1 perpendicular lines. Perpendicularity of lines - conditions of perpendicularity. Collection and use of personal information
The article deals with the issue of perpendicular lines in the plane and three-dimensional space. We will analyze in detail the definition of perpendicular lines and their designations with the examples given. Consider the conditions for applying the necessary and sufficient condition for the perpendicularity of two lines and consider in detail with an example.
The angle between intersecting lines in space can be right. Then the given lines are said to be perpendicular. When the angle between skew lines is a straight line, then the lines are also perpendicular. Hence it follows that perpendicular lines on the plane intersect, and perpendicular lines in spaces can be intersecting and skew.
That is, the concepts "lines a and b are perpendicular" and "lines b and a are perpendicular" are considered equal. This is where the concept of mutually perpendicular lines comes from. Summarizing the above, consider the definition.
Definition 1
Two lines are called perpendicular if the angle at their intersection is 90 degrees.
Perpendicularity is denoted by "⊥", and the notation becomes a ⊥ b, which means that line a is perpendicular to line b.
For example, the perpendicular lines in 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
You need to know the properties of perpendicularity, since most problems come down to checking it for subsequent solution. There are cases when about perpendicularity in question still in the condition of the assignment or when it is necessary to use proof. In order to prove perpendicularity, it is sufficient that the angle between the lines be right.
In order to determine their perpendicularity with the known equations of a rectangular coordinate system, it is necessary to apply the necessary and sufficient condition for the perpendicularity of lines. Let's look at the wording.
Theorem 1
In order for the lines a and b to be perpendicular, it is necessary and sufficient that the direction vector of the line has perpendicularity with respect to the direction vector of the given line b.
The proof itself is based on the definition of the directing 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 given equations of a straight line on a plane that define the lines a and b. We denote the direction vectors of lines a and b as 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 the vectors a → = (a x , a y) and b → = (b x , b y) is equal to zero, and the notation is a → , b → = a x b x + a y b y = 0 . We obtain that the necessary and sufficient condition for the perpendicularity of the lines a and blocated in a rectangular coordinate system O x y on the plane is a → , b → = a x b x + a y b y = 0 , where a → = (a x , a y) and b → = b x , b y are the direction vectors of the 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 lines on the plane of given 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 whether lines A B and A C are perpendicular or not.
Decision
The lines A B and A C have direction vectors A B → and A C → respectively. First, let's calculate A B → = (- 2 , - 3) , A C → = (- 6 , 4) . We obtain that the vectors A B → and A C → are perpendicular from the property of the scalar product of vectors equal to zero.
A B → , A C → = (- 2) (- 6) + (- 3) 4 = 0
It is obvious that the necessary and sufficient condition is satisfied, which means that A B and A C are perpendicular.
Answer: lines are perpendicular.
Example 2
Determine whether the given lines x - 1 2 = y - 7 3 and x = 1 + λ y = 2 - 2 · λ are perpendicular or not.
Decision
a → = (2 , 3) is the direction vector of the given line x - 1 2 = y - 7 3 ,
b → = (1 , - 2) is the direction vector of the line x = 1 + λ y = 2 - 2 · λ .
Let's proceed to the calculation of the scalar product of the vectors a → and b → . The expression will be written:
a → , b → = 2 1 + 3 - 2 = 2 - 6 ≠ 0
The result of the product is not equal to zero, we can conclude that the vectors are not perpendicular, which means that the lines are also not perpendicular.
Answer: lines are not perpendicular.
The necessary and sufficient condition for perpendicularity of lines a and b is applied for three-dimensional space, written as a → , b → = a x b x + a y b y + a z b z = 0 , where a → = (a x , a y , a z) and b → = (b x , b y , b z) are the direction vectors of lines a and b .
Example 3
Check the perpendicularity of 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 = 4 λ
Decision
The denominators from the canonical equations of the straight lines are considered to be the coordinates of the directing vector of the straight line. The direction vector coordinates from the parametric equation are the coefficients. It follows that a → = (2 , - 1 , 0) and b → = (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 → = 2 1 + (- 1) 2 + 0 4 = 0 .
The vectors are perpendicular because the product is zero. The necessary and sufficient condition is satisfied, which means that the lines are also perpendicular.
Answer: lines are perpendicular.
The check of perpendicularity can be carried out on the basis of other necessary and sufficient conditions for perpendicularity.
Theorem 2
Lines a and b on the plane are considered perpendicular when the normal vector of the line a is perpendicular to the vector b, this is the necessary and sufficient condition.
Proof 2
This condition is applicable when the equations of lines give a quick finding of the coordinates of the normal vectors of the given lines. That is, if there is a general equation of a straight line of the form A x + B y + C \u003d 0, an equation of a straight line in segments of the form x a + y b \u003d 1, an equation of a straight line with a slope of the form y \u003d k x + b, the coordinates of the vectors can be found.
Example 4
Find out if the lines 3 x - y + 2 = 0 and x 3 2 + y 1 2 = 1 are perpendicular.
Decision
Based on their equations, it is necessary to find the coordinates of the normal vectors of the straight lines. We get that n α → = (3 , - 1) is a normal vector for the line 3 x - y + 2 = 0 .
Let's simplify the equation x 3 2 + y 1 2 = 1 to the form 2 3 x + 2 y - 1 = 0 . Now the coordinates of the normal vector are clearly visible, which we write in this form n b → = 2 3 , 2 .
