distributive law) Unit orthogonal vectors. In any rectangular coordinate system you can enterunit pairwise orthogonal vectors , i j And k unit pairwise orthogonal vectors associated with coordinate axes: – with axle, X associated with coordinate axes: j And Y k – with axle Z
(unit pairwise orthogonal vectors . According to this definition: ) = (unit pairwise orthogonal vectors ,j ) = (X ,j ) = 0,
| , k =i | =| j | = 1.
| k | a Any vector a = can be expressed through these vectors in a unique way:x i+y j+z . k Another form of recording: = (a x, y, z ). Here, x, y z - coordinates a vector in this coordinate system. In accordance with the last relation and properties of unit orthogonal vectors ,j i, j
The scalar product of two vectors can be expressed differently. a = (a); b = (Let u, v, w a, b ) = ). Then ( xu + yv.
+ zw
The scalar product of two vectors is equal to the sum of the products of the corresponding coordinates. a = (can be expressed through these vectors in a unique way:, i+, j+ Vector length (modulus)
) is equal to: In addition, we now have the opportunity to conduct algebraic
operations on vectors, namely, addition and subtraction of vectors can be performed using coordinates: a + (b =) ;
a – a + (can be expressed through these vectors in a unique way:–x + u, y + v, z + w– u, y–v, z) .
w Cross product of vectors. [Vector artwork b ] a,a Andb vectors
There is another formula for the length of the vector [ a, b ] :
| [ a, b ] | = | a | | b | sin( a, b ) ,
i.e. length ( module ) vector product of vectorsa Andb is equal to the product of the lengths (modules) of these vectors and the sine of the angle between them. In other words: length (modulus) of the vector[ a, b ] numerically equal to the area of a parallelogram built on vectors a Andb .
Properties of a vector product.
I. Vector [ a, b ] perpendicular (orthogonal) both vectors a j b .
(Prove it, please!).
II.[ For any vectors b ] = – [the following relations are valid: ] .
III. [ mFor any vectors b ] = m[a, b ] .
IV. [ (Transitional law) ] = [ For any vectors c ] + [ a+b,c c ] .
V. [ For any vectors [ a, ] ] = b (a , c ) – c (a, b ) .
VI. [ [ For any vectors b ] , c ] = b (a , c ) – a (a, ) .
Necessary and sufficient condition for collinearity a, Another form of recording: = (a) And b = (Let) :
Necessary and sufficient condition for coplanarity a, a = (a), b = (Let) And c = (p, q, r) :
EXAMPLE The vectors are given: a = (1, 2, 3) and b = (– 2 , 0 ,4).
Calculate their dot and cross products and angle
between these vectors.
Solution. Using the appropriate formulas (see above), we obtain:
a). scalar product:
(a, b ) = 1 · (– 2) + 2 · 0 + 3 · 4 = 10 ;
b). vector product:
| " |
First of all, we need to understand the concept of a vector itself. In order to introduce the definition of a geometric vector, let us remember what a segment is. Let us introduce the following definition.
Definition 1
A segment is a part of a straight line that has two boundaries in the form of points.
A segment can have 2 directions. To denote the direction, we will call one of the boundaries of the segment its beginning, and the other boundary its end. The direction is indicated from its beginning to the end of the segment.
Definition 2
We will call a vector or a directed segment a segment for which it is known which of the boundaries of the segment is considered the beginning and which is its end.
Designation: In two letters: $\overline(AB)$ – (where $A$ is its beginning, and $B$ is its end).
In one small letter: $\overline(a)$ (Fig. 1).
Let us now introduce directly the concept of vector lengths.
Definition 3
The length of the vector $\overline(a)$ will be the length of the segment $a$.
Notation: $|\overline(a)|$
The concept of vector length is associated, for example, with such a concept as the equality of two vectors.
Definition 4
We will call two vectors equal if they satisfy two conditions: 1. They are codirectional; 1. Their lengths are equal (Fig. 2).
In order to define vectors, enter a coordinate system and determine the coordinates for the vector in the entered system. As we know, any vector can be decomposed in the form $\overline(c)=m\overline(i)+n\overline(j)$, where $m$ and $n$ are real numbers, and $\overline(i )$ and $\overline(j)$ are unit vectors on the $Ox$ and $Oy$ axis, respectively.
Definition 5
We will call the expansion coefficients of the vector $\overline(c)=m\overline(i)+n\overline(j)$ the coordinates of this vector in the introduced coordinate system. Mathematically:
$\overline(c)=(m,n)$
How to find the length of a vector?
In order to derive a formula for calculating the length of an arbitrary vector given its coordinates, consider the following problem:
Example 1
Given: vector $\overline(α)$ with coordinates $(x,y)$. Find: the length of this vector.
Let us introduce a Cartesian coordinate system $xOy$ on the plane. Let us set aside $\overline(OA)=\overline(a)$ from the origins of the introduced coordinate system. Let us construct projections $OA_1$ and $OA_2$ of the constructed vector on the $Ox$ and $Oy$ axes, respectively (Fig. 3).
The vector $\overline(OA)$ we have constructed will be the radius vector for point $A$, therefore, it will have coordinates $(x,y)$, which means
$=x$, $[OA_2]=y$
Now we can easily find the required length using the Pythagorean theorem, we get
$|\overline(α)|^2=^2+^2$
$|\overline(α)|^2=x^2+y^2$
$|\overline(α)|=\sqrt(x^2+y^2)$
Answer: $\sqrt(x^2+y^2)$.
Conclusion: To find the length of a vector whose coordinates are given, you need to find the root of the square of the sum of these coordinates.
Sample tasks
Example 2
Find the distance between points $X$ and $Y$, which have the following coordinates: $(-1.5)$ and $(7.3)$, respectively.
Any two points can be easily associated with the concept of a vector. Consider, for example, the vector $\overline(XY)$. As we already know, the coordinates of such a vector can be found by subtracting the corresponding coordinates of the starting point ($X$) from the coordinates of the end point ($Y$). We get that
First of all, we need to understand the concept of a vector itself. In order to introduce the definition of a geometric vector, let us remember what a segment is. Let us introduce the following definition.
Definition 1
A segment is a part of a straight line that has two boundaries in the form of points.
A segment can have 2 directions. To denote the direction, we will call one of the boundaries of the segment its beginning, and the other boundary its end. The direction is indicated from its beginning to the end of the segment.
Definition 2
We will call a vector or a directed segment a segment for which it is known which of the boundaries of the segment is considered the beginning and which is its end.
Designation: In two letters: $\overline(AB)$ – (where $A$ is its beginning, and $B$ is its end).
In one small letter: $\overline(a)$ (Fig. 1).
Let us now introduce directly the concept of vector lengths.
Definition 3
The length of the vector $\overline(a)$ will be the length of the segment $a$.
Notation: $|\overline(a)|$
The concept of vector length is associated, for example, with such a concept as the equality of two vectors.
Definition 4
We will call two vectors equal if they satisfy two conditions: 1. They are codirectional; 1. Their lengths are equal (Fig. 2).
In order to define vectors, enter a coordinate system and determine the coordinates for the vector in the entered system. As we know, any vector can be decomposed in the form $\overline(c)=m\overline(i)+n\overline(j)$, where $m$ and $n$ are real numbers, and $\overline(i )$ and $\overline(j)$ are unit vectors on the $Ox$ and $Oy$ axis, respectively.
