Values are called scalar (scalars) if they, after choosing a unit of measurement, are completely characterized by one number. Examples of scalars are angle, surface, volume, mass, density, electric charge, resistance, temperature.
A distinction should be made between two types of scalars: pure scalars and pseudoscalars.
3.1.1. Pure scalars.
Pure scalars are completely defined by one number, independent of the choice of the reference axes. Temperature and mass are examples of pure scalars.
3.1.2. Pseudoscalars.
Like pure scalars, pseudoscalars are defined using a single number, the absolute value of which does not depend on the choice of the reference axes. However, the sign of this number depends on the choice of positive directions on the coordinate axes.
Consider, for example, a rectangular parallelepiped, the projections of the edges of which on the rectangular coordinate axes are, respectively, equal.The volume of this parallelepiped is determined using the determinant
the absolute value of which does not depend on the choice of rectangular coordinate axes. However, if you change the positive direction on one of the coordinate axes, the determinant will change its sign. Volume is a pseudoscalar. Angle, area, surface are also pseudoscalars. Below (Sec. 5.1.8) we will see that the pseudoscalar is actually a tensor of a special kind.
Vector quantities
3.1.3. Axis.
The axis is an infinite straight line with a positive direction. Let such a straight line, and the direction from
considered positive. Consider a segment on this line and assume that the number measuring the length is equal to a (Fig. 3.1). Then the algebraic length of the segment is equal to a, the algebraic length of the segment is equal to - a.
If we take several parallel lines, then by defining a positive direction on one of them, we thereby define it on the rest. The situation is different if the lines are not parallel; then it is necessary to make special arrangements regarding the choice of the positive direction for each straight line.
3.1.4. Direction of rotation.
Let the axis. Rotation about the axis will be called positive or direct if it is carried out for an observer standing along the positive direction of the axis, right and left (Fig. 3.2). Otherwise, it is called negative or inverse.
3.1.5. Direct and reverse trihedrons.
Let some trihedron (rectangular or non-rectangular). Positive directions are chosen on the axes, respectively, from O to x, from O to y and from O to z.
Vector quantity
Vector quantity- a physical quantity that is a vector (tensor of rank 1). It is opposed, on the one hand, to scalar (rank 0 tensors), and on the other, to tensor quantities (strictly speaking, tensors of rank 2 or more). It can also be opposed to certain objects of a completely different mathematical nature.
In most cases, the term vector is used in physics to denote a vector in the so-called "physical space", i.e. in ordinary three-dimensional space in classical physics or in four-dimensional space-time in modern physics (in the latter case, the concept of a vector and a vector quantity coincide with the concept of a 4-vector and a 4-vector quantity).
The use of the phrase "vector quantity" is practically exhausted by this. As for the use of the term "vector", despite the default gravitation towards the same field of applicability, in a large number of cases it still goes far beyond such a framework. See below about this.
Use of terms vector and vector quantity in physics
In general, in physics, the concept of a vector almost completely coincides with that in mathematics. However, there is a terminological specificity associated with the fact that in modern mathematics this concept is somewhat excessively abstract (in relation to the needs of physics).
In mathematics, when pronouncing "vector" one understands rather a vector in general, i.e. any vector of any arbitrarily abstract linear space of any dimension and nature, which, if no special efforts are made, can even lead to confusion (not so much, of course, in essence, as for the convenience of word use). If it is necessary to concretize, in the mathematical style it is necessary either to speak rather long ("vector of such and such space"), or to keep in mind what is implied by the explicitly described context.
