MUTUALLY PERPENDICULAR
mutually perpendicular
Lopatin. Dictionary of the Russian language Lopatin. 2012
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mutually ... - MUTUALLY PERPENDICULAR in the Spelling Dictionary:
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, th, th; -ren, -rna. Perpendicular. Perpendicular lines. Position perpendicular to (adv.) To something. II noun perpendicularity, -and, ... - MUTUALLY
RELATIVELY PRIME NUMBERS, natural numbers that have no common divisors other than 1; eg 15 and ... - MUTUALLY in the Big Russian Encyclopedic Dictionary:
Mutually univocal correspondence, such a correspondence between the elements of two sets, for which each element of the first set corresponds to one definition. element of the second ... - PERPENDICULAR in the Complete Accentuated Paradigm by Zaliznyak:
perpendicular, perpendicular, perpendicular, perpendicular, perpendicular, perpendicular, perpendicular, perpendicular, perpendicular, perpendicular, perpendicular, perpendicular, perpendicular, perpendicular perpendicular, perpendicular, perpendicular, perpendicular, perpendicular, perpendicular, perpendicular, perpendicular, ... - PERPENDICULAR in the Thesaurus of Russian Business Vocabulary:
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which is perpendicular; plumb, forming right angles with a given line or ... - PERPENDICULAR in the Dictionary of Foreign Expressions:
which is perpendicular; plumb, forming right angles with a given line or ... - PERPENDICULAR in the Thesaurus of the Russian language:
Syn: transverse, intersecting, intersecting, orthogonal Ant: ... - PERPENDICULAR
Syn: transverse, intersecting, intersecting, orthogonal Ant: ... - MUTUALLY in the dictionary of Russian Synonyms:
mutually, bilaterally, bilaterally, borrowed, ... - PERPENDICULAR
adj. 1) Corresponding by value. with noun: the perpendicular associated with it. 2) a) Inherent to the perpendicular, characteristic of it. b) Located ... - MUTUALLY in the New Explanatory Dictionary of the Russian Language by Efremova:
adv. Corresponds by value with adj .:… - PERPENDICULAR in the Dictionary of the Russian language Lopatin:
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perpendicular; cr. f. -ren, ... - PERPENDICULAR in the Ozhegov Russian Language Dictionary:
being the perpendicular Perpendicular lines. Position perpendicular (adv.) To ... - PERPENDICULAR in the Explanatory Dictionary of the Russian Language by Ushakov:
perpendicular, perpendicular; perpendicular, perpendicular, perpendicular to what (mat.). Adj. to the perpendicular; which is perpendicular. Perpendicular line. One track is perpendicular to the other. One ... - PERPENDICULAR
perpendicular adj. 1) Corresponding by value. with noun: the perpendicular associated with it. 2) a) Inherent to the perpendicular, characteristic of it. b) ... - MUTUALLY in the Explanatory Dictionary of Efremova:
mutually adverb Corresponds by value with adj .:… - PERPENDICULAR
adj. 1.rel. with noun the perpendicular associated with it 2. Inherent to the perpendicular, characteristic of it. Ott. Perched at right angles ... - MUTUALLY in the New Dictionary of the Russian Language by Efremova:
- PERPENDICULAR
adj. 1.rel. with noun the perpendicular associated with it 2. Inherent to the perpendicular, characteristic of it. Ott. Located under the direct ... - MUTUALLY in the Big Modern Explanatory Dictionary of the Russian Language:
adv. qualities - are. Manifesting in relation to each other; ... - MUTUAL ADDITIONAL COLORS
two colors are called mutually complementary if their simultaneous action on the eye gives the impression of white; in other words, optical mixing of such two ... - MUTUAL ADDITIONAL COLORS in the Encyclopedia of Brockhaus and Efron:
? two colors are called mutually complementary if their simultaneous action on the eye gives the impression of white; in other words, optical mixing of such ... - WESTFALA BASSET in the Encyclopedia of Dogs:
_Hunting dogs_ Origin The dog is the result of crossing German hounds and Basset hounds. Description Height from 30 to 35 cm. Weight from ... - VECTOR PRODUCT OF VECTOR in the Big Encyclopedic Dictionary:
a by vector b vector p \u003d, or a b, equal in length to the area of \u200b\u200bthe parallelogram built on vectors a ... - NUMBER THEORY
theory, the science of whole numbers. The concept of an integer, as well as arithmetic operations on numbers, has been known since ancient times and ... - TOPOLOGY in the Great Soviet Encyclopedia, TSB:
(from the Greek topos - place and - logia) - a part of geometry dedicated to the study of the phenomenon of continuity (expressed, for example, in the concept ... - TILLER in the Great Soviet Encyclopedia, TSB:
(from the Dutch. roerpen, from roer - paddle, steering wheel and pen - pin), a lever mounted on the upper part of the steering axis; serves ... - TRANSFORMATION in the Great Soviet Encyclopedia, TSB:
one of the basic concepts of mathematics arising in the study of correspondences between classes of geometric objects, classes of functions, etc. For example, with geometric ... - LIGHT POLARIZATION in the Great Soviet Encyclopedia, TSB:
light, one of the fundamental properties of optical radiation (light), consisting in the inequality of different directions in the plane perpendicular to the light beam (direction ... - DISPLAY in the Great Soviet Encyclopedia, TSB:
(mat.) set A into set B, the correspondence, by virtue of which each element x of set A corresponds to a certain element y ... - GOTHIC in the Great Soviet Encyclopedia, TSB:
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(Greek geometria, from ge - Earth and metreo - Meru), a branch of mathematics that studies spatial relationships and forms, and others ... - GAUSSIAN PRINCIPLE in the Great Soviet Encyclopedia, TSB:
principle, the principle of least compulsion, one of the variational principles of mechanics, according to which for a mechanical system with ideal constraints (see ... - UNITED KINGDOM (STATE) in the Great Soviet Encyclopedia, TSB.
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E. is called what is contained in the body, which gives this body special properties, causes it to act mechanically on some ... - BODY COLOR
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(Matthew Flinders) - eng. traveler (1774-1814). Accompanying physician Bass in 1795 on his trip to the southeast. the coast of Australia; in … - GRAVITY in the Encyclopedic Dictionary of Brockhaus and Euphron:
Newton's law of universal T. can be formulated as follows: each atom interacts with each other atom, while the force of interaction ... - METAL PIPES in the Encyclopedic Dictionary of Brockhaus and Euphron.
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A substance or medium is called "transparent" in the usual sense of the word if it is possible to see objects through this substance or medium; in that … - AXLE NEUTRAL in the Encyclopedic Dictionary of Brockhaus and Euphron:
When the beam is bent, its fibers between two cross-sections are partly lengthened, partly shortened. In the case of a simple bend, according to the accepted theory ... - SHEET LOCATION in the Encyclopedic Dictionary of Brockhaus and Euphron:
(phyllotaxia). - In most plants, the leaves are located on the stems and branches so correctly that it is possible to establish common regarding their location ...
In this article, we will take a closer look at the plane and in three-dimensional space. We begin by defining perpendicular lines, show the notation and give examples. After that, we present a necessary and sufficient condition for the perpendicularity of two straight lines and analyze in detail the solutions of typical problems.
Page navigation.
Perpendicular lines - basic information.
Example.
Three points are given in the Oxy rectangular coordinate system. Are lines AB and AC perpendicular?
Decision.
Vectors and are direction vectors of straight lines AB and AC. Referring to the article, we calculate 
... The vectors and are perpendicular, since 
... Thus, the necessary and sufficient condition for the perpendicularity of lines AB and AC is satisfied. Consequently, straight lines AB and AC are perpendicular.
Answer:
Yes, straight lines are perpendicular.
Example.
Are straight and 
perpendicular?
Decision.
The directing vector of the straight line, a is the directing vector of the straight line ... Let's calculate the dot product of vectors and: 
... It is nonzero, therefore, the direction vectors of the lines are not perpendicular. That is, the condition of perpendicularity of straight lines is not met, therefore, the original straight lines are not perpendicular.
Answer:
No, straight lines are not perpendicular.
Similarly, a necessary and sufficient condition for the perpendicularity of straight lines a and b in the rectangular coordinate system Oxyz in three-dimensional space has the form 
where 
and 
- direction vectors of straight lines a and b, respectively.
Example.