Definition 5
We will call the expansion coefficients of the vector $\overline(c)=m\overline(i)+n\overline(j)$ the coordinates of this vector in the introduced coordinate system. Mathematically:
$\overline(c)=(m,n)$
How to find the length of a vector?
In order to derive a formula for calculating the length of an arbitrary vector given its coordinates, consider the following problem:
Example 1
Given: vector $\overline(α)$ with coordinates $(x,y)$. Find: the length of this vector.
Let us introduce a Cartesian coordinate system $xOy$ on the plane. Let us set aside $\overline(OA)=\overline(a)$ from the origins of the introduced coordinate system. Let us construct projections $OA_1$ and $OA_2$ of the constructed vector on the $Ox$ and $Oy$ axes, respectively (Fig. 3).
The vector $\overline(OA)$ we have constructed will be the radius vector for point $A$, therefore, it will have coordinates $(x,y)$, which means
$=x$, $[OA_2]=y$
Now we can easily find the required length using the Pythagorean theorem, we get
$|\overline(α)|^2=^2+^2$
$|\overline(α)|^2=x^2+y^2$
$|\overline(α)|=\sqrt(x^2+y^2)$
Answer: $\sqrt(x^2+y^2)$.
Conclusion: To find the length of a vector whose coordinates are given, you need to find the root of the square of the sum of these coordinates.
Sample tasks
Example 2
Find the distance between points $X$ and $Y$, which have the following coordinates: $(-1.5)$ and $(7.3)$, respectively.
Any two points can be easily associated with the concept of a vector. Consider, for example, the vector $\overline(XY)$. As we already know, the coordinates of such a vector can be found by subtracting the corresponding coordinates of the starting point ($X$) from the coordinates of the end point ($Y$). We get that
Oxy
ABOUT A OA.
, where 
OA 
.
Thus, 
.
Let's look at an example.
Example.
Solution.
:
Answer:
Oxyz in space.
A OA will be a diagonal.
In this case (since OA 
OA 
.
Thus, vector length 
.
Example.
Calculate Vector Length 
Solution.
, hence, 
Answer:
Straight line on a plane
General equation
Ax + By + C ( > 0).
Vector = (A; B) is a normal vector of a straight line.
In vector form: + C = 0, where is the radius vector of an arbitrary point on a line (Fig. 4.11).
Special cases:
1) By + C = 0- straight line parallel to the axis Ox;
2) Ax + C = 0- straight line parallel to the axis Oy;
3) Ax + By = 0- the straight line passes through the origin of coordinates;
4) y = 0- axis Ox;
5) x = 0- axis Oy.
Equation of a line in segments
Where a, b- the values of the segments cut off by the straight line on the coordinate axes.
Normal equation of a line(Fig. 4.11)
where is the angle formed normal to the line and the axis Ox; p- the distance from the origin to the straight line.
Reducing the general equation of a straight line to normal form:
Here is the normalized factor of the line; the sign is chosen opposite to the sign C, if and arbitrarily, if C=0.
Finding the length of a vector from coordinates.
We will denote the length of the vector by . Because of this notation, the length of a vector is often called the modulus of the vector.
Let's start by finding the length of a vector on a plane using coordinates.
Let us introduce a rectangular Cartesian coordinate system on the plane Oxy. Let a vector be specified in it and have coordinates . We obtain a formula that allows us to find the length of a vector through the coordinates and .
Let us postpone from the origin of coordinates (from the point ABOUT) vector . Let us denote the projections of the point A on the coordinate axes as and respectively and consider a rectangle with a diagonal OA.
By virtue of the Pythagorean theorem, the equality is true , where . From the definition of vector coordinates in a rectangular coordinate system, we can state that and , and by construction the length OA equal to the length of the vector, therefore, .
Thus, formula for finding the length of a vector according to its coordinates on the plane has the form .
If the vector is represented as an expansion in coordinate vectors , then its length is calculated using the same formula , since in this case the coefficients and are the coordinates of the vector in a given coordinate system.
Let's look at an example.
Example.
Find the length of the vector given in the Cartesian coordinate system.
Solution.
Immediately apply the formula to find the length of the vector from the coordinates :
Answer:
Now we get the formula for finding the length of the vector by its coordinates in a rectangular coordinate system Oxyz in space.
Let us plot the vector from the origin and denote the projections of the point A on the coordinate axes as and . Then we can construct a rectangular parallelepiped on the sides, in which OA will be a diagonal.
In this case (since OA– diagonal of a rectangular parallelepiped), from where . Determining the coordinates of a vector allows us to write equalities, and the length OA equal to the desired vector length, therefore, .
Thus, vector length in space is equal to the square root of the sum of the squares of its coordinates, that is, found by the formula .
Example.
Calculate Vector Length , where are the unit vectors of the rectangular coordinate system.
Solution.
We are given a vector decomposition into coordinate vectors of the form , hence, . Then, using the formula for finding the length of a vector from coordinates, we have .
In this article, we will begin to discuss one “magic wand” that will allow you to reduce many geometry problems to simple arithmetic. This “stick” can make your life much easier, especially when you feel unsure of constructing spatial figures, sections, etc. All this requires a certain imagination and practical skills. The method that we will begin to consider here will allow you to almost completely abstract from all kinds of geometric constructions and reasoning. The method is called "coordinate method". In this article we will consider the following questions:
- Coordinate plane
- Points and vectors on the plane
- Constructing a vector from two points
- Vector length (distance between two points)
- Coordinates of the middle of the segment
- Dot product of vectors
- Angle between two vectors
I think you've already guessed why the coordinate method is called that? That's right, it got this name because it operates not with geometric objects, but with their numerical characteristics (coordinates). And the transformation itself, which allows us to move from geometry to algebra, consists in introducing a coordinate system. If the original figure was flat, then the coordinates are two-dimensional, and if the figure is three-dimensional, then the coordinates are three-dimensional. In this article we will consider only the two-dimensional case. And the main goal of the article is to teach you how to use some basic techniques of the coordinate method (they sometimes turn out to be useful when solving problems on planimetry in Part B of the Unified State Exam). The next two sections on this topic are devoted to a discussion of methods for solving problems C2 (the problem of stereometry).
Where would it be logical to start discussing the coordinate method? Probably from the concept of a coordinate system. Remember when you first encountered her. It seems to me that in 7th grade, when you learned about the existence of a linear function, for example. Let me remind you that you built it point by point. Do you remember? You chose an arbitrary number, substituted it into the formula and calculated it that way. For example, if, then, if, then, etc. What did you get in the end? And you received points with coordinates: and. Next, you drew a “cross” (coordinate system), chose a scale on it (how many cells you will have as a unit segment) and marked the points you obtained on it, which you then connected with a straight line; the resulting line is the graph of the function.
There are a few points here that should be explained to you in a little more detail:
1. You choose a single segment for reasons of convenience, so that everything fits beautifully and compactly in the drawing.
2. It is accepted that the axis goes from left to right, and the axis goes from bottom to top
3. They intersect at right angles, and the point of their intersection is called the origin. It is indicated by a letter.
4. In writing the coordinates of a point, for example, on the left in parentheses there is the coordinate of the point along the axis, and on the right, along the axis. In particular, it simply means that at the point
5. In order to specify any point on the coordinate axis, you need to indicate its coordinates (2 numbers)
6. For any point lying on the axis,
7. For any point lying on the axis,
8. The axis is called the x-axis
9. The axis is called the y-axis
Now let's take the next step: mark two points. Let's connect these two points with a segment. And we’ll put the arrow as if we were drawing a segment from point to point: that is, we’ll make our segment directed!