In physics, on the other hand, it is almost always not about mathematical objects (possessing certain formal properties) in general, but about their certain specific ("physical") binding. Taking into account these considerations of concreteness with considerations of brevity and convenience, one can understand that the terminological practice in physics differs markedly from the mathematical one. However, it is not in obvious contradiction with the latter. This can be achieved with a few simple "tricks". First of all, they include the convention on the use of the term by default (when the context is not specifically specified). So, in physics, unlike mathematics, the word vector without additional clarifications is usually understood not as "some vector of any linear space in general", but first of all a vector associated with "ordinary physical space" (three-dimensional space of classical physics or four-dimensional space - the time of relativistic physics). For vectors of spaces that are not directly and directly related to "physical space" or "space-time", they use special names (sometimes including the word "vector", but with clarification). If a vector of some space that is not directly and directly related to "physical space" or "space-time" (and which is difficult to immediately somehow definitely characterize) is introduced into theory, it is often specifically described as an "abstract vector".
All that has been said to an even greater extent than to the term "vector" refers to the term "vector quantity". The default in this case even more rigidly implies a binding to "ordinary space" or space-time, and the use of abstract vector spaces with respect to elements is hardly ever encountered, at least such an application is seen as a rare exception (if not a reservation at all).
In physics, vectors most often, and vector quantities - almost always - are vectors of two similar classes:
Examples of vector physical quantities: speed, force, heat flux.
Genesis of vector quantities
How are physical "vector quantities" tied to space? First of all, it is striking that the dimension of vector quantities (in the usual sense of the use of this term, which is explained above) coincides with the dimension of the same "physical" (and "geometric") space, for example, space is three-dimensional and the vector of electric fields are three-dimensional. Intuitively, one can also notice that any vector physical quantity, no matter how vaguely connected it has with the usual spatial extent, nevertheless has a completely definite direction in this ordinary space.
However, it turns out that much more can be achieved by directly "reducing" the entire set of vector quantities of physics to the simplest "geometric" vectors, or rather even to one vector - the vector of elementary displacement, and it would be more correct to say - by producing all of them from it.
This procedure has two different (although essentially repeating each other in detail) realizations for the three-dimensional case of classical physics and for the four-dimensional space-time formulation, which is common in modern physics.
Classic 3D case
We will proceed from the usual three-dimensional "geometric" space in which we live and can move.
Let us take the vector of infinitesimal displacement as the initial and exemplary vector. It is pretty obvious that this is a normal "geometric" vector (like the final displacement vector).
We now note right away that multiplying a vector by a scalar always gives a new vector. The same can be said about the sum and difference of vectors. In this chapter, we will not differentiate between polar and axial vectors, so note that the cross product of two vectors also gives a new vector.
Also, the new vector gives the differentiation of the vector with respect to the scalar (since such a derivative is the limit of the ratio of the difference of vectors to the scalar). This can be said further about the derivatives of all higher orders. The same is true for integration over scalars (time, volume).
Now, note that, based on the radius vector r or from an elementary displacement d r, we easily understand that vectors are (since time is a scalar) kinematic quantities such as
From speed and acceleration, multiplied by a scalar (mass), appear
Since we are now also interested in pseudovectors, we note that
- using the Lorentz force formula, the electric field strength and the magnetic induction vector are tied to the force and velocity vectors.
Continuing this procedure, we find that all the vector quantities we know are now not only intuitive, but also formally tied to the original space. Namely, all of them, in a sense, are its elements, since are expressed in essence as linear combinations of other vectors (with scalar factors, possibly dimensional, but scalar, and therefore formally quite legal).
Modern four-dimensional case
The same procedure can be done based on 4D displacement. It turns out that all 4-vector quantities "originate" from 4-displacement, therefore being in some sense the same space-time vectors as the 4-displacement itself.
Types of vectors as applied to physics
- A polar or true vector is an ordinary vector.
- Axial vector (pseudovector) - in fact, is not a real vector, but formally it hardly differs from the latter, except that it changes direction to the opposite when the orientation of the coordinate system is changed (for example, when the coordinate system is mirrored). Examples of pseudo vectors: all quantities defined by the cross product of two polar vectors.
- There are several different equivalence classes for forces.
