EntrancePerpendicular lines. Complete lessons – Knowledge Hypermarket
Preliminary information about direct
The concept of a straight line, as well as the concept of a point, are the basic concepts of geometry. As you know, the basic concepts are not defined. This is no exception to the concept of a straight line. Therefore, let us consider the essence of this concept through its construction.
Take a ruler and, without lifting your pencil, draw a line of arbitrary length. We will call the resulting line a straight line. However, it should be noted here that this is not the entire straight line, but only part of it. The straight line itself is infinite at both ends.
We will denote straight lines by small Latin letter, or its two dots in parentheses (Fig. 1).
The concepts of a straight line and a point are connected by three axioms of geometry:
Axiom 1: For every arbitrary line there are at least two points that lie on it.
Axiom 2: You can find at least three points that do not lie on the same line.
Axiom 3: In 2 arbitrary points there is always a straight line, and this straight line is unique.
For two straight lines, their relative position is relevant. Three cases are possible:
- Two straight lines coincide. In this case, each point of one line will also be a point of the other line.
- Two lines intersect. In this case, only one point from one line will also belong to the other line.
- Two lines are parallel. In this case, each of these lines has its own set of points that are different from each other.
Perpendicularity of lines
Consider two arbitrary intersecting lines. Obviously, at the point of their intersection, 4 angles are formed. Then
Definition 1
We will call intersecting lines perpendicular if at least one angle formed by their intersection is equal to $90^0$ (Fig. 2).
Designation: $a⊥b$.
Consider the following problem:
Example 1
Find angles 1, 2 and 3 from the figure below
Angle 2 is vertical for the angle given to us, therefore
Angle 1 is adjacent to angle 2, therefore
$∠1=180^0-∠2=180^0-90^0=90^0$
Angle 3 is vertical to angle 1, therefore
$∠3=∠1=90^0$
From this problem we can make the following remark
Note 1
All angles between perpendicular lines are equal to $90^0$.
Fundamental theorem of perpendicular lines
Let us introduce the following theorem:
Theorem 1
Two lines that are perpendicular to the third will be disjoint.
Proof.
Let's look at Figure 3 according to the problem conditions.
Let us mentally divide this figure into two parts of the straight line $(ZP)$. Let's put the right side on the left. Then, since the lines $(NM)$ and $(XY)$ are perpendicular to the line $(PZ)$ and, therefore, the angles between them are right, the ray $NP$ will be superimposed entirely on the ray $PM$, and the ray $XZ $ will be superimposed entirely on the ray $YZ$.
Now, suppose the opposite: let these lines intersect. Without loss of generality, let us assume that they intersect on the left side, that is, let the ray $NP$ intersect with the ray $YZ$ at point $O$. Then, according to the construction described above, we will obtain that the ray $PM$ intersects with the ray $YZ$ at the point $O"$. But then we obtain that through two points $O$ and $O"$, there are two straight lines $(NM)$ and $(XY)$, which contradicts the axiom of 3 straight lines.
Therefore, the lines $(NM)$ and $(XY)$ do not intersect.
The theorem has been proven.
Sample task
Example 2
Given two lines that have an intersection point. Through a point that does not belong to any of them, two straight lines are drawn, one of which is perpendicular to one of the lines described above, and the other is perpendicular to the other of them. Prove that they are not the same.
Let's draw a picture according to the conditions of the problem (Fig. 4).
From the conditions of the problem we will have that $m⊥k,n⊥l$.
Let us assume the opposite, let the lines $k$ and $l$ coincide. Let it be straight $l$. Then, by condition, $m⊥l$ and $n⊥l$. Therefore, by Theorem 1, the lines $m$ and $n$ do not intersect. We have obtained a contradiction, which means that the lines $k$ and $l$ do not coincide.
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The article discusses the issue of perpendicular lines on the plane and three-dimensional space. We will analyze in detail the definition of perpendicular lines and their designations with the given examples. Let us consider the conditions for applying the necessary and sufficient condition for the perpendicularity of two straight lines and consider in detail using an example.
The angle between intersecting lines in space can be right. Then they say that the given lines are perpendicular. When the angle between intersecting lines is straight, then the lines are also perpendicular. It follows that perpendicular lines on a plane intersect, and perpendicular lines in space can be intersecting and crossing.
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. Having summarized the above, let's look at the definition.
Definition 1
Two lines are called perpendicular if the angle at their intersection makes 90 degrees.
Perpendicularity is denoted by “⊥”, and the notation takes the form a ⊥ b, which means line a is perpendicular to line b.