Are the lines specified in the rectangular coordinate system Oxyz in three-dimensional space perpendicular by the equations 
and?
Decision.
The numbers in the denominators of the canonical equations of the straight line in space are the corresponding coordinates of the directing vector of the straight line. And the coordinates of the directing vector of the straight line, which is given by the parametric equations of the straight line in space, are the coefficients at the parameter. Thus, 
and - direction vectors of given lines. Let's find out if they are perpendicular: 
... Since the dot product is zero, these vectors are perpendicular. This means that the condition of perpendicularity of the given lines is satisfied.
Answer:
Straight lines are perpendicular.
To check the perpendicularity of two straight lines in a plane, there are other necessary and sufficient perpendicularity conditions.
Theorem.
For the perpendicularity of the straight lines a and b on the plane, it is necessary and sufficient that the normal vector of the straight line a be perpendicular to the normal vector of the straight line b.
The voiced condition of perpendicularity of straight lines is convenient to use if the coordinates of normal vectors of straight lines can be easily found from the given equations of straight lines. This statement corresponds to the general equation of the line of the form 
, the equation of a straight line in segments and the equation of a straight line with a slope.
Example.
Make sure straight 
and are perpendicular.
Decision.
Given the equations of the straight lines, it is easy to find the coordinates of the normal vectors of these straight lines. Is the normal vector of the line ... We rewrite the equation as 
, from where the coordinates of the normal vector of this line are visible:.
The vectors and are perpendicular, since their dot product is zero: 
... Thus, the necessary and sufficient condition for the perpendicularity of the given lines is fulfilled, that is, they are really perpendicular.
In particular, if the straight line a on the plane is determined by the equation of a straight line with the slope of the form, and the straight line b is of the form, then the normal vectors of these straight lines have coordinates and, respectively, and the condition of perpendicularity of these straight lines is reduced to the following relation between the slopes.
Perpendicular is a frequently used word, the meaning of which is not well understood by many. This mini-article will tell you about the essence of the perpendicular.
What is a perpendicular
In simple terms, a perpendicular is a straight line that makes an angle of 90 with another line. The perpendicular concept is often used in geometry. You can often hear a sentence similar to this: “A perpendicular drawn to the base of a triangle divides a large triangle into two small ones. Find ... ”, etc. For example, consider a right-angled triangle with two legs (a and b) and a hypotenuse c.
In such a triangle:
- Leg a is perpendicular to leg b, since the angle between them is 90.
- Leg b is perpendicular to leg a, since the angle between them is 90.
In addition to geometry, this word can be used in different life situations. For example, if one road crosses another so that the angle is 90, they can be said to be perpendicular to each other.
From the above examples, a general rule can be deduced: If two planes intersect 90, they are perpendicular to each other.
The article deals with the question of perpendicular lines on a plane and three-dimensional space. Let us analyze in detail the definition of perpendicular lines and their designations with the given examples. Consider the conditions for applying the necessary and sufficient condition for the perpendicularity of two straight lines and consider in detail with an example.
The angle between intersecting straight lines in space can be right. Then they say that the data are perpendicular straight lines. When the angle between crossing straight lines is straight, then the straight lines are also perpendicular. It follows from this that perpendicular straight lines in the plane are intersecting, and perpendicular straight lines of space can be intersecting and crossing.
That is, the concepts “straight lines a and b are perpendicular” and “straight lines b and a are perpendicular” are considered equal. Hence the concept of mutually perpendicular straight lines came from. Summarizing the above, consider the definition.
Definition 1
Two straight lines are called perpendicular if the angle at their intersection is 90 degrees.
Perpendicularity is denoted by "⊥", and the notation takes the form a ⊥ b, which means that line a is perpendicular to line b.
For example, the perpendicular lines on the plane can be the sides of a square with a common vertex. In three-dimensional space, the lines O x, O z, O y are perpendicular in pairs: O x and O z, O x and O y, O y and O z.
Perpendicularity of lines - perpendicularity conditions
It is necessary to know the properties of perpendicularity, since most tasks boil down to checking it for subsequent solution. There are cases when perpendicularity is discussed even in the condition of the assignment or when it is necessary to use evidence. In order to prove the perpendicularity, it is enough that the angle between the straight lines is right.