Notes (edit)
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See what "Vector magnitude" is in other dictionaries:
vector quantity- - [Ya.N. Luginsky, M.S.Fezi Zhilinskaya, Y.S.Kabirov. English Russian Dictionary of Electrical Engineering and Power Engineering, Moscow, 1999] Subjects of electrical engineering, basic concepts EN vector quantity ... Technical translator's guide
vector quantity- vektorinis dydis statusas T sritis automatika atitikmenys: angl. vector quantity; vectorial quantity vok. Vektorgröße, f; vektorielle Größe, f rus. vector quantity, f pranc. grandeur vectorielle, f… Automatikos terminų žodynas
vector quantity- vektorinis dydis statusas T sritis fizika atitikmenys: angl. vector quantity; vectorial quantity vok. Vektorgröße, f; vektorielle Größe, f rus. vector quantity, f pranc. grandeur vectorielle, f… Fizikos terminų žodynas
Graphical representation of quantities varying according to the law of sine (cosine) and the relationship between them using directional segments of vectors. Vector diagrams are widely used in electrical engineering, acoustics, optics, vibration theory, etc. ... ... Wikipedia
Force request is redirected here; see also other meanings. Force Dimension LMT − 2 SI units ... Wikipedia
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This is a quantity that, as a result of experience, takes on one of many values, and the appearance of one or another value of this quantity before its measurement cannot be accurately predicted. The formal mathematical definition is as follows: let it be probabilistic ... ... Wikipedia
Vector and scalar functions of coordinates and time, which are characteristics of the electromagnetic field. Vector P. e. called vector quantity A, the rotor to the swarm is equal to the vector B of the magnetic induction of the field; rotA V. Scalar P. e. called scalar value φ, ... ... Big Encyclopedic Polytechnic Dictionary
The value that characterizes the rotate. the effect of force when it acts on tv. body. Distinguish M. with. relative to the center (point) and relative to the main. M. s. relative to the center O (Fig. a) is a vector quantity numerically equal to the product of the modulus of force F by ... ... Natural science. encyclopedic Dictionary
A vector value that characterizes the rate of change in the speed of a point in terms of its numerical value and in direction. In the rectilinear motion of a point, when its speed υ increases (or decreases) uniformly, numerically Y in time: ... ... Great Soviet Encyclopedia
We are surrounded by many different material objects. Material, because it is possible to touch, smell, see, hear, and much more that can be done. What these objects are, what happens to them, or will happen if you do something: throw, straighten, shove into the oven. Why is something happening to them and how exactly does it happen? All this is studying physics... Play a game: guess the object in the room, describe it in a few words, a friend must guess what it is. I indicate the characteristics of the intended subject. Adjectives: white, large, heavy, cold. Have you guessed? This is a refrigerator. The listed characteristics are not scientific measurements of your refrigerator. You can measure different things at the refrigerator. If the length, then it is large. If the color is white. If the temperature is cold. And if its mass, then it turns out that it is heavy. We imagine that one refrigerator can be explored from different angles. Mass, length, temperature - this is a physical quantity.
But this is just that small characteristic of the refrigerator that comes to mind instantly. Before buying a new refrigerator, you can also familiarize yourself with a number of physical quantities that allow you to judge whether it is better or worse, and why it is more expensive. Imagine the scale of how diverse everything around us is. And how varied the characteristics are.
Physical quantity designation
All physical quantities are usually denoted by letters, more often than the Greek alphabet. BUT! One and the same physical quantity can have several letter designations (in different literature).
And, conversely, the same letter can denote different physical quantities. 
Despite the fact that you might not have encountered such a letter, the meaning of a physical quantity, its participation in formulas remains the same.