For example, the sides of a square with a common vertex can be perpendicular lines on a plane. 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 - conditions of perpendicularity
It is necessary to know the properties of perpendicularity, since most problems come down to checking it for subsequent solution. There are cases when perpendicularity is discussed in the task conditions or when it is necessary to use a proof. In order to prove perpendicularity, it is enough that the angle between the lines is right.
In order to determine their perpendicularity with the known equations of the rectangular coordinate system, it is necessary to apply the necessary and sufficient condition for the perpendicularity of the lines. Let's look at the wording.
Theorem 1
In order for lines a and b to be perpendicular, it is necessary and sufficient that the direction vector of the line be perpendicular to the direction vector of the given line b.
The proof itself is based on determining the direction vector of a line and on determining the perpendicularity of lines.
Evidence 1
Let a rectangular cartesian system coordinates O x y with given equations of a line on the plane, which determine the lines a and b. We denote the direction vectors of straight 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 only possible when the scalar product of the vectors a → = (a x , a y) and b → = (b x , b y) is equal to zero, and the entry has the form 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 b, located in the 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 lines a and b.
The condition is applicable when it is necessary to find the coordinates of 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 the rectangular coordinate system O x y. Determine whether lines A B and A C are perpendicular or not.
Solution
Direct 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: straight 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.
Solution
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 move on to calculating the scalar product of 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 the lines are also not perpendicular.
Answer: the lines are not perpendicular.
The necessary and sufficient condition for the 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 the lines a and b.
Example 3
Check the perpendicularity of lines in a rectangular coordinate system of three-dimensional space, given by equations x 2 = y - 1 = z + 1 0 and x = λ y = 1 + 2 λ z = 4 λ
Solution
Denominators from canonical equations straight lines are considered to be 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 → = (2, - 1, 0) and b → = (1, 2, 4) are direction vectors of the given lines. To identify their perpendicularity, let’s find the scalar product of the vectors.
The expression will take the form a → , b → = 2 · 1 + (- 1) · 2 + 0 · 4 = 0 .
The vectors are perpendicular because the product is zero. The necessary and sufficient condition is met, which means the lines are also perpendicular.
Answer: straight lines are perpendicular.
The perpendicularity check can be carried out based on other necessary and sufficient conditions of perpendicularity.
Theorem 2
Lines a and b on a plane are considered perpendicular when the normal vector of line a is perpendicular to vector b, this is a necessary and sufficient condition.
Evidence 2
This condition is applicable when the equations of lines provide a quick way to find the coordinates of normal vectors of given lines. That is, if there is a general equation of a line of the form A x + B y + C = 0, an equation of a line in segments of the form x a + y b = 1, an equation of a line with an angular coefficient of the form y = k x + b, it is possible to find the coordinates of the vectors.
Example 4
Find out whether the lines 3 x - y + 2 = 0 and x 3 2 + y 1 2 = 1 are perpendicular.
Solution
Based on their equations, it is necessary to find the coordinates of the normal vectors of the lines. We obtain that n α → = (3, - 1) is normal vector for the straight 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 .
Vectors n a → = (3, - 1) and n b → = 2 3, 2 will be perpendicular, since their scalar product will ultimately give a value equal to 0. We get n a → , n b → = 3 · 2 3 + (- 1) · 2 = 0 .
The necessary and sufficient condition has been met.
Answer: straight lines are perpendicular.
When a line a on a plane is defined using an equation with a slope y = k 1 x + b 1, and a 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 whether the lines y = - 3 7 x and y = 7 3 x - 1 2 are perpendicular.
Solution
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 angular coefficients gives the value - 1, - 3 7 · 7 3 = - 1, that is, the lines are perpendicular.
Answer: the given lines are perpendicular.
There is one more condition used to determine the perpendicularity of lines on a plane.
Theorem 3
For the lines a and b to be perpendicular on a plane, a necessary and sufficient condition is that the direction vector of one of the lines is collinear with the normal vector of the second line.
Evidence 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 another. In other words, one 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 whether the given lines x - y - 1 = 0 and x 0 = y - 4 2 are perpendicular.
Solution
We find that the normal vector of the straight line x - y - 1 = 0 has coordinates n a → = (1, - 1), and b → = (0, 2) is the direction vector of the straight 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 number t such that the equality n a → = t · b → holds. Hence the conclusion that the lines are not perpendicular.
Answer: the lines are not perpendicular.
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Anaz. mutually perpendicular if l is perpendicular to every line lying on a. For a generalization of the concept of perpendicularity, see Art. Orthogonality.
Mathematical encyclopedia. - M.: Soviet Encyclopedia. I. M. Vinogradov. 1977-1985.
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