Remember what another directional segment is called? That's right, it's called a vector!
So if we connect dot to dot, and the beginning will be point A, and the end will be point B, then we get a vector. You also did this construction in 8th grade, remember?
It turns out that vectors, like points, can be denoted by two numbers: these numbers are called vector coordinates. Question: Do you think it is enough for us to know the coordinates of the beginning and end of a vector to find its coordinates? It turns out that yes! And this is done very simply:
Thus, since in a vector the point is the beginning and the end is the end, the vector has the following coordinates:
For example, if, then the coordinates of the vector
Now let's do the opposite, find the coordinates of the vector. What do we need to change for this? Yes, you need to swap the beginning and end: now the beginning of the vector will be at the point, and the end will be at the point. Then:
Look carefully, what is the difference between vectors and? Their only difference is the signs in the coordinates. They are opposites. This fact is usually written like this:
Sometimes, if it is not specifically stated which point is the beginning of the vector and which is the end, then vectors are denoted not by two capital letters, but by one lowercase letter, for example: , etc.
Now a little practice yourself and find the coordinates of the following vectors:
Examination:
Now solve a slightly more difficult problem:
A vector with a starting point 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)
- Vectors can be added to each other
- Vectors can be subtracted from each other
- Vectors can be multiplied (or divided) by an arbitrary non-zero number
- 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 the right triangle. Well, the Pythagorean theorem for sure. The required segment is the hypotenuse of this triangle, and the segments are the legs. What are the coordinates of the point? Yes, they are easy to find from the picture: Since the segments are parallel to the axes and, respectively, their lengths are easy to find: if we denote the lengths of the segments respectively by, then
Now let's use the Pythagorean theorem. We know the lengths of the legs, we will find the hypotenuse:
Thus, the distance between two points is the root of the sum of the 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
1. And this is for attentiveness) We have already found the coordinates of the vectors earlier: . Then the vector has coordinates. The square of its length will be equal to:
2. Find the coordinates of the vector
Then the square of its length is
Nothing complicated, right? Simple arithmetic, nothing more.
The following problems cannot be classified unambiguously; they are more about general erudition and the ability to draw simple pictures.
1. Find the sine of the angle from the cut, connecting the point, with the abscissa axis.
And
How are we going to proceed here? We need to find the sine of the angle between and the axis. Where can we look for sine? That's right, in a right triangle. So what do we need to do? Build this triangle!
Since the coordinates of the point are and, then the segment is equal to, and the segment. We need to find the sine of the angle. Let me remind you that sine is the ratio of the opposite side to the hypotenuse, then
What's left for us to do? Find the hypotenuse. You can do this in two ways: using the Pythagorean theorem (the legs are known!) or using the formula for the distance between two points (in fact, the same thing as the first method!). I'll go the second way:
Answer:
The next task will seem even easier to you. She is on the coordinates of the point.
Task 2. From the point the per-pen-di-ku-lyar is lowered onto the ab-ciss axis. Nai-di-te abs-cis-su os-no-va-niya per-pen-di-ku-la-ra.
Let's make a drawing:
The base of a perpendicular is the point at which it intersects the x-axis (axis), for me this is a point. The figure shows that it has coordinates: . We are interested in the abscissa - that is, the “x” component. She is equal.
Answer: .
Task 3. In the conditions of the previous problem, find the sum of the distances from the point to the coordinate axes.
The task is generally elementary if you know what the distance from a point to the axes is. You know? I hope, but still remind you:
So, in my drawing just above, have I already drawn one such perpendicular? Which axis is it on? To the axis. And what is its length then? She is equal. Now draw a perpendicular to the axis yourself and find its length. It will be equal, right? Then their sum is equal.
Answer: .
Task 4. In the conditions of task 2, find the ordinate of a point symmetrical to the point relative to the abscissa axis.
I think it is intuitively clear to you what symmetry is? Many objects have it: many buildings, tables, airplanes, many geometric shapes: ball, cylinder, square, rhombus, etc. Roughly speaking, symmetry can be understood as follows: a figure consists of two (or more) identical halves. This symmetry is called axial symmetry. What then is an axis? This is exactly the line along which the figure can, relatively speaking, be “cut” into equal halves (in this picture the axis of symmetry is straight):
Now let's get back to our task. We know that we are looking for a point that is symmetrical about the axis. Then this axis is the axis of symmetry. This means that we need to mark a point such that the axis cuts the segment into two equal parts. Try to mark such a point yourself. Now compare with my solution:
Did it work out the same way for you? Fine! We are interested in the ordinate of the found point. It is equal
Answer:
Now tell me, after thinking for a few seconds, what will be the abscissa of a point symmetrical to point A relative to the ordinate? What is your answer? Correct answer: .
In general, the rule can be written like this:
A point symmetrical to a point relative to the abscissa axis has the coordinates:
A point symmetrical to a point relative to the ordinate axis has coordinates:
Well, now it's completely scary task: find the coordinates of a point symmetrical to the point relative to the origin. You first think for yourself, and then look at my drawing!
Answer:
Now parallelogram problem:
Task 5: The points appear ver-shi-na-mi pa-ral-le-lo-gram-ma. Find or-di-on-that point.
You can solve this problem in two ways: logic and the coordinate method. I'll use the coordinate method first, and then I'll tell you how you can solve it differently.
It is quite clear that the abscissa of the point is equal to. (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 triangles. Find the length of its midline, parallel.
Do you remember what the middle line of a triangle is? Then this task is elementary for you. If you don’t remember, I’ll remind you: the middle line of a triangle is the line that connects the midpoints of opposite sides. It is parallel to the base and equal to half of it.
The base is a segment. We had to look for its length earlier, it is equal. Then the length of the middle line is half as large and equal.
Answer: .
Comment: this problem can be solved in another way, which we will turn to a little later.
In the meantime, here are a few problems for you, practice on them, they are very simple, but they help you get better at using the coordinate method!
1. The points are the top of the tra-pe-tions. Find the length of its midline.
2. Points and appearances ver-shi-na-mi pa-ral-le-lo-gram-ma. Find or-di-on-that point.
3. Find the length from the cut, connecting the point and
4. Find the area behind the colored figure on the co-ordi-nat plane.
5. A circle with a center in na-cha-le ko-or-di-nat passes through the point. Find her ra-di-us.
6. Find-di-te ra-di-us of the circle, describe-san-noy about the right-angle-no-ka, the tops of something have a co-or -di-na-you are so-responsible
Solutions:
1. It is known that the midline of a trapezoid is equal to half the sum of its bases. The base is equal, and the base. Then
Answer:
2. The easiest way to solve this problem is to note that (parallelogram rule). Calculating the coordinates of vectors is not difficult: . When adding vectors, the coordinates are added. Then it has coordinates. The point also has these coordinates, since the origin of the vector is the point with the coordinates. We are interested in the ordinate. She is equal.
Answer:
3. We immediately act according to the formula for the distance between two points:
Answer:
4. Look at the picture and tell me which two figures the shaded area is “sandwiched” between? It is sandwiched between two squares. Then the area of the desired figure is equal to the area of the large square minus the area of the small one. The side of a small square is a segment connecting the points and Its length is
Then the area of the small square is
We do the same with a large square: its side is a segment connecting the points and its length is equal to
Then the area of the large square is
We find the area of the desired figure using the formula:
Answer:
5. If a circle has the origin as its center and passes through a point, then its radius will be exactly equal to the length of the segment (make a drawing and you will understand why this is obvious). Let's find the length of this segment:
Answer:
6. It is known that the radius of a circle circumscribed about a rectangle is equal to half its diagonal. Let's find the length of any of the two diagonals (after all, in a rectangle they are equal!)