Vector and scalar quantities
In physics, there are two types of physical quantities: vector and scalar. Their main difference is that vector physical quantities have the direction... What does a physical quantity mean has a direction? For example, the number of potatoes in a bag, we will call ordinary numbers, or scalars. Temperature is another example of such a quantity. Other very important quantities in physics have direction, for example, speed; we must specify not only the speed of movement of the body, but also the path along which it moves. Momentum and force also have direction, as well as displacement: when someone takes a step, you can tell not only how far he has stepped, but also where he is walking, that is, determine the direction of his movement. Vector values are best remembered.
Why is an arrow being drawn over the letters?
Draw an arrow only above the letters of vector physical quantities. According to the way in mathematics they denote vector! Addition and subtraction operations on these physical quantities are performed according to the mathematical rules for operations with vectors. The expression "modulus of speed" or "absolute value" means precisely "modulus of the velocity vector", that is, the numerical value of the speed without taking into account the direction - the sign "plus" or "minus".
Designation of vector quantities
The main thing to remember
1) What is a vector quantity;
2) How the scalar value differs from the vector value;
3) Vector physical quantities;
4) Designation of a vector quantity
The two words that scare the student - vector and scalar - are not really scary. If you approach the topic with interest, then everything can be understood. In this article, we will consider which quantity is vector and which is scalar. More precisely, we will give examples. Each student, probably, paid attention to the fact that in physics some quantities are indicated not only by a symbol, but also by an arrow from above. What do they mean? This will be discussed below. Let's try to figure out how it differs from a scalar one.
Examples of vectors. How are they designated
What is meant by a vector? What characterizes the movement. It doesn't matter whether in space or on a plane. What quantity is vector in general? For example, an airplane flies at a certain speed at a certain altitude, has a specific mass, and starts moving from the airport with the required acceleration. What is related to aircraft movement? What made him fly? Acceleration, speed, of course. Vector quantities from the physics course are illustrative examples. To put it bluntly, a vector quantity is associated with movement, displacement.
Water also moves at a certain speed from the height of the mountain. See? The movement is carried out not by volume or mass, but by speed. The tennis player allows the ball to move with the racket. It sets the acceleration. By the way, the force applied in this case is also a vector quantity. Because it is obtained due to the given speeds and accelerations. Strength is also capable of changing, carrying out concrete actions. The wind that sways the leaves in the trees is also an example. Since there is speed.
Positive and negative values
A vector quantity is a quantity that has a direction in the surrounding space and a modulus. The scary word appeared again, this time module. Imagine that you need to solve a problem where a negative acceleration value will be recorded. It would seem that negative meanings do not exist in nature. How can speed be negative?
The vector has such a concept. This applies, for example, to forces that are applied to the body, but have different directions. Remember the third where action equals reaction. The guys are pulling the rope. One team in blue shirts, the other in yellow. The latter are stronger. Let's assume that their force vector is positive. At the same time, the former cannot pull the rope, but they try. An opposing force arises.
Vector or scalar?
Let's talk about the difference between a vector value and a scalar one. Which parameter has no direction, but has its own meaning? Let's list some scalar values below:
Do they all have direction? No. Which quantity is vector and which is scalar can be shown only by illustrative examples. In physics, there are such concepts not only in the section "Mechanics, dynamics and kinematics", but also in the paragraph "Electricity and magnetism". The Lorentz force is all vector quantities.
Vector and scalar in formulas
In physics textbooks, there are often formulas that have an arrow on top. Remember Newton's second law. Force ("F" with an arrow on top) is equal to the product of mass ("m") and acceleration ("a" with an arrow on top). As mentioned above, force and acceleration are vector quantities, but mass is scalar.
Unfortunately, not all publications have a designation for these values. Probably, this was done to simplify, so that schoolchildren would not be misled. It is best to buy those books and reference books in which vectors are indicated in formulas.
The illustration will show which value is vector. It is recommended to pay attention to pictures and diagrams in physics lessons. Vector quantities have a direction. Where is directed Of course, down. This means that the arrow will be shown in the same direction.
