A vector in geometry is a directed segment or an ordered pair of points in Euclidean space. Ortom vector is the unit vector of a normed vector space or a vector whose norm (length) is equal to one.

You will need

  • Geometry knowledge.

Instruction

First you need to calculate the length vector. As you know, the length (modulus) vector is equal to the square root of the sum of the squares of the coordinates. Let a vector with coordinates be given: a(3, 4). Then its length is |a| = (9 + 16)^1/2 or |a|=5.

To find ort vector a, it is necessary to divide each of it by its length. The result will be a vector, which is called the ort or unit vector. For vector a(3, 4) ort will be the vector a(3/5, 4/5). Vector a` will be single for vector a.

To check whether the ort was found correctly, you can do the following: find the length of the received ort, if it is equal to one, then everything is found correctly, if not, then an error has crept into the calculations. Let's check whether the ort a` is found correctly. Length vector a` is equal to: a` = (9/25 + 16/25)^1/2 = (25/25)^1/2 = 1. So, the length vector a` is equal to one, so the unit vector is found correctly.

A change in the coordinate x2 - x1 is usually denoted by the symbol Δx12 (it reads “delta x one, two”). This entry means that over the time interval from the moment t1 to the moment t2, the change in the coordinate of the body Δx12 = x2 - x1. Thus, if the body moved in the positive direction of the X axis of the chosen coordinate system (x2 > x1), then Δx12 >

On fig. 45 shows a point body B, which moves in the negative direction of the X axis. During the time interval from t1 to t2, it moves from a point with a larger coordinate x1 to a point with a smaller coordinate x2. As a result, the change in the coordinate of point B over the considered time interval Δx12 = x2 - x1 = (2 - 5) m = -3 m. The displacement vector in this case will be directed in the negative direction of the X axis, and its module |Δx12| is 3 m. From the examples considered, the following conclusions can be drawn.

In the examples considered (see Figs. 44 and 45), the body moved all the time in one direction.

How to find the displacement modulus in physics? (Maybe there is some universal formula?)

Therefore, the distance traveled by him is equal to the modulus of the body coordinate change and the displacement modulus: s12 = |Δx12|.

Let us determine the change in the coordinate and the displacement of the body over the time interval from t0 = 0 to t2 = 7 s. In accordance with the definition, the change in the coordinate Δx02 = x2 - x0 = 2 m >

Now let's determine the path that the body has traveled for the same period of time from t0 = 0 to t2 = 7 s. First, the body traveled 8 m in one direction (which corresponds to the modulus of change of coordinate Δx01), and then 6 m in the opposite direction (this value corresponds to the modulus of change of coordinate Δx12). This means that the total body has passed 8 + 6 = 14 (m). According to the definition of the path, in the time interval from t0 to t2, the body traveled the path s02 = 14 m.

Results

The movement of a point over a period of time is a directed segment of a straight line, the beginning of which coincides with the initial position of the point, and the end coincides with the final position of the point.

Questions

Exercises

Vectors, actions with vectors

pythagorean theorems cosine theorem

The length of the vector will be denoted by . The modulus of a number has a similar designation, and the length of a vector is often called the modulus of a vector.

, where .

In this way, .

Consider an example.

:

.

In this way, vector length .

Calculate Vector Length

, Consequently,

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Let's consider examples.

.

moving

:

:

.

.



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In this way, .


or ,
or ,

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So far, we have considered only rectilinear uniform motion. In this case, the point bodies moved in the chosen reference frame either in the positive or in the negative direction of the X coordinate axis. We found that, depending on the direction of motion of the body, for example, during the time interval from the moment t1 to the moment t2, the change in the coordinate of the body (x2 - x1 ) can be positive, negative or equal to zero (if x2 = x1).

A change in the coordinate x2 - x1 is usually denoted by the symbol Δx12 (it reads “delta x one, two”). This entry means that over the time interval from the moment t1 to the moment t2, the change in the coordinate of the body Δx12 = x2 - x1. Thus, if the body moved in the positive direction of the X axis of the selected coordinate system (x2 > x1), then Δx12 > 0. If the movement occurred in the negative direction of the X axis (x21), then Δx12

It is convenient to determine the result of the movement using a vector quantity. This vector quantity is displacement.

The movement of a point over a period of time is a directed segment of a straight line, the beginning of which coincides with the initial position of the point, and the end coincides with the final position of the point.

Like any vector quantity, displacement is characterized by module and direction.

We will write the point displacement vector for the time interval from t1 to t2 in the following way: Δx12.

Let us explain what has been said with an example. Let some point A (dotted head) move in the positive direction of the X axis and move from a point with coordinate x1 to a point with a larger coordinate x2 over a period of time from t1 to t2 (Fig. 44). In this case, the displacement vector is directed in the positive direction of the X axis, and its module is equal to the change in the coordinate for the considered time interval: Δx12 = x2 - x1 = (5 - 2) m = 3 m.

On fig. 45 shows a point body B that moves in the negative direction of the X axis.