The vectors n a → = (3 , - 1) and n b → = 2 3 , 2 will be perpendicular, since their scalar product will eventually give a value equal to 0 . We get n a → , n b → = 3 2 3 + (- 1) 2 = 0 .
The necessary and sufficient condition was fulfilled.
Answer: lines are perpendicular.
When the line a on the plane is defined using the slope equation y = k 1 x + b 1 , and the line b - y = k 2 x + b 2 , it follows that the normal vectors will have coordinates (k 1 , - 1) and (k 2 , - 1) . The perpendicularity condition itself reduces to k 1 · k 2 + (- 1) · (- 1) = 0 ⇔ k 1 · k 2 = - 1 .
Example 5
Find out if the lines y = - 3 7 x and y = 7 3 x - 1 2 are perpendicular.
Decision
The straight line y = - 3 7 x has a slope equal to - 3 7 , and the straight line y = 7 3 x - 1 2 - 7 3 .
The product of the slope coefficients gives the value - 1, - 3 7 · 7 3 = - 1, that is, the lines are perpendicular.
Answer: given lines are perpendicular.
There is another condition used to determine the perpendicularity of lines in the plane.
Theorem 3
For the lines a and b to be perpendicular in the plane, a necessary and sufficient condition is the collinearity of the direction vector of one of the lines with the normal vector of the second line.
Proof 3
The condition is applicable when it is possible to find the direction vector of one 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 general equation 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 = 0 and x 0 = y - 4 2 are perpendicular.
Decision
We get that the normal vector of the line x - y - 1 = 0 has coordinates n a → = (1 , - 1) , and b → = (0 , 2) is the direction vector of the line x 0 = y - 4 2 .
This shows that the vectors n a → = (1, - 1) and b → = (0, 2) are not collinear, because the collinearity condition is not satisfied. There is no such number t that the equality n a → = t · b → holds true. Hence the conclusion that the lines are not perpendicular.
Answer: lines are not perpendicular.
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Perpendicularity is the relationship between various objects in Euclidean space - lines, planes, vectors, subspaces, and so on. In this material, we will take a closer look at perpendicular lines and the characteristic features related to them. Two lines can be called perpendicular (or mutually perpendicular) if all four angles formed by their intersection are exactly ninety degrees.
There are certain properties of perpendicular lines realized on a plane:
Construction of perpendicular lines
Perpendicular lines are built on a plane using a square. Any draftsman should keep in mind that an important feature of every square is that it necessarily has a right angle. To create two perpendicular lines, we need to match one of the two sides right angle our
drawing square with a given straight line and draw a second straight 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 any two other lines lying in the same plane and also perpendicular in it. In addition, if only two straight lines can be perpendicular on a plane, then in three-dimensional space there are already three. Moreover, the number of perpendicular lines (or planes) can be further increased.
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A straight line (line segment) is indicated by two capital letters of the Latin alphabet or one small letter. The point is indicated only by a capital Latin letter.
The lines may not intersect, intersect, or coincide. Intersecting lines have only one common point, non-intersecting lines have no common point, and coinciding lines have all points in common.
Definition. Two lines that intersect at right angles are called perpendicular. The perpendicularity of straight lines (or their segments) is denoted by the sign of perpendicularity "⊥".
For example:
Your AB and CD(Fig. 1) intersect at the point O and ∠ AOC = ∠WOS = ∠AOD = ∠BOD= 90°, then AB ⊥ CD.
If a AB ⊥ CD(Fig. 2) and intersect at the point AT, then ∠ ABC = ∠ABD= 90°
Properties of perpendicular lines
1. Through a dot BUT(Fig. 3) only one perpendicular line can be drawn AB to a straight line CD; other lines passing through the point BUT and crossing CD, are called oblique lines (Fig. 3, straight lines AE and AF).
2. From a point A you can drop a perpendicular to a straight line CD; length of the perpendicular (length of the segment AB) drawn from the point BUT directly CD, is the shortest distance from A before CD(Fig. 3).
Two lines are called perpendicular if they intersect at a right angle.
Line a intersects line b at a right angle at point A. You can hover using the perpendicular icon: a ⊥ b. It reads like this: line a is perpendicular to line b.
It should be noted that an adjacent angle and a vertical angle with a right angle are also right angles.
Through each point of a line, one can draw a line perpendicular to it, and only one.
Proof.
Let b be a given line and point A belongs to this line. Let's take some ray b1 on the line b with the starting point at A. Let's set aside the angle (a1b1) equal to 90° from the ray b1. By definition, the line containing ray a1 will be perpendicular to line b.
Suppose there exists another line perpendicular to the line b and passing through the point A. Take on this line a ray c1 emanating from the point A and lying in the same half-plane as the ray a1. Then ∠ (a1b1) = ∠ (c1b1) = 90 º. But according to axiom 8, only one angle equal to 90 º can be set aside in this half-plane. Therefore, it is impossible to draw another line perpendicular to line b through point A into the given half-plane. The theorem has been proven. 
A perpendicular to a given line is a segment of a line perpendicular to a given line that has one of their ends at the point of intersection. This end of the segment is called the base of the perpendicular. AB is the perpendicular to line a. Point A is the base of the perpendicular.