Vectors can be graphically represented by directional line segments. The length is chosen on a certain scale to indicate the magnitude of the vector , and the direction of the segment represents vector direction . For example, if we assume that 1 cm represents 5 km/h, then a northeasterly wind of 15 km/h will be represented by a 3 cm directional line, as shown in the figure.

Vector in the plane it is a directed segment. Two vectors equal if they have the same value and direction.

Consider a vector drawn from point A to point B. The point is called starting point vector, and the point B is called end point. The symbolic notation for this vector is (read as “vector AB”). Vectors are also denoted by bold letters, such as U, V, and W. The four vectors in the figure on the left have the same length and direction. Therefore they represent equal winds; that is,

In the context of vectors, we use = to denote their equality.

length, or magnitude expressed as ||. To determine if vectors are equal, we find their magnitudes and directions.

Example 1 The vectors u, , w are shown in the figure below. Prove that u = w.

Solution First we find the length of each vector using the distance formula:
|u| = √ 2 + (4 - 3) 2 = √9 + 1 = √10,
|| = √ 2 + 2 = √9 + 1 = √10 ,
|w| = √(4 - 1) 2 + [-1 - (-2)] 2 = √9 + 1 = √10 .
From here
|u| = | = |w|.
The vectors u, , and w, as you can see from the figure, seem to have the same direction, but we will check their slope. If the lines they are on have the same slope, then the vectors have the same direction. Calculate slopes:
Since u, , and w have the same magnitude and the same direction,
u = w.

Keep in mind that equal vectors only require the same magnitude and the same direction, not being in the same place. The topmost figure is an example of the equality of vectors.

Suppose a person takes 4 steps to the east and then 3 steps to the north. The person will then be 5 steps away from the starting point in the direction shown on the left. A vector 4 units long and with a right direction represents 4 steps east and a vector 3 units long up represents 3 steps north. Sum of these two vectors is a vector of 5 steps of magnitude and in the direction shown. The amount is also called resulting two vectors.

In general, two non-zero vectors u and v can be added geometrically by positioning the start point of vector v to the end point of vector u, and then finding a vector that has the same start point as vector u and the same end point as vector v as shown in the figure below.

The sum is a vector represented by a directed segment from point A of vector u to end point C of vector v. Thus, if u = and v = , then
u+v=+=

We can also describe vector addition as placing the starting points of vectors together, building a parallelogram, and finding the diagonal of the parallelogram. (pictured below.) This addition is sometimes referred to as parallelogram rule addition of vectors. Vector addition is commutative. As shown in the figure, both vectors u + v and v + u are represented by the same directed segment.

If two forces F 1 and F 2 act on the same object, resulting force is the sum F 1 + F 2 of these two separate forces.

Example Two forces of 15 newtons and 25 newtons act on the same object perpendicular to each other. Find their sum, or resultant force, and the angle it makes with the greater force.

Solution Let's draw the condition of the problem, in this case a rectangle, using v or to represent the result. To find its value, we use the Pythagorean theorem:
|v| 2 = 152 + 252 Here |v| denotes the length or magnitude of v.
|v| = √152 + 252
|v| ≈ 29.2.
To find the direction, note that since OAB is a right angle,
tanθ = 15/25 = 0.6.
Using a calculator, we find θ, the angle that the large force makes with the net force:
θ = tan - 1 (0.6) ≈ 31°
The resulting one has a magnitude of 29.2 and an angle of 31° with the greater force.

Pilots can correct the direction of their flight if there is a side wind. Wind and aircraft speed can be represented as winds.

Example 3. Aircraft speed and direction. The aircraft is moving along an azimuth of 100° at a speed of 190 km/h, while the wind speed is 48 km/h and its azimuth is 220°. Find the absolute speed of the aircraft and the direction of its movement, taking into account the wind.

Solution Let's do a drawing first. The wind is represented and the aircraft's velocity vector is . The resulting velocity vector is v, the sum of the two vectors. The angle θ between v and is called drift angle .


Note that COA = 100° - 40° = 60°. Then the value of CBA is also equal to 60° (opposite angles of the parallelgram are equal). Since the sum of all the angles of a parallelogram is 360° and COB and OAB are of the same magnitude, each must be 120°. By cosine rule in OAB, we have
|v| 2 = 48 2 + 190 2 - 2.48.190.cos120°
|v| 2 = 47.524
|v| = 218
Then |v| equals 218 km/h. According to sine rule , in the same triangle,
48 /sinθ = 218 /sin 120°,
or
sinθ = 48.sin120°/218 ≈ 0.1907
θ ≈ 11°
Then, θ = 11°, to the nearest integer angle. The absolute speed is 218 km / h, and the direction of its movement, taking into account the wind: 100 ° - 11 °, or 89 °.

