Perpendicular direct and their properties. Perpendicular direct what is called perpendicular direct

The article discusses the issue of perpendicular direct on the plane and three-dimensional space. The definition of perpendicular direct and their designations with the above examples will describe in detail. Consider the conditions for applying the necessary and sufficient condition perpendicularity of the two direct and consider in detail on the example.

The angle between intersecting straight in space can be direct. Then they say that the data is direct perpendicular. When the angle between cross-go straight straight, then direct is also perpendicular. It follows that perpendicular straight lines on the plane intersecting, and perpendicular direct spaces can be intersecting and crossing.

That is, the concepts of "straight a and b perpendicular" and "straight b and a perpendicular" are considered equal. Hence the concept of mutually perpendicular direct. Summarizing the foregoing, consider the definition.

Definition 1.

Two straight lines are called perpendicular if the angle during their intersection gives 90 degrees.

Perpendicularity is denoted by "⊥", and the recording takes the form A ⊥ B, which means straight a perpendicular to the direct b.

For example, perpendicular straight on the plane can be the side of the square with a total vertex. In the three-dimensional space, the straight lines o x, o z, o y are perpendicular to pairs: o x and o z, o x and o y, o y and o z.

Perpendicularity of direct - perpendicularity conditions

The properties of perpendicularity need to know, since most tasks are reduced to its verification for the subsequent solution. There are cases when the perpendicularity is in question still in the condition of the task or when it is necessary to use proof. In order to prove perpendicularity, it is enough for the angle between the straight line.

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 direct. Consider the wording.

Theorem 1.

In order for straight a and b to be perpendicular to, it is necessary and enough that the guide vector straight has perpendicularity with respect to the guide vector of a given straight line b.

The proof itself is based on the definition of the direct vectors and on the definition of the perpendicularity of the direct.

Proof 1.

Let the rectangular decartian coordinate system of the coordinates of the coordinates of the coordinates with the specified equations are straight on the plane, which define straight a and b. Direct vectors of straight lines a and b are denoted by A → and B →. From the equation of direct a and b, the perpendicularity of the vectors A → and B → is necessary and sufficient condition. This is possible only with a scalar product of vectors A → \u003d (a x, a y) and b → \u003d (b x, b y) equal to zero, and the recording is viewed A →, B → \u003d A x · b x + A y · b y \u003d 0. We obtain that we obtain the necessary and sufficient condition for the perpendicularity of direct a and b, which are in the rectangular coordinate system about x y on the plane, is A →, B → \u003d AX · BX + AY · BY \u003d 0, where A → \u003d (AX, AY) And B → \u003d BX, BY is the direct vectors of the straight lines a and b.

The condition is applicable when it is necessary to find the coordinates of the guide vectors or in the presence of canonical or parametric equations of direct on the plane of the specified direct a and b.

Example 1.

Three points A (8, 6), B (6, 3), C (2, 10) are specified in the rectangular coordinate system about x y. Determine direct and in and and with perpendicular or not.

Decision

Straight A B and A C have guide vectors A B → and A C → respectively. To begin with, I calculate A B → \u003d (- 2, - 3), A C → \u003d (- 6, 4). We obtain that vectors A b → and a C → perpendicular to the property of 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 performed, it means that in and and perpendicular.

Answer:direct perpendicular.

Example 2.

Determine the specified straight x - 1 2 \u003d y - 7 3 and x \u003d 1 + λ y \u003d 2 - 2 · λ perpendicular or not.

Decision

a → \u003d (2, 3) is the guide vector of a given straight line x - 1 2 \u003d y - 7 3,

b → \u003d (1, - 2) is a straight line guide vector x \u003d 1 + λ y \u003d 2 - 2 · λ.

Let us turn to the calculation of the scalar product of vectors A → and B →. The expression will be recorded:

a →, B → \u003d 2 · 1 + 3 · - 2 \u003d 2 - 6 ≠ 0

The result of the work is not zero, it can be concluded that the vectors are not perpendicular to, it means that the straight lines are also not perpendicular.

Answer:straight are not perpendicular.

Required and sufficient condition for the perpendicularity of direct a and b is used for three-dimensional space, written in the form A →, B → \u003d AX · BX + AY · BY + AZ · BZ \u003d 0, where A → \u003d (AX, AY, AZ) and B → \u003d (BX, BZ) are guide vectors of direct a and b.

Example 3.