In order to determine their perpendicularity for the known equations of a rectangular coordinate system, it is necessary to apply the necessary and sufficient condition for the perpendicularity of straight lines. Consider the wording.
Theorem 1
For straight lines a and b to be perpendicular, it is necessary and sufficient that the direction vector of the straight line be perpendicular to the direction vector of the given straight line b.
The proof itself is based on the definition of the direction vector of the line and on the definition of the perpendicularity of the lines.
Proof 1
Let a rectangular Cartesian coordinate system O x y be introduced with the given equations of a straight line on a plane, which define straight lines a and b. The direction vectors of lines a and b will be denoted by a → and b →. From the equation of lines a and b, a necessary and sufficient condition is the perpendicularity of the vectors a → and b →. This is possible only when the scalar product of vectors a → \u003d (a x, a y) and b → \u003d (b x, b y) is equal to zero, and the notation is a →, b → \u003d a x b x + a y b y \u003d 0. We obtain that a →, b → \u003d ax bx + ay by \u003d 0, where a → \u003d (ax, ay), is a necessary and sufficient condition for the perpendicularity of the straight lines a and b located in the rectangular coordinate system O x y on the plane and b → \u003d bx, by are direction vectors of lines a and b.
The condition is applicable when it is necessary to find the coordinates of the direction vectors or in the presence of canonical or parametric equations of straight lines on the plane of given straight lines a and b.
Example 1
Three points A (8, 6), B (6, 3), C (2, 10) are given in a rectangular coordinate system O x y. Determine if lines A B and A C are perpendicular or not.
Decision
Lines А В and А С have direction vectors A B → and A C → respectively. First, let's calculate A B → \u003d (- 2, - 3), A C → \u003d (- 6, 4). We obtain that the vectors A B → and A C → are perpendicular from the property on the scalar product of vectors equal to zero.
A B →, A C → \u003d (- 2) (- 6) + (- 3) 4 \u003d 0
It is obvious that the necessary and sufficient condition is satisfied, which means that AB and AC are perpendicular.
Answer:straight lines are perpendicular.
Example 2
Determine if the given lines x - 1 2 \u003d y - 7 3 and x \u003d 1 + λ y \u003d 2 - 2 · λ are perpendicular or not.
Decision
a → \u003d (2, 3) is the direction vector of the given line x - 1 2 \u003d y - 7 3,
b → \u003d (1, - 2) is the direction vector of the line x \u003d 1 + λ y \u003d 2 - 2 λ.
Let's proceed to calculating the scalar product of vectors a → and b →. The expression will be written:
a →, b → \u003d 2 1 + 3 - 2 \u003d 2 - 6 ≠ 0
The result of the product is not zero, we can conclude that the vectors are not perpendicular, which means that the straight lines are also not perpendicular.
Answer:straight lines are not perpendicular.
The necessary and sufficient condition of perpendicularity of lines a and b is applied for three-dimensional space, is written as a →, b → \u003d ax bx + ay by + az bz \u003d 0, where a → \u003d (ax, ay, az) and b → \u003d (bx, by, bz) are direction vectors of lines a and b.
Example 3
Check the perpendicularity of straight lines in a rectangular coordinate system of three-dimensional space, given by the equations x 2 \u003d y - 1 \u003d z + 1 0 and x \u003d λ y \u003d 1 + 2 λ z \u003d 4 λ
Decision
The denominators from the canonical equations of straight lines are considered the coordinates of the directing vector of the straight line. The coordinates of the direction vector from the parametric equation are coefficients. It follows that a → \u003d (2, - 1, 0) and b → \u003d (1, 2, 4) are the direction vectors of the given lines. To identify their perpendicularity, we find the scalar product of vectors.
The expression becomes a →, b → \u003d 2 1 + (- 1) 2 + 0 4 \u003d 0.
The vectors are perpendicular because the product is zero. The necessary and sufficient condition is fulfilled, so the lines are also perpendicular.
Answer:straight lines are perpendicular.
The squareness check may be performed based on other necessary and sufficient squareness conditions.
Theorem 2
Lines a and b on the plane are considered perpendicular if the normal vector of the line a is perpendicular to the vector b, this is a necessary and sufficient condition.