Answer:
Well, did you cope with everything? It wasn't very difficult to figure it out, was it? There is only one rule here - be able to make a visual picture and simply “read” all the data from it.
We have very little left. There are literally two more points that I would like to discuss.
Let's try to solve this simple problem. Let two points and be given. Find the coordinates of the midpoint of the segment. The solution to this problem is as follows: let the point be the desired middle, then it has coordinates:
That is: coordinates of the middle of the segment = the arithmetic mean of the corresponding coordinates of the ends of the segment.
This rule is very simple and usually does not cause difficulties for students. Let's see in what problems and how it is used:
1. Find-di-te or-di-na-tu se-re-di-ny from-cut, connect-the-point and
2. The points appear to be the top of the world. Find-di-te or-di-na-tu points per-re-se-che-niya of his dia-go-na-ley.
3. Find-di-te abs-cis-su center of the circle, describe-san-noy about the rectangular-no-ka, the tops of something have co-or-di-na-you so-responsibly-but.
Solutions:
1. The first problem is simply a classic. We proceed immediately to determine the middle of the segment. It has coordinates. The ordinate is equal.
Answer:
2. It is easy to see that this quadrilateral is a parallelogram (even a rhombus!). You can prove this yourself by calculating the lengths of the sides and comparing them with each other. What do I know about parallelograms? Its diagonals are divided in half by the point of intersection! Yeah! So what is the point of intersection of the diagonals? This is the middle of any of the diagonals! I will choose, in particular, the diagonal. Then the point has coordinates The ordinate of the point is equal to.
Answer:
3. What does the center of the circle circumscribed about the rectangle coincide with? It coincides with the point of intersection 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 around the triangle, the tops of something have a co-or-di -no misters
2. Find-di-te or-di-on-that center of the circle, describe-san-noy about the triangle-no-ka, the tops of which have coordinates
3. What kind of ra-di-u-sa should there be a circle with a center at a point so that it touches the ab-ciss axis?
4. Find-di-those or-di-on-that point of re-se-ce-tion of the axis and from-cut, connect-the-point and
Answers:
Was everything successful? I really hope for it! Now - the last push. Now be especially careful. The material that I will now explain is directly related not only to simple problems on the coordinate method from Part B, but is also found everywhere in Problem C2.
Which of my promises have I not yet kept? Remember what operations on vectors I promised to introduce and which ones I ultimately introduced? Are you sure I haven't forgotten anything? Forgot! I forgot to explain what vector multiplication means.
There are two ways to multiply a vector by a vector. Depending on the chosen method, we will get objects of different natures:
The cross product is done quite cleverly. We will discuss how to do it and why it is needed in the next article. And in this one we will focus on the scalar product.
There are two ways that allow us to calculate it:
As you guessed, the result should be the same! So let's look at the first method first:
Dot product via coordinates
Find: - generally accepted notation for scalar product
The formula for calculation is as follows:
That is, the scalar product = the sum of the products of vector coordinates!
Example:
Find-di-te
Solution:
Let's find the coordinates of each of the vectors:
We calculate the scalar product using the formula:
Answer:
See, absolutely nothing complicated!
Well, now try it yourself:
· Find a scalar pro-iz-ve-de-nie of centuries and
Did you manage? Maybe you noticed a small catch? Let's check:
Vector coordinates, as in the previous problem! Answer: .
In addition to the coordinate product, 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:
- Calculate the scalar product through coordinates
- Find the lengths of the vectors and multiply them
- Divide the result of point 1 by the result of point 2
Let's practice with examples:
1. Find the angle between the eyelids and. Give the answer in grad-du-sah.
2. In the conditions of the previous problem, find the cosine between the vectors
Let's do this: I'll help you solve the first problem, and try to do the second yourself! Agree? Then let's begin!
1. These vectors are our old friends. We have already calculated their scalar product and it was equal. Their coordinates are: , . Then we find their lengths:
Then we look for the cosine between the vectors:
What is the cosine of the angle? This is the corner.
Answer:
Well, now solve the second problem yourself, and then compare! I will give just a very short solution:
2. has coordinates, has coordinates.
Let be the angle between the vectors and, then
Answer:
It should be noted that problems directly on vectors and the coordinate method in Part B of the exam paper are quite rare. However, the vast majority of C2 problems can be easily solved by introducing a coordinate system. So you can consider this article the foundation on the basis of which we will make quite clever constructions that we will need to solve complex problems.
COORDINATES AND VECTORS. AVERAGE LEVEL
You and I continue to study the coordinate method. In the last part, we derived a number of important formulas that allow you to:
- Find vector coordinates
- Find the length of a vector (alternatively: the distance between two points)
- Add and subtract vectors. Multiply them by a real number
- Find the midpoint of a segment
- Calculate dot product of vectors
- Find the angle between vectors
Of course, the entire coordinate method does not fit into these 6 points. It underlies such a science as analytical geometry, which you will become familiar with at university. I just want to build a foundation that will allow you to solve problems in a single state. exam. We have dealt with the tasks of Part B. Now it’s time to move to a whole new level! This article will be devoted to a method for solving those C2 problems in which it would be reasonable to switch to the coordinate method. This reasonableness is determined by what is required to be found in the problem and what figure is given. So, I would use the coordinate method if the questions are:
- Find the angle between two planes
- Find the angle between a straight line and a plane
- Find the angle between two straight lines
- Find the distance from a point to a plane
- Find the distance from a point to a line
- Find the distance from a straight line to a plane
- 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:
- Rectangular parallelepiped
- Pyramid (triangular, quadrangular, hexagonal)
Also from my experience it is inappropriate to use the coordinate method for:
- Finding cross-sectional areas
- Calculation of volumes of bodies
However, it should immediately be noted that the three “unfavorable” situations for the coordinate method are quite rare in practice. In most tasks, it can become your savior, especially if you are not very good at three-dimensional constructions (which can sometimes be quite intricate).
What are all the figures I listed above? They are no longer flat, like, for example, a square, a triangle, a circle, but voluminous! Accordingly, we need to consider not a two-dimensional, but a three-dimensional coordinate system. It is quite easy to construct: just in addition to the abscissa and ordinate axis, we will introduce another axis, the applicate axis. The figure schematically shows their relative position:
All of them are mutually perpendicular and intersect at one point, which we will call the origin of coordinates. As before, we will denote the abscissa axis, the ordinate axis - , and the introduced applicate axis - .
If previously each point on the plane was characterized by two numbers - the abscissa and the ordinate, then each point in space is already described by three numbers - the abscissa, the ordinate, and the applicate. For example:
Accordingly, the abscissa of a point is equal, the ordinate is , and the applicate is .
Sometimes the abscissa of a point is also called the projection of a point onto the abscissa axis, the ordinate - the projection of a point onto the ordinate axis, and the applicate - the projection of a point onto the applicate axis. Accordingly, if a point is given, then a point with coordinates:
called the projection of a point onto a plane
called the projection of a point onto a plane
A natural question arises: are all the formulas derived for the two-dimensional case valid in space? The answer is yes, they are fair and have the same appearance. For a small detail. I think you've already guessed which one it is. In all formulas we will have to add one more term responsible for the applicate axis. Namely.