Physics is studied in depth in technical universities. In many disciplines, teachers talk about which quantities are scalar and vector. Such knowledge is required in the areas: construction, transport, natural sciences.
When studying various branches of physics, mechanics and technical sciences, there are quantities that are completely determined by specifying their numerical values, more precisely, which are completely determined using the number obtained as a result of their measurement by a homogeneous quantity taken as a unit. Such quantities are called scalar or, in short, scalars. Scalar quantities, for example, are length, area, volume, time, mass, body temperature, density, work, electrical capacity, etc. Since a scalar quantity is determined by a number (positive or negative), it can be plotted on the corresponding coordinate axis. For example, they often build an axis of time, temperature, length (distance traveled) and others.
In addition to scalar quantities, in various problems there are quantities, for the determination of which, in addition to the numerical value, it is also necessary to know their direction in space. Such quantities are called vector... Physical examples of vector quantities can be the displacement of a material point moving in space, the speed and acceleration of this point, as well as the force acting on it, the strength of the electric or magnetic field. Vector quantities are used, for example, in climatology. Consider a simple example from climatology. If we say that the wind is blowing at a speed of 10 m / s, then we will thereby introduce a scalar value of the wind speed, but if we say that the north wind is blowing at a speed of 10 m / s, then in this case the wind speed will already be a vector quantity.
Vector quantities are depicted using vectors.
For the geometric representation of vector quantities, directional segments are used, that is, segments with a fixed direction in space. In this case, the length of the segment is equal to the numerical value of the vector quantity, and its direction coincides with the direction of the vector quantity. The directional segment characterizing a given vector quantity is called geometric vector or just vector.
The concept of a vector plays an important role both in mathematics and in many areas of physics and mechanics. Many physical quantities can be represented using vectors, and this representation very often helps to generalize and simplify formulas and results. Vector quantities and vectors representing them are often identified with each other: for example, they say that force (or speed) is a vector.
Elements of vector algebra are used in such disciplines as: 1) electrical machines; 2) automated electric drive; 3) electric lighting and irradiation; 4) undeveloped alternating current circuits; 5) applied mechanics; 6) theoretical mechanics; 7) physics; 8) hydraulics: 9) machine parts; 10) sopromat; 11) management; 12) chemistry; 13) kinematics; 14) statics, etc.
2. Definition of the vector. A straight line segment is specified by two equal points - its ends. But you can consider a directed segment defined by an ordered pair of points. About these points it is known which of them is the first (beginning), and which is the second (end).
A directed segment is understood as an ordered pair of points, the first of which, point A, is called its beginning, and the second, B, is called its end.
Then under vector in the simplest case, the directed segment itself is understood, and in other cases, different vectors are different equivalence classes of directed segments, determined by some specific equivalence relation. Moreover, the equivalence relation can be different, determining the type of the vector ("free", "fixed", etc.). Simply put, within an equivalence class, all directed line segments included in it are treated as perfectly equal, and each can equally represent the entire class.
Vectors play an important role in the study of infinitesimal transformations of space.
Definition 1. A directed segment (or, which is the same, an ordered pair of points) we will call vector... The direction on the segment is usually marked with an arrow. An arrow is placed above the letter designation of the vector when writing, for example: (in this case, the letter corresponding to the beginning of the vector must be placed in front). In books, vector letters are often typed in bold, for example: a.
The so-called zero vector, in which the beginning and the end coincide, will also be referred to vectors.
A vector whose beginning coincides with its end is called zero. The zero vector is denoted or just 0.
The distance between the beginning and the end of the vector is called its the length(and module and absolute value). The length of the vector is denoted by | | or | |. The length of the vector, or the modulus of the vector, is the length of the corresponding directed segment: | | =.
The vectors are called collinear, if they are located on one straight line or on parallel lines, in short, if there is a line to which they are parallel.