Proof 2
This condition is applicable when the equations of the straight lines give a quick determination of the coordinates of the normal vectors of the given straight lines. That is, in the presence of a general equation of a straight line of the form A x + B y + C \u003d 0, equations of a straight line in segments of the form x a + y b \u003d 1, equations of a straight line with a slope of the form y \u003d k x + b coordinates of vectors can be found.
Example 4
Find out if the lines 3 x - y + 2 \u003d 0 and x 3 2 + y 1 2 \u003d 1 are perpendicular.
Decision
Based on their equations, it is necessary to find the coordinates of the normal vectors of straight lines. We obtain that n α → \u003d (3, - 1) is the normal vector for the line 3 x - y + 2 \u003d 0.
Simplify the equation x 3 2 + y 1 2 \u003d 1 to the form 2 3 x + 2 y - 1 \u003d 0. Now the coordinates of the normal vector are clearly visible, which we write in this form n b → \u003d 2 3, 2.
The vectors n a → \u003d (3, - 1) and n b → \u003d 2 3, 2 will be perpendicular, since their dot product will end up with a value of 0. We get n a →, n b → \u003d 3 2 3 + (- 1) 2 \u003d 0.
The necessary and sufficient condition was fulfilled.
Answer:straight lines are perpendicular.
When the straight line a in the plane is defined using the equation with the slope y \u003d k 1 x + b 1, and the straight line b - y \u003d k 2 x + b 2, it follows that normal vectors will have coordinates (k 1, - 1) and (k 2, - 1). The perpendicularity condition itself is reduced to k 1 k 2 + (- 1) (- 1) \u003d 0 ⇔ k 1 k 2 \u003d - 1.
Example 5
Find out if the lines y \u003d - 3 7 x and y \u003d 7 3 x - 1 2 are perpendicular.
Decision
The line y \u003d - 3 7 x has a slope equal to - 3 7, and the line y \u003d 7 3 x - 1 2 - 7 3.
The product of the slopes gives the value - 1, - 3 7 · 7 3 \u003d - 1, that is, the straight lines are perpendicular.
Answer:given straight lines are perpendicular.
There is one more condition used to determine the perpendicularity of straight lines in a plane.
Theorem 3
For the perpendicularity of straight lines a and b on the plane, a necessary and sufficient condition is the collinearity of the direction vector of one of the straight lines with the normal vector of the second straight line.
Proof 3
The condition is applicable when it is possible to find the direction vector of one straight line and the coordinates of the normal vector of the other. In other words, one straight line is given by a canonical or parametric equation, and the other by a general equation of a straight line, an equation in segments, or an equation of a straight line with a slope.
Example 6
Determine if the given lines x - y - 1 \u003d 0 and x 0 \u003d y - 4 2 are perpendicular.
Decision
We get that the normal vector of the line x - y - 1 \u003d 0 has coordinates n a → \u003d (1, - 1), and b → \u003d (0, 2) is the direction vector of the line x 0 \u003d y - 4 2.
This shows that the vectors n a → \u003d (1, - 1) and b → \u003d (0, 2) are not collinear, because the collinearity condition is not satisfied. There is no such number t that the equality n a → \u003d t · b → holds. Hence the conclusion that straight lines are not perpendicular.
Answer:straight lines are not perpendicular.
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Perpendicularity is the relationship between various objects in Euclidean space - straight lines, planes, vectors, subspaces, and so on. In this article, we will take a closer look at the perpendicular straight lines and characteristic features related to them. Two straight lines can be called perpendicular (or mutually perpendicular) if all four angles formed by their intersection are strictly ninety degrees.
There are certain properties of perpendicular straight lines realized on a plane:
Drawing perpendicular lines
Perpendicular lines are drawn on a plane using a square. Any draftsman should keep in mind that an important feature of each square is that it necessarily has a right angle. To create two perpendicular lines, we need to combine one of the two sides of the right angle of our
drawing square with a given line and draw a second line along the second side of this right angle. This will create two perpendicular lines.
Three-dimensional space
An interesting fact is that perpendicular lines can be realized, and in this case, two lines will be called such if they are parallel, respectively, to some other two lines lying in the same plane and also perpendicular in it. In addition, if on a plane only two straight lines can be perpendicular, then in three-dimensional space there are already three. Moreover, the number of perpendicular lines (or planes) can be further increased.