1. If two points are given: , then:
- Vector coordinates:
- Distance between two points (or vector length)
- The midpoint of the segment has coordinates
2. If two vectors are given: and, then:
- Their scalar product is equal to:
- The cosine of the angle between the vectors is equal to:
However, space is not so simple. As you understand, adding one more coordinate introduces significant diversity into the spectrum of figures “living” in this space. And for further narration I will need to introduce some, roughly speaking, “generalization” of the straight line. This “generalization” will be a plane. What do you know about plane? Try to answer the question, what is a plane? It's very difficult to say. However, we all intuitively imagine what it looks like:
Roughly speaking, this is a kind of endless “sheet” stuck into space. “Infinity” should be understood that the plane extends in all directions, that is, its area is equal to infinity. However, this “hands-on” explanation does not give the slightest idea about the structure of the plane. And it is she who will be interested in us.
Let's remember one of the basic axioms of geometry:
- a straight line passes through two different points on a plane, and only one:
Or its analogue in space:
Of course, you remember how to derive the equation of a line from two given points; it’s not at all difficult: if the first point has coordinates: and the second, then the equation of the line will be as follows:
You took this in 7th grade. In space, the equation of a line looks like this: let us be given two points with coordinates: , then the equation of the line passing through them has the form:
For example, a line passes through points:
How should this be understood? This should be understood as follows: a point lies on a line if its coordinates satisfy the following system:
We will not be very interested in the equation of a line, but we need to pay attention to the very important concept of the direction vector of a line. - any non-zero vector lying on a given line or parallel to it.
For example, both vectors are direction vectors of a straight line. Let be a point lying on a line and let be its direction vector. Then the equation of the straight 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 already know how to use a technique that is usually studied in an analytical geometry course. So let's get started.
The equation of a plane is not too different from the equation of a straight line on a plane, namely, it has the form:
some numbers (not all equal to zero), but variables, for example: etc. As you can see, the equation of a plane is not very different from the equation of a straight line (linear function). However, remember what you and I argued? We said that if we have three points that do not lie on the same line, then the equation of the plane can be uniquely reconstructed from them. But how? I'll try to explain it to you.
Since the equation of the plane is:
And the points belong to this plane, then when substituting the coordinates of each point into the equation of the plane we should obtain the correct identity:
Thus, there is a need to solve three equations with unknowns! Dilemma! However, you can always assume that (to do this you need to divide by). Thus, we get three equations with three unknowns:
However, we will not solve such a system, but will write out the mysterious expression that follows from it:
Equation of a plane passing through three given points
\[\left| (\begin(array)(*(20)(c))(x - (x_0))&((x_1) - (x_0))&((x_2) - (x_0))\\(y - (y_0) )&((y_1) - (y_0))&((y_2) - (y_0))\\(z - (z_0))&((z_1) - (z_0))&((z_2) - (z_0)) \end(array)) \right| = 0\]
Stop! What is this? Some very unusual module! However, the object that you see in front of you has nothing to do with the module. This object is called a third-order determinant. From now on, when you deal with the method of coordinates on a plane, you will very often encounter these same determinants. What is a third order determinant? Oddly enough, it's just a number. It remains to understand what specific number we will compare with the determinant.
Let's first write the third-order determinant in a more general form:
Where are some numbers. Moreover, by the first index we mean the row number, and by the index we mean the column number. For example, it means that this number is at the intersection of the second row and third column. Let's pose the following question: how exactly will we calculate such a determinant? That is, what specific number will we compare to it? For the third-order determinant there is a heuristic (visual) triangle rule, it looks like this:
- 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
- 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
- Then the determinant is equal to the difference between the values obtained at the step and
If we write all this down in numbers, we get the following expression:
However, you don’t need to remember the method of calculation in this form; it’s enough to just keep in your head the triangles and the very idea of what adds up to what and what is then subtracted from what).
Let's illustrate the triangle method with an example:
1. Calculate the determinant:
Let's figure out what we add and what we subtract:
Terms that come with a plus:
This is the main diagonal: the product of the elements is equal to
The first triangle, "perpendicular to the main diagonal: the product of the elements is equal to
Second triangle, "perpendicular to the main diagonal: the product of the elements is equal to
Add up three numbers:
Terms that come with a minus
This is a side diagonal: the product of the elements is equal to
The first triangle, “perpendicular to the secondary diagonal: the product of the elements is equal to
The second triangle, “perpendicular to the secondary diagonal: the product of the elements is equal to
Add up three numbers:
All that remains to be done is to subtract the sum of the “plus” terms from the sum of the “minus” terms:
Thus,
As you can see, there is nothing complicated or supernatural in calculating third-order determinants. It’s just important to remember about triangles and not make arithmetic errors. Now try to calculate it yourself:
Task: find the distance between the indicated points:
- The first triangle perpendicular to the main diagonal:
- Second triangle perpendicular to the main diagonal:
- Sum of terms with plus:
- The first triangle perpendicular to the secondary diagonal:
- Second triangle perpendicular to the side diagonal:
- Sum of terms with minus:
- 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:
- Construct the equation of a plane passing through three points:
Answers:
Did everything coincide? Again, if there are certain difficulties, then my advice is this: take three points from your head (with a high degree of probability they will not lie on the same straight line), build a plane based on them. And then you check yourself online. For example, on the site:
However, with the help of determinants we will construct not only the equation of the plane. Remember, I told you that not only dot product is defined for vectors. There is also a vector product, as well as a mixed product. And if the scalar product of two vectors is a number, then the vector product of two vectors will be a vector, and this vector will be perpendicular to the given ones:
Moreover, its module will be equal to the area of a parallelogram built on the vectors and. We will need this vector to calculate the distance from a point to a line. How can we calculate the vector product of vectors and, if their coordinates are given? The third-order determinant comes to our aid again. However, before I move on to the algorithm for calculating the vector product, I have to make a small digression.
This digression concerns basis vectors.
They are shown schematically in the figure:
Why do you think they are called basic? The fact is that :
Or in the picture:
The validity of this formula is obvious, because:
Vector artwork
Now I can start introducing the cross product:
The vector product of two vectors is a vector, which is calculated according to the following rule:
Now let's give some examples of calculating the cross product:
Example 1: Find the cross product of vectors:
Solution: I make up a determinant:
And I calculate it:
Now from writing through basis vectors, I will return to the usual vector notation:
Thus:
Now try it.
Ready? We check:
And traditionally two tasks for control:
- Find the vector product of the following vectors:
- Find the vector product of the following vectors:
Answers:
Mixed product of three vectors
The last construction I'll need is the mixed product of three vectors. It, like a scalar, is a number. There are two ways to calculate it. - through a determinant, - through a mixed product.
Namely, let us be given three vectors:
Then the mixed product of three vectors, denoted by, can be calculated as:
1. - that is, the mixed product is the scalar product of a vector and the vector product of two other vectors
For example, the mixed product of three vectors is:
Try to calculate it yourself using the vector product and make sure that the results match!
And again, two examples for independent solutions:
Answers:
Selecting a coordinate system
Well, now we have all the necessary foundation of knowledge to solve complex stereometric geometry problems. However, before proceeding directly to examples and algorithms for solving them, I believe that it will be useful to dwell on the following question: how exactly choose a coordinate system for a particular figure. After all, it is the choice of the relative position of the coordinate system and the figure in space that will ultimately determine how cumbersome the calculations will be.