Check the perpendicularity of the direct in the rectangular coordinate system of the three-dimensional space specified 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 the direct are considered the coordinates of the guide vector direct. The coordinates of the guide vector from the parametric equation are coefficients. It follows from this that A → \u003d (2, - 1, 0) and B → \u003d (1, 2, 4) are the guide vectors of the specified direct. To identify their perpendicularity, we find a scalar product of vectors.

The expression will take the form A →, B → \u003d 2 · 1 + (- 1) · 2 + 0 · 4 \u003d 0.

The vectors are perpendicular, as the work is zero. The necessary and sufficient condition is fulfilled, it means directly perpendicular.

Answer:direct perpendicular.

Checking perpendicularity can be carried out on the basis of other necessary and sufficient conditions of perpendicularity.

Theorem 2.

Straight A and B on the plane are considered perpendicular to the perpendicularity of the normal vector direct A with a vector B, this is the necessary and sufficient condition.

Proof 2.

This condition is applicable when the equations of direct give the coordinates of normal vectors of the specified direct. That is, if there is a general equation of a direct form A x + b y + c \u003d 0, the equations are straight in segments of the form x a + y b \u003d 1, the equations of the straight line with the angular coefficient of the form y \u003d k x + b the coordinates of the vectors can be found.

Example 4.

Find out, perpendicular to the straight 3 x - y + 2 \u003d 0 and x 3 2 + y 1 2 \u003d 1.

Decision

Based on their equations, it is necessary to find the coordinates of normal vectors of direct. We obtain that N α → \u003d (3, - 1) is a normal vector for a straight line 3 x - y + 2 \u003d 0.

We 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, which we write in this form N b → \u003d 2 3, 2 are clearly visible.

Vectors n A → \u003d (3, - 1) and N b → \u003d 2 3, 2 will be perpendicular, since their scalar product will eventually value equal to 0. We obtain n a →, n b → \u003d 3 · 2 3 + (- 1) · 2 \u003d 0.

The necessary and sufficient condition was fulfilled.

Answer:direct perpendicular.

When straight A on the plane is determined using an equation with an angular coefficient Y \u003d k 1 x + b 1, and straight b - y \u003d k 2 x + b 2, it follows that normal vectors will have coordinates (K 1, - 1) and (k 2, - 1). The condition of perpendicularity is reduced to k 1 · k 2 + (- 1) · (- 1) \u003d 0 ⇔ k 1 · k 2 \u003d - 1.

Example 5.

Find out whether the straight lines y \u003d - 3 7 x and y \u003d 7 3 x - 1 2 are perpendicular.

Decision

Straight y \u003d - 3 7 x has an angular coefficient equal to 3 7, and straight y \u003d 7 3 x - 1 2 - 7 3.

The product of angular coefficients gives value to 1, - 3 7 · 7 3 \u003d - 1, that is, the direct are perpendicular.

Answer:the specified direct perpendicular.

There is another condition used to determine the perpendicularity of direct on the plane.

Theorem 3.

For perpendicularity of direct a and b on the plane, the required and sufficient condition is the collinearity of the guide vector of one of the straight line with the normal vector of the second straight.

Proof 3.

The condition is applicable when there is the possibility of finding the guide vector of one straight and coordinates of the normal vector. In other words, one direct is defined by a canonical or parametric equation, and the other with a direct equation, equation in segments or a straight line with an angular coefficient.

Example 6.

Determine whether the specified straight lines x - y - 1 \u003d 0 and x 0 \u003d y - 4 2 perpendicular.

Decision

We obtain that the normal vector straight line X is y - 1 \u003d 0 has coordinates N a → \u003d (1, - 1), and B → \u003d (0, 2) - the guide vector is straight x 0 \u003d y - 4 2.

It can be seen that vectors n a → \u003d (1, - 1) and b → \u003d (0, 2) are not collinear, because the condition of collinearity is not performed. There is no such number T so that the equality N A → \u003d T · B →. Hence the conclusion that the straight lines are not perpendicular.

Answer:straight are not perpendicular.

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Perpendicular straight lines form a whole formation of figures, constructions and calculations in geometry. Without an understanding of perpendicular straight lines fails to solve such figures as right triangle, rectangle, square or rectangular trapezium. Therefore, it is worth paying special attention to these concepts.

