(tensors of rank 0), on the other hand, tensor quantities (strictly speaking, tensors of rank 2 or more). It can also be contrasted with 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”, that is, in the usual three-dimensional space of 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”, it, despite the default inclination to the same field of applicability, in large quantities cases still go very far beyond such limits. See below for details.
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 overly abstract (in relation to the needs of physics).
In mathematics, when pronouncing “vector,” they mean rather a vector in general, that is, any vector of any abstract linear space of any dimension and nature, which, unless special efforts are made, can even lead to confusion (not so much, of course, in essence, as for ease of use). If it is necessary to be more specific, in the mathematical style one has to either speak at quite a length (“vector of such and such a space”), or keep in mind what is implied by the explicitly described context.
In physics, we are almost always talking not about mathematical objects (possessing certain formal properties) in general, but about their specific (“physical”) connection. Taking these considerations of specificity into account with considerations of brevity and convenience, it can be understood that terminological practice in physics differs markedly from that of mathematics. However, it is not in obvious contradiction with the latter. This can be achieved with a few simple “tricks”. First of all, these include the agreement on the use of the term by default (when the context is not specifically specified). Thus, in physics, unlike mathematics, the word vector without additional clarification usually means not “some vector of any linear space in general,” but primarily a vector associated with “ordinary physical space” (the three-dimensional space of classical physics or the four-dimensional space -time of relativistic physics). For vectors of spaces that are not directly and directly related to “physical space” or “space-time”, special names are used (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 characterize in any specific way) is introduced into theory, it is often specifically described as an “abstract vector”.
Everything that was said in to a greater extent, than to the term "vector", refers to the term "vector quantity". The silence in this case even more strictly implies a binding to “ordinary space” or space-time, and the use of abstract vector spaces in relation to elements is almost never encountered, at least such a use seems to be the rarest exception (if not a reservation at all).
In physics, vectors most often, and vector quantities - almost always - are called vectors of two classes that are similar to each other:
Examples of vector physical quantities: speed, force, heat flow.
Genesis of vector quantities
How are physical “vector quantities” related to space? First of all, what is striking is that the dimension vector quantities(in the usual sense of using 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 electric field vector is three-dimensional. Intuitively, one can also notice that any vector physical quantity, no matter what vague connection it has with ordinary spatial extension, nevertheless has a very 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 deriving them all from it.
This procedure has two different (although essentially repeating each other in detail) implementations for the three-dimensional case of classical physics and for the four-dimensional space-time formulation common to modern physics.
Classic 3D case
We will start from the usual three-dimensional “geometric” space in which we live and can move.
Let's take the vector of infinitesimal displacement as the initial and reference vector. It's pretty obvious that this is a regular "geometric" vector (just like a finite displacement vector).
Let us now immediately note 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 make a difference between polar and axial vectors, so we note that both vector product two vectors 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 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 elementary displacement d r, we easily understand that vectors are (since time is a scalar) such kinematic quantities as
From speed and acceleration, multiplied by a scalar (mass), we get
Since we are now 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 discover that all vector quantities known to us are now not only intuitively, but also formally, tied to the original space. Namely, all of them, in a sense, are its elements, since they are expressed essentially as linear combinations of other vectors (with scalar factors, perhaps dimensional, but scalar, and therefore formally quite legal).
Quantities are called scalar (scalars) if, after choosing a unit of measurement, they are completely characterized by one number. Examples of scalar quantities are angle, surface, volume, mass, density, electric charge, resistance, temperature.
It is necessary to distinguish between two types of scalar quantities: pure scalars and pseudoscalars.
3.1.1. Pure scalars.
Pure scalars are completely defined by a single number, independent of the choice of reference axes. Examples of pure scalars are temperature and mass.
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 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 sign. Volume is a pseudoscalar. Angle, area, and surface are also pseudoscalars. Below (Section 5.1.8) we will see that a pseudoscalar is actually a tensor of a special kind.
Vector quantities
3.1.3. Axis.
An axis is an infinite straight line on which the positive direction is chosen. Let such a straight line, and the direction from
is considered positive. Let's 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, having determined the positive direction on one of them, we thereby determine it on the rest. The situation is different if the lines are not parallel; then you need to specifically agree on the choice of the positive direction for each straight line.
3.1.4. Direction of rotation.
Let the axis. We will call rotation about an axis positive or direct if it is carried out for an observer standing along the positive direction of the axis, to the right and to the left (Fig. 3.2). Otherwise it is called negative or inverse.
3.1.5. Direct and inverse trihedra.
Let it be some trihedron (rectangular or non-rectangular). Positive directions are selected on the axes respectively from O to x, from O to y and from O to z.
Scalar and vector quantities
- Vector calculus (for example, displacement (s), force (F), acceleration (a), velocity (V) energy (E)).
scalar quantities that are completely determined by specifying their numerical values (length (L), area (S), volume (V), time (t), mass (m), etc.);
- Scalar quantities: temperature, volume, density, electric potential, potential energy of a body (for example, in a gravity field). Also the modulus of any vector (for example, those listed below).
Vector quantities: radius vector, speed, acceleration, electric field strength, magnetic field strength. And many others :)
- a vector quantity has a numerical expression and direction: speed, acceleration, force, electromagnetic induction, displacement, etc., and a scalar quantity has only a numerical expression: volume, density, length, width, height, mass (not to be confused with weight), temperature
- vector, for example, speed (v), force (F), displacement (s), impulse (p), energy (E). An arrow-vector is placed above each of these letters. that's why they are vector. and scalar ones are mass (m), volume (V), area (S), time (t), height (h)
- Vector movements are linear, tangential movements.