Let me remind you that in this section we consider the following figures:
- Rectangular parallelepiped
- Straight prism (triangular, hexagonal...)
- Pyramid (triangular, quadrangular)
- Tetrahedron (same as triangular pyramid)
For a rectangular parallelepiped or cube, I recommend you the following construction:
That is, I will place the figure “in the corner”. The cube and parallelepiped are very good figures. For them, you can always easily find the coordinates of its vertices. For example, if (as shown in the picture)
then the coordinates of the vertices are as follows:
Of course, you don’t need to remember this, but remembering how best to position a cube or rectangular parallelepiped is advisable.
Straight prism
The prism is a more harmful figure. It can be positioned in space in different ways. However, the following option seems to me the most acceptable:
Triangular prism:
That is, we place one of the sides of the triangle entirely on the axis, and one of the vertices coincides with the origin of coordinates.
Hexagonal prism:
That is, one of the vertices coincides with the origin, and one of the sides lies on the axis.
Quadrangular and hexagonal pyramid:
The situation is similar to a cube: we align two sides of the base with the coordinate axes, and align one of the vertices with the origin of coordinates. The only slight difficulty will be to calculate the coordinates of the point.
For a hexagonal pyramid - the same as for a hexagonal prism. The main task will again be to find the coordinates of the vertex.
Tetrahedron (triangular pyramid)
The situation is very similar to the one I gave for a triangular prism: one vertex coincides with the origin, one side lies on the coordinate axis.
Well, now you and I are finally close to starting to solve problems. From what I said at the very beginning of the article, you could draw the following conclusion: most C2 problems are divided into 2 categories: angle problems and distance problems. First, we will look at the problems of finding an angle. They are in turn divided into the following categories (as complexity increases):
Problems for finding angles
- Finding the angle between two straight lines
- Finding the angle between two planes
Let's look at these problems sequentially: let's start by finding the angle between two straight lines. Well, remember, haven’t you and I solved similar examples before? Do you remember, we already had something similar... We were looking for the angle between two vectors. Let me remind you, if two vectors are given: and, then the angle between them is found from the relation:
Now our goal is to find the angle between two straight lines. Let's look at the “flat picture”:
How many angles did we get when two straight lines intersected? Just a few things. True, only two of them are not equal, while the others are vertical to them (and therefore coincide with them). So which angle should we consider the angle between two straight lines: or? Here the rule is: the angle between two straight lines is always no more than degrees. That is, from two angles we will always choose the angle with the smallest degree measure. That is, in this picture the angle between two straight lines is equal. In order not to bother each time with finding the smallest of two angles, cunning mathematicians suggested using a modulus. Thus, the angle between two straight lines is determined by the formula:
You, as an attentive reader, should have had a question: where, exactly, do we get these very numbers that we need to calculate the cosine of an angle? Answer: we will take them from the direction vectors of the lines! Thus, the algorithm for finding the angle between two straight lines is as follows:
- We apply formula 1.
Or in more detail:
- We are looking for the coordinates of the direction vector of the first straight line
- We are looking for the coordinates of the direction vector of the second straight line
- We calculate the modulus of their scalar product
- We are looking for the length of the first vector
- We are looking for the length of the second vector
- Multiply the results of point 4 by the results of point 5
- We divide the result of point 3 by the result of point 6. We get the cosine of the angle between the lines
- If this result allows us to accurately calculate the angle, we look for it
- 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, all its faces (including the base) are regular triangles. Since we are not given the length of the side, I can take it to be equal. I think you understand that the angle will not actually depend on how much our tetrahedron is “stretched”?. I will also draw the height and median in the tetrahedron. Along the way, I will draw its base (it will also be useful to us).
I need to find the angle between and. What do we know? We only know the coordinate of the point. This means that we need to find the coordinates of the points. Now we think: a point is the point of intersection of the altitudes (or bisectors or medians) of the triangle. And a point is a raised point. The point is the middle of the segment. Then we finally need to find: the coordinates of the points: .
Let's start with the simplest thing: the coordinates of a point. Look at the figure: It is clear that the applicate of a point is equal to zero (the point lies on the plane). Its ordinate is equal (since it is the median). It is more difficult to find its abscissa. However, this is easily done based on the Pythagorean theorem: Consider a triangle. Its hypotenuse is equal, and one of its legs is equal Then:
Finally we have: .
Now let's find the coordinates of the point. It is clear that its applicate is again equal to zero, and its ordinate is the same as that of the point, that is. Let's find its abscissa. This is done quite trivially if you remember that the heights of an equilateral triangle by the point of intersection are divided in proportion, counting from the top. Since: , then the required abscissa of the point, equal to the length of the segment, is equal to: . Thus, the coordinates of the point are:
Let's find the coordinates of the point. It is clear that its abscissa and ordinate coincide with the abscissa and ordinate of the point. And the applicate is equal to the length of the segment. - this is one of the legs of the triangle. The hypotenuse of a triangle is a segment - a leg. It is sought for reasons that I have highlighted in bold:
The point is the middle of the segment. Then we need to remember the formula for the coordinates of the midpoint of the segment:
That's it, now we can look for the coordinates of the direction vectors:
Well, everything is ready: we substitute all the data into the formula:
Thus,
Answer:
You should not be scared by such “scary” answers: for C2 tasks this is common practice. I would rather be surprised by the “beautiful” answer in this part. Also, as you noticed, I practically did not resort to anything other than the Pythagorean theorem and the property of altitudes of an equilateral triangle. That is, to solve the stereometric problem, I used the very minimum of stereometry. The gain in this is partially “extinguished” by rather cumbersome calculations. But they are quite algorithmic!
2. Let us depict a regular hexagonal pyramid along with the coordinate system, as well as its base:
We need to find the angle between the lines and. Thus, our task comes down to finding the coordinates of the points: . We will find the coordinates of the last three using a small drawing, and we will find the coordinate of the vertex through the coordinate of the point. There's a lot of work to do, but we need to get started!
a) Coordinate: it is clear that its applicate and ordinate are equal to zero. Let's find the abscissa. To do this, consider a right triangle. Alas, in it we only know the hypotenuse, which is equal. We will try to find the leg (for it is clear that double the length of the leg will give us the abscissa of the point). How can we look for it? Let's remember what kind of figure we have at the base of the pyramid? This is a regular hexagon. What does it mean? This means that all sides and all angles are equal. We need to find one such angle. Any ideas? There are a lot of ideas, but there is a formula:
The sum of the angles of a regular n-gon is .
Thus, the sum of the angles of a regular hexagon is equal to degrees. Then each of the angles is equal to:
Let's look at the picture again. It is clear that the segment is the bisector of the angle. Then the angle is equal to degrees. Then:
Then where from.
Thus, has coordinates
b) Now we can easily find the coordinate of the point: .
c) Find the coordinates of the point. Since its abscissa coincides with the length of the segment, it is equal. Finding the ordinate is also not very difficult: if we connect the dots and designate the point of intersection of the line as, say, . (do it yourself simple construction). Then Thus, the ordinate of point B is equal to the sum of the lengths of the segments. Let's look at the triangle again. Then
Then since Then the point has coordinates
d) Now let's find the coordinates of the point. Consider the rectangle and prove that Thus, the coordinates of the point are:
e) It remains to find the coordinates of the vertex. It is clear that its abscissa and ordinate coincide with the abscissa and ordinate of the point. Let's find the applica. Since, then. Consider a right triangle. According to the conditions of the problem, a side edge. This is the hypotenuse of my triangle. Then the height of the pyramid is a leg.