What is perpendicular direct

When crossing two straight lines, 4 corners are formed. The definition of perpendicular straight lines sounds like this: it is straight, the angle between which is 90 degrees. Corners only 4, full angle is 360 degrees. If one of the corners is 90 degrees, then 3 others will be 90.

In order for the segments to be perpendicular, two conditions should be performed: the segments must intersect, and the crossing angle between them should be 90 degrees.

Fig. 1. Perpendicular lines.

Properties

Perpendicular direct is not so many properties. All of them do not require evidence, as they proceed from the definition of perpendicularity.

• If each of the two direct perpendicular to the third, then these direct are parallel. And in parallel, they are due to the fact that the resulting one-sided corners will in the amount of giving 180 degrees. So, straight parallel to 3 signs of parallelism. This property can be proved on any of the three signs of parallelism.
• The perpendicular segment from the point to direct or the segment will be called the distance from the point to direct.
• The distance from direct to the line is also perpendicular, lowered from any point one straight to another direct.
• If throughout the length of two direct distances between them does not change, then the direct will be parallel.

Figures with perpendicular straight

One of the first figures with whom a person gets acquainted is a square and a rectangle.

Straight angles are pleasant to the human look, so very often the square or rectangle is used as a form for tabletops, chairs, bedside tables and other items. The whole people surrounding the world is made up of parallel and perpendicular lines.

Fig. 2. Square.

Since the time of ancient Greece, a rectangular triangle is known. The shape of a rectangular triangle was taken by various navigation devices, besides, a long time to study the properties of the rectangular triangle was given to Pythagores. It is his authorship that the Pythagora Theorem is belonging, which is extremely demanded in solutions to tasks.

There is a rectangular trapezium, which one of the sides is rectangular to both bases. And the planometry is at all pissed by perpendiculars in space: the correct prism, the rectangular pyramid and the most ordinary cube.

In addition, in any triangle you can spend the height that is necessary for finding the area of \u200b\u200bthe figure. The perpendicular for finding the area is useful in the parallelogram, and the rectangular triangle and the square have a height of their parties, which is why the area of \u200b\u200bthese figures is much easier to find.

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Perpendicularity is called the relationship between a variety of objects in the Euclidean space - straight, planes, vectors, subspaces, and so on. In this material, we will carefully consider perpendicular direct and characteristic features, relating to them. Two straight lines can be called perpendicular (or mutuallypendicular), if all four corners that are formed by their intersection are strictly ninety-degrees.

There are certain properties of perpendicular direct, implemented on the plane:

Construction of perpendicular lines

Perpendicular straight lines are built on the plane with the help of the square. Any drawler must keep in mind that an important feature of each square is that it necessarily has a straight corner. To create two perpendicular straight, we need to combine one of two sides direct corner OUR

a drawing kit with a given straight and conduct the second straight along the second side of this direct angle. Thus, two perpendicular straight lines will be created.

Three-dimensional space

Interesting the fact that perpendicular direct can be implemented and in this case will be called two straight lines, if they are parallel to any two other direct, lying in the same plane and also perpendicular to it. In addition, if only two straight lines can be perpendicular on the plane, then three in three-dimensional space. Moreover, the number of perpendicular lines (or planes) can be even more increased.

The straight line (cut straight) is denoted by two large letters of the Latin alphabet or one small letter. The point is indicated only by a large Latin letter.

Direct may not intersect, intersect or coincide. Intersecting straight lines have only one common point, non-vigorous direct - not a single dot point, in the coinciding direct all points are common.

Definition. Two straight, intersecting at right angles are called perpendicular. The perpendicularity of direct (or their segments) is indicated by the sign of perpendicularity "⊥".

For example:

Your AB and CD (Fig. 1) intersect at the point ABOUT and ∠ AOS = ∠Rest = ∠AOD = ∠Bod. \u003d 90 °, then AB ⊥ CD.

If a AB ⊥ CD (Fig. 2) and intersect at the point IN, then ∠ ABC. = ∠ABD. \u003d 90 °

Properties perpendicular lines

1. Through the point BUT (Fig. 3) can be carried out only one perpendicular straight AU to direct CD; The rest of the straight, passing through the point BUT and crossing Cd.are called sloping straight (Fig. 3, direct AE and AF.).

2. From the point A. You can lower the perpendicular to the straight CD; Perpendicular Length (Cut Length AU) spent from the point BUT on straight CD- this is the shortest distance from A. before CD (Fig. 3).