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, we will consider the essence of this concept through its construction.
Take a ruler and, without tearing off the 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 a part of it. The straight line itself is infinite at both ends.
Straight lines will be denoted by a small Latin letter, or by two of its dots in parentheses (Fig. 1).
The concepts of a line and a point are related by three axioms of geometry:
Axiom 1: For every arbitrary straight line, there are at least two points that lie on it.
Axiom 2: You can find at least three points that will not lie on the same straight line.
Axiom 3: A straight line always passes through 2 arbitrary points, and this straight line is unique.
For two straight lines, their relative position is relevant. Three cases are possible:
- The two lines coincide. In this case, each point of one will also be a point of another straight line.
- Two lines intersect. In this case, only one point from one straight line will also belong to another straight line.
- Two straight 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, 4 corners are formed at the point of their intersection. Then
Definition 1
Intersecting straight lines will be called 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 picture below
Angle 2 is vertical for the given angle, hence
Angle 1 is adjacent to angle 2, hence
$∠1=180^0-∠2=180^0-90^0=90^0$
Angle 3 is vertical to angle 1, hence
$∠3=∠1=90^0$
From this task we can make the following remark
Remark 1
All angles between perpendicular lines are equal to $ 90 ^ 0 $.
The main theorem of perpendicular lines
Let us introduce the following theorem:
Theorem 1
Two lines that are perpendicular to the third will be disjoint.
Proof.
Consider Figure 3 by the statement of the problem.
Let's mentally divide this picture into two parts of the straight line $ (ZP) $. We will overlay the right part on the left one. Then, since the straight lines $ (NM) $ and $ (XY) $ are perpendicular to the straight line $ (PZ) $ and, therefore, the angles between them are right, then the ray $ NP $ is imposed entirely on the ray $ PM $, and the ray $ XZ $ will be imposed entirely on ray $ YZ $.
Now, suppose the opposite: let these lines intersect. Without loss of generality, suppose that they intersect on the left side, that is, let ray $ NP $ intersect ray $ YZ $ at point $ O $. Then, by the construction described above, we will get that the ray $ PM $ also intersects the ray $ YZ $ at the point $ O "$. But then we get that through two points $ O $ and $ O" $, two lines $ (NM) $ and $ (XY) $, which contradicts the 3-line axiom.
Therefore, the lines $ (NM) $ and $ (XY) $ do not intersect.
The theorem is proved.
Example task
Example 2
You are given two straight lines that have an intersection point. Through a point that does not belong to either of them, two straight lines are drawn, one of which is perpendicular to one of the above-described straight lines, and the other is perpendicular to the other of them. Prove they don't match.
Let's draw a picture according to the condition of the problem (Fig. 4).
From the condition of the problem we will have that $ m⊥k, n⊥l $.
Suppose the opposite, let the lines $ k $ and $ l $ coincide. Let it be the straight line $ l $. Then, by hypothesis, $ 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 straight lines $ k $ and $ l $ do not coincide.
Perpendicularity is the relationship between various objects in Euclidean space - lines, planes, vectors, subspaces, and so on. In this article, we will take a closer look at the perpendicular lines and specific traits 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 align one of the two sides of the right angle of 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 straight lines can be realized and in this case, two straight lines will be called such if they are parallel, respectively, to some other two straight 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.
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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. Let us 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 that the perpendicular straight lines in the plane are intersecting, and the perpendicular straight lines of the 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. This is where 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 when they intersect 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 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
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 with 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 the 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 → = (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 a →, b → = ax bx + ay by = 0, where a → = (ax, ay) and b → = 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 whether lines A B and A C are perpendicular or not.
Solution
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 on 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 AB and AC 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 us proceed 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 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.
A necessary and sufficient condition of perpendicularity of lines a and b is applied for three-dimensional space, written as a →, b → = ax bx + ay by + az bz = 0, where a → = (ax, ay, az) and b → = (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 = y - 1 = z + 1 0 and x = λ y = 1 + 2 λ z = 4 λ
Solution
The denominators from the canonical equations of the 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 → = (2, - 1, 0) and b → = (1, 2, 4) are direction vectors of the given lines. To identify their perpendicularity, we find the scalar product of vectors.
The expression will take the form a →, b → = 2 1 + (- 1) 2 + 0 4 = 0.
The vectors are perpendicular since the product is zero. The necessary and sufficient condition is satisfied, so the lines are also perpendicular.
Answer: straight lines are perpendicular.
The squareness check can 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 straight lines give a quick determination of the coordinates of the normal vectors of given straight lines. That is, in the presence of a general equation of a straight line of the form A x + B y + C = 0, equations of a straight line in segments of the form x a + y b = 1, equations of a straight line with a slope of the form y = k x + b, the coordinates of 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.
Solution
Based on their equations, it is necessary to find the coordinates of the normal vectors of straight lines. We obtain that n α → = (3, - 1) is the normal vector for the line 3 x - y + 2 = 0.
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 dot product will end up with a value of 0. We get n a →, n b → = 3 2 3 + (- 1) 2 = 0.
The necessary and sufficient condition has been fulfilled.
Answer: straight lines are perpendicular.
When the straight line a on the plane is defined using the equation with the slope y = k 1 x + b 1, and the straight 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.
Solution
The line y = - 3 7 x has a slope equal to - 3 7, and the line y = 7 3 x - 1 2 - 7 3.
The product of the slopes gives the value - 1, - 3 7 · 7 3 = - 1, that is, the straight lines are perpendicular.
Answer: the 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 the 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 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.
Solution
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. Hence the conclusion that straight lines are not perpendicular.
Answer: straight lines are not perpendicular.
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A straight line (a segment of a straight line) is designated by two capital letters of the Latin alphabet or one small letter. A point is indicated only by a capital Latin letter.
Lines may not intersect, intersect or coincide. Intersecting straight lines have only one common point, non-intersecting straight lines - no common point, coinciding straight lines have all points in common.
Definition. Two straight lines intersecting at right angles are called perpendicular. The perpendicularity of straight lines (or their segments) is denoted by the perpendicularity sign "⊥".
For instance:
Your AB and CD(fig. 1) intersect at the point O and ∠ AOC = ∠VOS = ∠AOD = ∠BOD= 90 °, then AB ⊥ CD.
If AB ⊥ CD(fig. 2) and intersect at the point V, then ∠ ABC = ∠ABD= 90 °
Properties of perpendicular lines
1. Through point A(fig. 3) only one perpendicular line can be drawn AB to straight CD; the rest of the lines passing through the point A and crossing CD, are called oblique straight lines (Fig. 3, straight lines AE and AF).
2. From point A you can lower the perpendicular to a straight line CD; perpendicular length (segment length AB) drawn from the point A on a straight line CD, is the shortest distance from A before CD(fig. 3).