Then the point has coordinates:
Well, that's it, I have the coordinates of all the points that interest me. I am looking for the coordinates of the directing vectors of straight lines:
We are looking for the angle between these vectors:
Answer:
Again, in solving this problem I did not use any sophisticated techniques other than the formula for the sum of the angles of a regular n-gon, as well as the definition of the cosine and sine of a right triangle.
3. Since we are again not given the lengths of the edges in the pyramid, I will consider them equal to one. Thus, since ALL edges, and not just the side ones, are equal to each other, then at the base of the pyramid and me there is a square, and the side faces are regular triangles. Let us draw such a pyramid, as well as its base on a plane, noting all the data given in the text of the problem:
We are looking for the angle between and. I will make very brief calculations when I search for the coordinates of the points. You will need to “decipher” them:
b) - the middle of the segment. Its coordinates:
c) I will find the length of the segment using the Pythagorean theorem in a triangle. I can find it using the Pythagorean theorem in a triangle.
Coordinates:
d) - the middle of the segment. Its coordinates are
e) Vector coordinates
f) Vector coordinates
g) Looking for the angle:
A cube is the simplest figure. I'm sure you'll figure it out on your own. The answers to problems 4 and 5 are as follows:
Finding the angle between a straight line and a plane
Well, the time for simple puzzles is over! Now the examples will be even more complicated. To find the angle between a straight line and a plane, we will proceed as follows:
- Using three points we construct an equation of the plane
,
using a third order determinant. - Using two points, we look for the coordinates of the directing vector of the straight line:
- 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 have already had to deal with triangular and quadrangular pyramids, but not yet with prisms.
Solutions:
1. Let us depict a prism, as well as its base. Let's combine it with the coordinate system and note all the data that is given in the problem statement:
I apologize for some non-compliance with the proportions, but for solving the problem this is, in fact, not so important. The plane is simply the "back wall" of my prism. It is enough to simply guess that the equation of such a plane has the form:
However, this can be shown directly:
Let's choose arbitrary three points on this plane: for example, .
Let's create the equation of the plane:
Exercise for you: calculate this determinant yourself. Did you succeed? Then the equation of the plane looks like:
Or simply
Thus,
To solve the example, I need to find the coordinates of the direction vector of the straight line. Since the point coincides with the origin of coordinates, the coordinates of the vector will simply coincide with the coordinates of the point. To do this, 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 you can easily find the last coordinate from the picture from the point). Then we compose the equation of the plane:
We calculate:
We are looking for the coordinates of the guiding vector: It is clear that its coordinates coincide with the coordinates of the point, isn’t it? How to find coordinates? These are the coordinates of the point, raised along the applicate axis by one! . Then we look for the desired angle:
Answer:
3. Draw a regular hexagonal pyramid, and then draw a plane and a straight line in it.
Here it’s even problematic to draw a plane, not to mention solving this problem, but the coordinate method doesn’t care! Its versatility is its main advantage!
The plane passes through three points: . We are looking for their coordinates:
1) . Find out the coordinates for the last two points yourself. You will 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:
- Using three points we look for the equation of the first plane:
- Using the other three points we look for the equation of the second plane:
- We apply the formula:
As you can see, the formula is very similar to the two previous ones, with the help of which we looked for angles between straight lines and between a straight line and a plane. So it won’t be difficult for you to remember this one. Let's move on to the analysis of the tasks:
1. The side of the base of the right triangular prism is equal, and the dia-go-nal of the side face is equal. Find the angle between the plane and the plane of the axis of the prism.
2. In the right four-corner pi-ra-mi-de, all the edges of which are equal, find the sine of the angle between the plane and the plane bone, passing through the point per-pen-di-ku-lyar-but straight.
3. In a regular four-corner prism, the sides of the base are equal, and the side edges are equal. There is a point on the edge from-me-che-on so that. Find the angle between the planes and
4. In a right quadrangular prism, the sides of the base are equal, and the side edges are equal. There is a point on the edge from the point so that Find the angle between the planes and.
5. In a cube, find the co-si-nus of the angle between the planes and
Problem solutions:
1. I draw a regular (an equilateral triangle at the base) triangular prism and mark on it the planes that appear in the problem statement:
We need to find the equations of two planes: The equation of the base is trivial: you can compose the corresponding determinant using three points, but I will compose the equation right away:
Now let’s find the equation Point has coordinates Point - Since is the median and altitude of the triangle, it is easily found using the Pythagorean theorem in the triangle. Then the point has coordinates: Let's find the applicate of the point. To do this, consider a right triangle
Then we get the following coordinates: We compose the equation of the plane.
We calculate the angle between the planes:
Answer:
2. Making a drawing:
The most difficult thing is to understand what kind of mysterious plane this is, passing perpendicularly through the point. Well, the main thing is, what is it? The main thing is attentiveness! In fact, the line is perpendicular. The straight line is also perpendicular. Then the plane passing through these two lines will be perpendicular to the line, and, by the way, pass through the point. This plane also passes through the top of the pyramid. Then the desired plane - And the plane has already been given to us. We are looking for the coordinates of the points.
We find the coordinate of the point through the point. From the small picture it is easy to deduce that the coordinates of the point will be as follows: What now remains to be found to find the coordinates of the top of the pyramid? You also need to calculate its height. This is done using the same Pythagorean theorem: first prove that (trivially from small triangles forming a square at the base). Since by condition, we have:
Now everything is ready: vertex coordinates:
We compose the equation of the plane:
You are already an expert in calculating determinants. Without difficulty you will receive:
Or otherwise (if we multiply both sides by the root of two)
Now let's find the equation of the plane:
(You haven’t forgotten how we get the equation of a plane, right? If you don’t understand where this minus one came from, then go back to the definition of the equation of a plane! It just always turned out before that my plane belonged to the origin of coordinates!)
We calculate the determinant:
(You may notice that the equation of the plane coincides with the equation of the line passing through the points and! Think about why!)
Now let's calculate the angle:
We need to find the sine:
Answer:
3. Tricky question: what do you think a rectangular prism is? This is just a parallelepiped that you know well! Let's make a drawing right away! You don’t even have to depict the base separately; it’s of little use here:
The plane, as we noted earlier, is written in the form of an equation:
Now let's create a plane
We immediately create the equation of the plane:
Looking for an angle:
Now the answers to the last two problems:
Well, now is the time to take a little break, because you and I are great and have done a great job!
Coordinates and vectors. Advanced level
In this article we will discuss with you another class of problems that can be solved using the coordinate method: distance calculation problems. Namely, we will consider the following cases:
- 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, once we have all the necessary data, we apply the formula:
You should already know how we construct the equation of a plane from the previous problems that I discussed in the last part. Let's get straight to the tasks. The scheme is as follows: 1, 2 - I help you decide, and in some detail, 3, 4 - only the answer, you carry out the solution yourself and compare. Let's start!
Tasks:
1. Given a cube. The length of the edge of the cube is equal. Find the distance from the se-re-di-na from the cut to the plane
2. Given the right four-coal pi-ra-mi-yes, the side of the side is equal to the base. Find the distance from the point to the plane where - se-re-di-on the edges.
3. In the right triangular pi-ra-mi-de with the os-no-va-ni-em, the side edge is equal, and the hundred-ro-on the os-no-va- nia is equal. Find the distance from the top to the plane.