Scalar motions are closed motions that screen vector motions.
Vector movements are transmitted through scalar ones, as through intermediaries, just as current is transmitted from atom to atom through a conductor. - Scalar quantities: temperature, volume, density, electric potential, potential energy of a body (for example, in a gravity field). Also the modulus of any vector (for example, those listed below).
Vector quantities: radius vector, speed, acceleration, electric field strength, magnetic field strength. And many others:-
- A scalar quantity (scalar) is a physical quantity that has only one characteristic: a numerical value.
A scalar quantity can be positive or negative.
Examples of scalar quantities: mass, temperature, path, work, time, period, frequency, density, energy, volume, electrical capacity, voltage, current, etc.
Mathematical operations with scalar quantities are algebraic operations.
Vector quantity
A vector quantity (vector) is a physical quantity that has two characteristics: module and direction in space.
Examples of vector quantities: speed, force, acceleration, tension, etc.
Geometrically, a vector is depicted as a directed segment of a straight line, the length of which is scaled to the modulus of the vector.
In physics courses, we often encounter quantities for which it is enough to know only numerical values to describe them. For example, mass, time, length.
Quantities that are characterized only by a numerical value are called scalar or scalars.
In addition to scalar quantities, quantities are used that have both a numerical value and a direction. For example, speed, acceleration, force.
Quantities that are characterized by numerical value and direction are called vector or vectors.
Vector quantities are indicated by the corresponding letters with an arrow at the top or in bold. For example, the force vector is denoted by \(\vec F\) or F . The numerical value of a vector quantity is called the modulus or length of the vector. The value of the force vector is denoted by F or \(\left|\vec F \right|\).
Vector image
Vectors are represented by directed segments. The beginning of the vector is the point from which the directed segment begins (point A in Fig. 1), the end of the vector is the point at which the arrow ends (point B in Fig. 1).
Rice. 1.
The two vectors are called equal, if they have the same length and are directed in the same direction. Such vectors are represented by directed segments having the same lengths and directions. For example, in Fig. 2 shows the vectors \(\vec F_1 =\vec F_2\).
Rice. 2. When two or more vectors are depicted in one drawing, the segments are constructed on a pre-selected scale. For example, in Fig. Figure 3 shows vectors whose lengths are \(\upsilon_1\) = 2 m/s, \(\upsilon_2\) = 3 m/s.
Rice. 3. Method for specifying a vector
On a plane, a vector can be specified in several ways:
1. Specify the coordinates of the beginning and end of the vector. For example, the vector \(\Delta\vec r\) in Fig. 4 is given by the coordinates of the beginning of the vector – (2, 4) (m), the end – (6, 8) (m).
Rice. 4. 2. Indicate the magnitude of the vector (its value) and the angle between the direction of the vector and some pre-selected direction on the plane. Often for such a direction in positive side axis 0 X. Angles measured from this direction counterclockwise are considered positive. In Fig. 5 vector \(\Delta\vec r\) is given by two numbers b and \(\alpha\) , indicating the length and direction of the vector.
Rice. 5. In physics, there are several categories of quantities: vector and scalar.
What is a vector quantity?
A vector quantity has two main characteristics: direction and module. Two vectors will be the same if their absolute value and direction are the same. To denote a vector quantity, letters with an arrow above them are most often used. An example of a vector quantity is force, velocity, or acceleration.
In order to understand the essence of a vector quantity, one should consider it from geometric point vision. A vector is a segment that has a direction. The length of such a segment correlates with the value of its modulus. A physical example of a vector quantity is displacement material point, moving in space. Parameters such as the acceleration of this point, the speed and forces acting on it, the electromagnetic field will also be displayed as vector quantities.
If we consider a vector quantity regardless of direction, then such a segment can be measured. But the resulting result will reflect only partial characteristics of the quantity. To fully measure it, the value should be supplemented with other parameters of the directional segment.
In vector algebra there is a concept zero vector. This concept means a point. As for the direction of the zero vector, it is considered uncertain. To denote the zero vector, the arithmetic zero is used, typed in bold.
If we analyze all of the above, we can conclude that all directed segments define vectors. Two segments will define one vector only if they are equal. When comparing vectors, the same rule applies as when comparing scalar quantities. Equality means complete agreement in all respects.
What is a scalar quantity?
Unlike a vector, a scalar quantity has only one parameter - this its numerical value. It is worth noting that the analyzed value can have both a positive numerical value and a negative one.
Examples include mass, voltage, frequency or temperature. With such quantities you can perform various arithmetic operations: addition, division, subtraction, multiplication. A scalar quantity does not have such a characteristic as direction.
A scalar quantity is measured by a numerical value, so it can be displayed on coordinate axis. For example, very often the axis of the distance traveled, temperature or time is constructed.
Main differences between scalar and vector quantities
From the descriptions given above, it is clear that the main difference between vector quantities and scalar quantities is their characteristics. A vector quantity has a direction and magnitude, while a scalar quantity has only a numerical value. Of course, a vector quantity, like a scalar quantity, can be measured, but such a characteristic will not be complete, since there is no direction.
In order to more clearly imagine the difference between a scalar quantity and a vector quantity, an example should be given. To do this, let’s take such an area of knowledge as climatology. If we say that the wind is blowing at a speed of 8 meters per second, then a scalar quantity will be introduced. But if we say that the north wind blows at a speed of 8 meters per second, then we are talking about a vector value.
Vectors play a huge role in modern mathematics, as well as in many areas of mechanics and physics. Most physical quantities can be represented as vectors. This allows us to generalize and significantly simplify the formulas and results used. Often vector values and vectors are identified with each other. For example, in physics you may hear that speed or force is a vector.