During the time interval from t1 to t2, it moves from a point with a larger x1 coordinate to a point with a smaller x2 coordinate. As a result, the change in the coordinate of point B over the considered time interval Δx12 = x2 - x1 = (2 - 5) m = -3 m. The displacement vector in this case will be directed in the negative direction of the X axis, and its module |Δx12| is 3 m. From the examples considered, the following conclusions can be drawn.

The direction of movement in rectilinear movement in one direction is the same as the direction of movement.

The modulus of the displacement vector is equal to the modulus of the change in the coordinates of the body over the considered period of time.

In everyday life, the concept of "path" is used to describe the end result of movement. Usually the path is denoted by the symbol S.

The path is the entire distance traveled by a point body during the considered period of time.

Like any distance, the path is a non-negative value. For example, the path traveled by point A in the considered example (see Fig. 44) is three meters. The path traveled by point B is also three meters.

In the examples considered (see Figs. 44 and 45), the body moved all the time in one direction. Therefore, the distance traveled by him is equal to the modulus of the body coordinate change and the displacement modulus: s12 = |Δx12|.

If the body moved all the time in the same direction, then the distance traveled by it is equal to the displacement modulus and the coordinate change modulus.

The situation will change if the body during the considered period of time changes the direction of motion.

On fig. 46 shows how the point body moved from the moment t0 = 0 to the moment t2 = 7 s. Until the moment t1 = 4 s, the movement proceeded uniformly in the positive direction of the X axis. As a result, the change in the coordinate Δx01 = x1 - x0 = (11 - 3) m = -8 m. After that, the body began to move in the negative direction of the X axis until the moment t2 = 7 s. At the same time, the change in its coordinates Δx12 = x2 - x1 = (5 - 11) m = -6 m. The graph of this movement is shown in fig. 47.

Let us determine the change in the coordinate and the displacement of the body over the time interval from t0 = 0 to t2 = 7 s. In accordance with the definition, the change in coordinate Δx02 = x2 - x0 = 2 m > 0. Therefore, the displacement Δx02 is directed in the positive direction of the X axis, and its module is 2 m.

Now let's determine the path that the body has traveled for the same period of time from t0 = 0 to t2 = 7 s. First, the body traveled 8 m in one direction (which corresponds to the modulus of change of coordinate Δx01), and then 6 m in the opposite direction (this value corresponds to the modulus of change of coordinate Δx12).

Trajectory

This means that the total body has passed 8 + 6 = 14 (m). According to the definition of the path, in the time interval from t0 to t2, the body traveled the path s02 = 14 m.

The analyzed example allows us to conclude:

In the case when the body during the considered period of time changes the direction of its movement, the path (the entire distance traveled by the body) is greater than both the modulus of the body's displacement and the modulus of the change in the body's coordinates.

Now imagine that the body after the moment of time t2 = 7 s continued its movement in the negative direction of the X axis until the moment t3 = 8 s in accordance with the law shown in Fig. 47 dotted line. As a result, at the time t3 = 8 s, the coordinate of the body became x3 = 3 m. It is easy to determine that in this case, the movement of the body over the time interval from t0 to t3 s is equal to Δx13 = 0.

It is clear that if we only know the movement of the body during its movement, then we cannot say how the body moved during this time. For example, if it were only known about the body that its initial and final coordinates are equal, then we would say that during the movement the displacement of this body is zero. It would be impossible to say anything more concrete about the nature of the motion of this body. The body could, under such conditions, generally stand still for the entire period of time.

The movement of the body over a certain period of time depends only on the initial and final coordinates of the body and does not depend on how the body moved during this period of time.

Results

The movement of a point over a period of time is a directed segment of a straight line, the beginning of which coincides with the initial position of the point, and the end coincides with the final position of the point.

The displacement of a point body is determined only by the final and initial coordinates of the body and does not depend on how the body moved during the considered period of time.

The path is the entire distance traveled by a point body during the considered period of time.

If the body in the process of movement did not change the direction of movement, then the path traveled by this body is equal to the modulus of its displacement.

If the body during the considered period of time changed the direction of its movement, the path is greater than both the displacement of the body and the modulus of the change in the body's coordinates.

The path is always non-negative. It is equal to zero only if during the entire considered period of time the body was at rest (stood still).

Questions

  1. What is movement? What does it depend on?
  2. What is a path? What does it depend on?
  3. How does the path differ from moving and changing the coordinate for the same period of time during which the body moved in a straight line without changing the direction of movement?

Exercises

  1. Using the law of motion in graphical form, presented in fig. 47, describe the nature of the movement of the body (direction, speed) at different time intervals: from t0 to t1, from t1 to t2, from t2 to t3.
  2. The dog Proton ran out of the house at the time t0 = 0, and then at the command of its owner at the time t4 = 4 s rushed back. Knowing that the Proton ran in a straight line all the time and the modulus of its speed |v| \u003d 4 m / s, determine graphically: a) the change in the coordinates and the path of the Proton over the time interval from t0 \u003d 0 to t6 \u003d 6 s; b) the path of the Proton over the time interval from t2 = 2 s to t5 = 5 s.

Vectors, actions with vectors

Finding the length of a vector, examples and solutions.