4. In a right hexagonal prism, all edges are equal. Find the distance from a point to a plane.
Solutions:
1. Draw a cube with single edges, construct a segment and a plane, denote the middle of the segment with a letter
.
First, let's start with the easy one: find the coordinates of the point. Since then (remember the coordinates of the middle of the segment!)
Now we compose the equation of the plane using three points
\[\left| (\begin(array)(*(20)(c))x&0&1\\y&1&0\\z&1&1\end(array)) \right| = 0\]
Now I can start finding the distance:
2. We start again with a drawing on which we mark all the data!
For a pyramid, it would be useful to draw its base separately.
Even the fact that I draw like a chicken with its paw will not prevent us from solving this problem with ease!
Now it's easy to find the coordinates of a point
Since the coordinates of the point, then
2. Since the coordinates of point a are the middle of the segment, then
Without any problems, we can find the coordinates of two more points on the plane. We create an equation for the plane and simplify it:
\[\left| (\left| (\begin(array)(*(20)(c))x&1&(\frac(3)(2))\\y&0&(\frac(3)(2))\\z&0&(\frac( (\sqrt 3 ))(2))\end(array)) \right|) \right| = 0\]
Since the point has coordinates: , we calculate the distance:
Answer (very rare!):
Well, did you figure it out? It seems to me that everything here is just as technical as in the examples that we looked at in the previous part. So I am sure that if you have mastered that material, then it will not be difficult for you to solve the remaining two problems. I'll just give you the answers:
Calculating the distance from a straight line to a plane
In fact, there is nothing new here. How can a straight line and a plane be positioned relative to each other? They have only one possibility: to intersect, or a straight line is parallel to the plane. What do you think is the distance from a straight line to the plane with which this straight line intersects? It seems to me that it is clear here that such a distance is equal to zero. Not an interesting case.
The second case is trickier: here the distance is already non-zero. However, since the line is parallel to the plane, then each point of the line is equidistant from this plane:
Thus:
This means that my task has been reduced to the previous one: we are looking for the coordinates of any point on a straight line, looking for the equation of the plane, and calculating the distance from the point to the plane. In fact, such tasks are extremely rare in the Unified State Examination. I managed to find only one problem, and the data in it were such that the coordinate method was not very applicable to it!
Now let's move on to another, much more important class of problems:
Calculating the distance of a point to a line
What do we need?
1. Coordinates of the point from which we are looking for the distance:
2. Coordinates of any point lying on a line
3. Coordinates of the directing vector of the straight line
What formula do we use?
What the denominator of this fraction means should be clear to you: this is the length of the directing vector of the straight line. This is a very tricky numerator! The expression means the modulus (length) of the vector product of vectors and How to calculate the vector product, we studied in the previous part of the work. Refresh your knowledge, we will need it very much now!
Thus, the algorithm for solving problems will be as follows:
1. We are looking for the coordinates of the point from which we are looking for the distance:
2. We are looking for the coordinates of any point on the line to which we are looking for the distance:
3. Construct a vector
4. Construct a directing vector of a straight line
5. Calculate the vector product
6. We look for the length of the resulting vector:
7. Calculate the distance:
We have a lot of work to do, and the examples will be quite complex! So now focus all your attention!
1. Given a right triangular pi-ra-mi-da with a top. The hundred-ro-on the basis of the pi-ra-mi-dy is equal, you are equal. Find the distance from the gray edge to the straight line, where the points and are the gray edges and from veterinary.
2. The lengths of the ribs and the straight-angle-no-go par-ral-le-le-pi-pe-da are equal accordingly and Find the distance from the top to the straight line
3. In a right hexagonal prism, all edges are equal, find the distance from a point to a straight line
Solutions:
1. We make a neat drawing on which we mark all the data:
We have a lot of work to do! First, I would like to describe in words what we will look for and in what order:
1. Coordinates of points and
2. Point coordinates
3. Coordinates of points and
4. Coordinates of vectors and
5. Their cross product
6. Vector length
7. Length of the vector product
8. Distance from to
Well, we have a lot of work ahead of us! Let's get to it with our sleeves rolled up!
1. To find the coordinates of the height of the pyramid, we need to know the coordinates of the point. Its applicate is zero, and its ordinate is equal to its abscissa is equal to the length of the segment. Since is the height of an equilateral triangle, it is divided in the ratio, counting from the vertex, from here. Finally, we got the coordinates:
Point coordinates
2. - middle of the segment
3. - middle of the segment
Midpoint of the segment
4.Coordinates
Vector coordinates
5. Calculate the vector product:
6. Vector length: the easiest way to replace is that the segment is the midline of the triangle, which means it is equal to half the base. So.
7. Calculate the length of the vector product:
8. Finally, we find the distance:
Ugh, that's it! I’ll tell you honestly: solving this problem using traditional methods (through construction) would be much faster. But here I reduced everything to a ready-made algorithm! I think the solution algorithm is clear to you? Therefore, I will ask you to solve the remaining two problems yourself. Let's compare the answers?
Again, I repeat: it is easier (faster) to solve these problems through constructions, rather than resorting to the coordinate method. I demonstrated this 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 of the mixed product (we introduced it in the previous part), and the denominator is, as in the previous formula (the modulus of the vector product of the direction vectors of the straight lines, the distance between which we are looking for).
I'll remind you that
Then the formula for the distance can be rewritten as:
This is a determinant divided by a determinant! Although, to be honest, I have no time for jokes here! This formula, 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 a right triangular prism, all the edges of which are equal, find the distance between the straight lines and.
2. Given a right triangular prism, all the edges of the base are equal to the section passing through the body rib and se-re-di-well ribs are a square. Find the distance between the straight lines and
I decide the first, and based on it, you decide the second!
1. I draw a prism and mark straight lines and
Coordinates of point C: then
Point coordinates
Vector coordinates
Point coordinates
Vector coordinates
Vector coordinates
\[\left((B,\overrightarrow (A(A_1)) \overrightarrow (B(C_1)) ) \right) = \left| (\begin(array)(*(20)(l))(\begin(array)(*(20)(c))0&1&0\end(array))\\(\begin(array)(*(20) (c))0&0&1\end(array))\\(\begin(array)(*(20)(c))(\frac((\sqrt 3 ))(2))&( - \frac(1) (2))&1\end(array))\end(array)) \right| = \frac((\sqrt 3 ))(2)\]
We calculate the vector product between vectors and
\[\overrightarrow (A(A_1)) \cdot \overrightarrow (B(C_1)) = \left| \begin(array)(l)\begin(array)(*(20)(c))(\overrightarrow i )&(\overrightarrow j )&(\overrightarrow k )\end(array)\\\begin(array )(*(20)(c))0&0&1\end(array)\\\begin(array)(*(20)(c))(\frac((\sqrt 3 ))(2))&( - \ frac(1)(2))&1\end(array)\end(array) \right| - \frac((\sqrt 3 ))(2)\overrightarrow k + \frac(1)(2)\overrightarrow i \]
Now we calculate its length:
Answer:
Now try to carefully complete the second task. The answer to it will be: .
Coordinates and vectors. Brief description and basic formulas
A vector is a directed segment. - the beginning of the vector, - the end of the vector.
A vector is denoted by or.
Absolute value vector - the length of the segment representing the vector. Denoted as.
Vector coordinates:
,
where are the ends of the vector \displaystyle a .
Sum of vectors: .
Product of vectors:
Dot product of vectors:
The scalar product of vectors is equal to the product of their absolute values and the cosine of the angle between them:
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