Given a vector w, we can find two other vectors u and v whose sum is w. The vectors u and v are called components w and the process of finding them is called decomposition , or a representation of a vector by its vector components.

When we decompose a vector, we usually look for perpendicular components. Very often, however, one component will be parallel to the x-axis and the other will be parallel to the y-axis. Therefore, they are often called horizontal and vertical vector components. In the figure below, the vector w = is decomposed as the sum of u = and v = .

The horizontal component of w is u and the vertical component is v.

Example 4 The w vector has a magnitude of 130 and a slope of 40° relative to the horizontal. Decompose the vector into horizontal and vertical components.

Solution First we draw a picture with horizontal and vertical vectors u and v, whose sum is w.

From ABC, we find |u| and |v| using the definitions of cosine and sine:
cos40° = |u|/130, or |u| = 130.cos40° ≈ 100,
sin40° = |v|/130, or |v| = 130.sin40° ≈ 84.
Then, the horizontal w component is 100 to the right and the vertical w component is 84 upwards.

"Geometry "Area of ​​the trapezoid"" - Think. The area of ​​the trapezoid. AH=. 1. AD = 4 cm. Base. Find the area of ​​trapezoid ABCD. Find the area of ​​a rectangular trapezoid. Geometry. Repeat the proof of the theorem. Divide the polygon into triangles. Problem with a solution.

"Determination of axial symmetry" - Build points A "and B". Axial symmetry. Figure. Missing coordinates. Building a segment. Line segment. Axis of symmetry. Symmetry in poetry. Construction of a triangle. Points that lie on the same perpendicular. Building a point. Symmetry. Triangles. Build triangles. Draw a point. Plot points. Figures with one axis of symmetry. Straight. Figures with two axes of symmetry. Symmetry in nature.

"Quadangles, their signs and properties" - Tests. Rhombus corners. A rectangle with all sides equal. Types of quadrilaterals. Learn about types of quadrilaterals. A quadrilateral whose vertices are at the midpoints of the sides. Quadrangles. Quadrilaterals, their signs and properties. Trapeze. Parallelogram. Parallelogram properties. Diagonals. What two equal triangles can form a square? Rectangle. Square. Types of trapezoid.

"Theorem of the inscribed angle" - The study of new material. The circles intersect. Answer. Updating students' knowledge. Check yourself. Circle radius. Correct answer. The radius of the circle is 4 cm. Consolidation of the studied material. Sharp corner. Find the angle between the chords. Triangle. Inscribed angle theorem. The concept of an inscribed angle. Find the angle between them. What is the name of the angle with the vertex at the center of the circle? Solution. Knowledge update.

"Construction of a tangent to a circle" - Circle. Mutual arrangement of a straight line and a circle. Circle and line. Diameter. Common points. Chord. Solution. Circle and line have one common point. Tangent to a circle. Repetition. Tangent segment theorem.

"Geometry "Similar Triangles"" - Two triangles are called similar. Sine, cosine and tangent values ​​for 30°, 45°, 60° angles. Find the area of ​​an isosceles right triangle. Theorem on the ratio of the areas of similar triangles. Similar triangles. The second sign of the similarity of triangles. Continuation of the sides. Sine, cosine and tangent values. proportional cuts. The two sides of the triangle are connected by a segment that is not parallel to the third.

This chapter is devoted to the development of the vector apparatus of geometry. Using vectors, you can prove theorems and solve geometric problems. Examples of this use of vectors are given in this chapter. But the study of vectors is also useful because they are widely used in physics to describe various physical quantities, such as, for example, speed, acceleration, force.

Many physical quantities, for example, force, displacement of a material point, speed, are characterized not only by their numerical value, but also by their direction in space. These physical quantities are called vector quantities(or short vectors).

Consider an example. Let a force of 8 N act on the body. In the figure, the force is represented by a segment with an arrow (Fig. 240). The arrow indicates the direction of the force, and the length of the segment corresponds to the numerical value of the force on the selected scale. So, in figure 240, a force of 1 N is shown as a segment 0.6 cm long, therefore a force of 8 N is depicted as a segment 4.8 cm long.

Rice. 240

Abstracting from the specific properties of physical vector quantities, we come to the geometric concept of a vector.

Consider an arbitrary segment. Its ends are also called boundary points of the segment.

Two directions can be specified on a segment: from one boundary point to another and vice versa.