By definition, a vector is a directed segment, and the length of this segment on a given scale is the length of the vector. Thus, the problem of finding the length of a vector in the plane and in space is reduced to finding the length of the corresponding segment. To solve this problem, we have at our disposal all the means of geometry, although in most cases it is enough pythagorean theorems. With its help, you can get a formula for calculating the length of a vector from its coordinates in a rectangular coordinate system, as well as a formula for finding the length of a vector from the coordinates of its start and end points. When a vector is a side of a triangle, then its length can be found from cosine theorem, if the lengths of the other two sides and the angle between them are known.

Finding the length of a vector by coordinates.

The length of the vector will be denoted by .

physical dictionary (kinematics)

The modulus of a number has a similar designation, and the length of a vector is often called the modulus of a vector.

Let's start by finding the length of the vector on the plane by the coordinates.

Let us introduce a rectangular Cartesian coordinate system Oxy on the plane. Let a vector be given in it and it has coordinates . Let's get a formula that allows you to find the length of the vector through the coordinates and .

Set aside from the origin of coordinates (from the point O) the vector . Let us denote the projections of the point A on the coordinate axes as and respectively, and consider a rectangle with a diagonal OA.

By virtue of the Pythagorean theorem, the equality , where . From the definition of the coordinates of a vector in a rectangular coordinate system, we can assert that and , and by construction, the length of OA is equal to the length of the vector , therefore, .

In this way, formula for finding the length of a vector in its coordinates on the plane has the form .

If the vector is represented as a decomposition in coordinate vectors , then its length is calculated by the same formula , since in this case the coefficients and are the coordinates of the vector in the given coordinate system.

Consider an example.

Find the length of the vector given in Cartesian coordinates.

Immediately apply the formula to find the length of the vector by coordinates :

Now we get a formula for finding the length of a vector by its coordinates in the Oxyz rectangular coordinate system in space.

We set aside the vector from the origin and denote the projections of point A on the coordinate axes as and . Then we can build on the sides and a rectangular parallelepiped, in which OA will be a diagonal.

In this case (since OA is the diagonal of a rectangular parallelepiped), whence . Determining the coordinates of the vector allows us to write the equalities , and the length OA is equal to the desired length of the vector, therefore, .

In this way, vector length in space is equal to the square root of the sum of the squares of its coordinates, that is, is found by the formula .

Calculate Vector Length , where are the orts of the rectangular coordinate system.

We are given the expansion of a vector in terms of coordinate vectors of the form , Consequently, . Then, according to the formula for finding the length of a vector by coordinates, we have .

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The length of the vector in terms of the coordinates of its start and end points.

But how to find the length of a vector if the coordinates of its start and end points are given?

In the previous paragraph, we received formulas for finding the length of a vector from its coordinates in the plane and in three-dimensional space. Then we can use them if we find the coordinates of the vector by the coordinates of its start and end points.

Thus, if points and are given on the plane, then the vector has coordinates and its length is calculated by the formula , and the formula for finding the length of the vector by the coordinates of the points and three-dimensional space has the form .

Let's consider examples.

Find the length of the vector if in a rectangular Cartesian coordinate system .

You can immediately apply the formula for finding the length of a vector by the coordinates of the start and end points on the plane :

The second solution is to determine the coordinates of the vector through the coordinates of the points and apply the formula :

.

Determine for what values ​​the length of the vector is , if .

The length of the vector by the coordinates of the start and end points can be found as

Equating the obtained value of the length of the vector to , we calculate the required ones:

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Finding the length of a vector using the cosine theorem.

Most problems on finding the length of a vector are solved in coordinates. However, when the coordinates of the vector are not known, one has to look for other solutions.

Let the lengths of two vectors be known, and the angle between them (or the cosine of the angle), and it is required to find the length of the vector or . In this case, using the law of cosines in the triangle ABC, you can calculate the length of the side BC, which is equal to the desired length of the vector.

Let's take a look at the solution of the example to clarify what has been said.

The lengths of the vectors and are 3 and 7, respectively, and the angle between them is . Calculate the length of the vector .

The length of the vector is equal to the length of the side BC in the triangle ABC. From the condition, we know the lengths of the sides AB and AC of this triangle (they are equal to the lengths of the corresponding vectors), as well as the angle between them, so we have enough data to apply the cosine theorem:

In this way, .

So, to find the length of a vector by coordinates, we use the formulas
or ,
by the coordinates of the start and end points of the vector —
or ,
in some cases, the cosine theorem leads to a result.

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  • Bugrov Ya.S., Nikolsky S.M. Higher Mathematics. Volume One: Elements of Linear Algebra and Analytic Geometry.
  • Atanasyan L.S., Butuzov V.F., Kadomtsev S.B., Poznyak E.G., Yudina I.I. Geometry. Grades 7 - 9: a textbook for educational institutions.
  • Atanasyan L.S., Butuzov V.F., Kadomtsev S.B., Kiseleva L.S., Poznyak E.G. Geometry. Textbook for 10-11 grades of high school.

Lecture Search

Vector scalar square

What happens if a vector is multiplied by itself?

The number is called scalar square vector , and are denoted as .

In this way, vector scalar squareis equal to the square of the length of the given vector:

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