The vectors are called coplanar, if there is a plane to which they are parallel, they can be represented by vectors lying on the same plane. A null vector is considered collinear to any vector, since it has no definite direction. Its length, of course, is zero. Obviously, any two vectors are coplanar; but, of course, not every three vectors in space are coplanar. Since vectors parallel to each other are parallel to the same plane, collinear vectors are even more coplanar. Of course, the converse is not true: coplanar vectors may or may not be collinear. By virtue of the above condition, the zero vector is collinear with any vector and coplanar with any pair of vectors, i.e. if at least one of the three vectors is zero, then they are coplanar.
2) The word "coplanar" means in essence: "having a common plane", that is, "located in the same plane." But since we are talking here about free vectors that can be transferred (without changing the length and direction) in an arbitrary way, we must call coplanar vectors parallel to the same plane, because in this case they can be transferred so that they are located in one plane.
To shorten the speech, let's agree in one term: if several free vectors are parallel to the same plane, then we will say that they are coplanar. In particular, two vectors are always coplanar; to be convinced of this, it is enough to postpone them from the same point. It is clear, further, that the direction of the plane in which two given vectors are parallel is quite definite if these two vectors are not parallel to each other. Any plane to which these coplanar vectors are parallel will be referred to simply as the plane of these vectors.
Definition 2. The two vectors are called equal if they are collinear, the same direction, and have equal lengths.
It must always be remembered that the equality of the lengths of two vectors does not mean the equality of these vectors.
By the very meaning of the definition, two vectors, which are separately equal to the third, are equal to each other. Obviously, all zero vectors are equal to each other.
This definition directly implies that, having chosen any point A ", we can construct (and, moreover, only one) vector A" B "equal to some given vector, or, as they say, transfer the vector to point A".
Comment... For vectors, there is no concept of "more" or "less", i.e. they are equal or not equal.
A vector whose length is equal to one is called single vector and is denoted by e. The unit vector, the direction of which coincides with the direction of the vector a, is called orthom vector and is denoted by a.
3. On another definition of a vector... Note that the concept of equality of vectors differs significantly from the concept of equality, for example, of numbers. Each number is equal only to itself, in other words, two equal numbers under all circumstances can be considered the same number. With vectors, as we can see, the situation is different: by definition, there are different, but equal vectors. Although in most cases we will not need to distinguish between them, it may well turn out that at some point we will be interested in just the vector, and not another, equal to it vector A "B".
In order to simplify the concept of equality of vectors (and remove some of the difficulties associated with it), sometimes they go to complicate the definition of a vector. We will not use this complicated definition, but we will formulate it. To avoid confusion, we will write "Vector" (with a capital letter) to denote the concept defined below.
Definition 3... Let a directed segment be given. The set of all directed segments equal to a given one in the sense of Definition 2 is called Vector.
Thus, each directed line segment defines a Vector. It is easy to see that two directed segments define the same Vector if and only if they are equal. For Vectors, as well as for numbers, equality means coincidence: two Vectors are equal if and only if they are one and the same Vector.
With a parallel space transfer, a point and its image form an ordered pair of points and define a directed segment, and all such directed segments are equal in the sense of Definition 2. Therefore, a parallel space transfer can be identified with a Vector composed of all these directed segments.
It is well known from the initial physics course that force can be represented by a directional segment. But it cannot be depicted by a Vector, since the forces depicted by equal directed segments perform, generally speaking, different actions. (If the force acts on an elastic body, then the directed segment representing it cannot be transferred even along the straight line on which it lies.)
This is just one of the reasons why, along with the Vectors, that is, sets (or, as they say, classes) of equal directed segments, it is necessary to consider individual representatives of these classes. In these circumstances, the application of Definition 3 is complicated by the large number of reservations. We will adhere to Definition 1, and in the general sense it will always be clear whether we are talking about a well-defined vector, or any one equal to it can be substituted in its place.
In connection with the definition of the vector, it is worth explaining the meaning of some words found in the literature.