To choose one of these directions, we call one boundary point of the segment the beginning of the segment, and the other - the end of the segment and we will assume that the segment is directed from the beginning to the end.

Definition

In the figures, a vector is depicted as a segment with an arrow showing the direction of the vector. Vectors are denoted by two uppercase Latin letters with an arrow above them, for example . The first letter indicates the beginning of the vector, the second - the end (Fig. 242).

Rice. 242

Figure 243, a shows the vectors points A, C, E are the beginnings of these vectors, and B, D, F are their ends. Vectors are often denoted by one lowercase Latin letter with an arrow above it: (Fig. 243, b).

Rice. 243

For what follows, it is expedient to agree that any point of the plane is also a vector. In this case, the vector is called zero. The beginning of the zero vector coincides with its end. In the figure, such a vector is represented by a single point. If, for example, the point representing the zero vector is denoted by the letter M, then this zero vector can be denoted as follows: (Fig. 243, a). The zero vector is also denoted by the symbol In Figure 243 vectors are non-zero, and the vector is zero.

The length or modulus of a non-zero vector is the length of the segment AB. The length of a vector (vector ) is denoted as follows: . The length of the null vector is considered to be zero:

The lengths of the vectors shown in figures 243, a and 243, 6 are as follows:

(each cell in figure 243 has a side equal to the unit of measurement of the segments).

Vector equality

Before defining equal vectors, let's look at an example. Consider the motion of a body in which all its points move at the same speed and in the same direction.

The speed of each point M of the body is a vector quantity, so it can be represented by a directed segment, the beginning of which coincides with the point M (Fig. 244). Since all points of the body move with the same speed, all directed segments representing the speeds of these points have the same direction and their lengths are equal.

Rice. 244

This example tells us how to determine the equality of vectors.

Let us first introduce the concept of collinear vectors.

Non-zero vectors are called collinear, if they lie either on the same line or on parallel lines; the zero vector is considered collinear to any vector.

In Figure 245, the vectors (vector zero) are collinear, and the vectors and are also non-collinear.

Rice. 245

If two non-zero vectors and are collinear, then they can be directed either in the same way or oppositely. In the first case, the vectors and are called co-directional, and in the second opposite directions 1 .

The co-direction of vectors and is denoted as follows: If the vectors and are oppositely directed, then this is denoted as follows: Figure 245 shows both co-directed and oppositely directed vectors:

The beginning of the zero vector coincides with its end, so the zero vector does not have any particular direction. In other words, any direction can be considered the direction of the zero vector. We agree to assume that the zero vector is codirectional with any vector. Thus, in figure 245, etc.

Non-zero collinear vectors have properties that are illustrated in Figure 246, a - c.




Rice. 246

We now give the definition of equal vectors.

Definition

Thus, the vectors and are equal if . The equality of vectors and is denoted as follows:

Postponing a vector from a given point

If point A is the beginning of the vector, then they say that the vector is postponed from point A(Fig. 247). Let us prove the following assertion:

from any point M, you can postpone a vector equal to a given vector, and moreover, only one.




Rice. 247

Indeed, if is a null vector, then the required vector is the vector . Let's assume that the vector is non-zero, and points A and B are its beginning and end. Let us draw a line p parallel to AB through the point M (Fig. 248; if M is a point of the line AB, then we take the line AB itself as the line p). On the line p, we set aside the segments MN and MN", equal to the segment AB, and choose from the vectors one that is co-directed with the vector (in Figure 248 vector). This vector is the desired vector, equal to the vector . It follows from the construction that there is only one such vector.




Rice. 248

Comment

Equal vectors plotted from different points are often denoted by the same letter. This is how, for example, equal velocity vectors of different points are indicated in Figure 244. Sometimes such vectors are said to be the same vector, but plotted from different points.

Practical tasks

738. Mark points A, B and C that do not lie on one straight line. Draw all non-zero vectors whose beginning and end coincide with any two of these points. Write down all the resulting vectors and indicate the beginning and end of each vector.

739. Having chosen a suitable scale, draw vectors depicting the flight of an airplane, first 300 km south from city A to B, and then 500 km east from city B to C. Then draw a vector that depicts the movement from the start point to the end point.

740. Draw vectors so that:

741. Draw two non-collinear vectors and . Draw several vectors: a) co-directional with the vector ; b) co-directional with the vector ; c) oppositely directed to the vector ; d) oppositely directed to the vector .

742. Draw two vectors: a) having equal lengths and non-colinear; b) having equal lengths and co-directional; c) having equal lengths and opposite directions. In which case the resulting vectors are equal?

Answer In case b).