Vector− a purely mathematical concept that is only used in physics or other applied sciences and which allows you to simplify the solution of some complex problems.
Vector− directed straight segment.
In a course of elementary physics one has to operate with two categories of quantities − scalar and vector.
Scalar quantities (scalars) are quantities characterized by a numerical value and sign. The scalars are length − l, mass − m, path − s, time − t, temperature − T, electric charge − q, energy − W, coordinates, etc.
All algebraic operations (addition, subtraction, multiplication, etc.) apply to scalar quantities.
Example 1.
Determine the total charge of the system, consisting of the charges included in it, if q 1 = 2 nC, q 2 = −7 nC, q 3 = 3 nC.
Full system charge
q = q 1 + q 2 + q 3 = (2 − 7 + 3) nC = −2 nC = −2 × 10 −9 C.
Example 2.
For quadratic equation kind
ax 2 + bx + c = 0;
x 1,2 = (1/(2a)) × (−b ± √(b 2 − 4ac)).
Vector Quantities (vectors) are quantities, to determine which it is necessary to indicate, in addition to the numerical value, the direction. Vectors − speed v, strength F, impulse p, electric field strength E, magnetic induction B etc.
The numerical value of a vector (modulus) is denoted by a letter without a vector symbol or the vector is enclosed between vertical bars r = |r|.
Graphically, the vector is represented by an arrow (Fig. 1),
The length of which on a given scale is equal to its magnitude, and the direction coincides with the direction of the vector.
Two vectors are equal if their magnitudes and directions coincide.
Vector quantities are added geometrically (according to the rule of vector algebra).
Finding a vector sum from given component vectors is called vector addition.
The addition of two vectors is carried out according to the parallelogram or triangle rule. Sum vector
c = a + b
equal to the diagonal of a parallelogram built on vectors a And b. Module it
с = √(a 2 + b 2 − 2abcosα) (Fig. 2). 
At α = 90°, c = √(a 2 + b 2 ) is the Pythagorean theorem.
The same vector c can be obtained using the triangle rule if from the end of the vector a set aside vector b. Trailing vector c (connecting the beginning of the vector a and the end of the vector b) is the vector sum of terms (component vectors a And b).
The resulting vector is found as the trailing line of the broken line whose links are the component vectors (Fig. 3). 
Example 3.
Add two forces F 1 = 3 N and F 2 = 4 N, vectors F 1 And F 2 make angles α 1 = 10° and α 2 = 40° with the horizon, respectively
F = F 1 + F 2(Fig. 4). 
The result of the addition of these two forces is a force called the resultant. Vector F directed along the diagonal of a parallelogram built on vectors F 1 And F 2, both sides, and is equal in modulus to its length.
Vector module F find by the cosine theorem
F = √(F 1 2 + F 2 2 + 2F 1 F 2 cos(α 2 − α 1)),
F = √(3 2 + 4 2 + 2 × 3 × 4 × cos(40° − 10°)) ≈ 6.8 H.
If
(α 2 − α 1) = 90°, then F = √(F 1 2 + F 2 2 ).
Angle which is vector F is equal to the Ox axis, we find it using the formula
α = arctan((F 1 sinα 1 + F 2 sinα 2)/(F 1 cosα 1 + F 2 cosα 2)),
α = arctan((3.0.17 + 4.0.64)/(3.0.98 + 4.0.77)) = arctan0.51, α ≈ 0.47 rad.
The projection of vector a onto the Ox (Oy) axis is a scalar quantity depending on the angle α between the direction of the vector a and Ox (Oy) axis. (Fig. 5) 
Vector projections a on the Ox and Oy axes of the rectangular coordinate system. (Fig. 6) 
In order to avoid mistakes when determining the sign of the projection of a vector onto an axis, it is useful to remember the following rule: if the direction of the component coincides with the direction of the axis, then the projection of the vector onto this axis is positive, but if the direction of the component is opposite to the direction of the axis, then the projection of the vector is negative. (Fig. 7) 
Subtraction of vectors is an addition in which a vector is added to the first vector, numerically equal to the second, in the opposite direction
a − b = a + (−b) = d(Fig. 8). 
Let it be necessary from the vector a subtract vector b, their difference − d. To find the difference of two vectors, you need to go to the vector a add vector ( −b), that is, a vector d = a − b will be a vector directed from the beginning of the vector a to the end of the vector ( −b) (Fig. 9). 
In a parallelogram built on vectors a And b both sides, one diagonal c has the meaning of the sum, and the other d− vector differences a And b(Fig. 9).
Product of a vector a by scalar k equals vector b= k a, whose modulus is k times greater than the modulus of the vector a, and the direction coincides with the direction a for positive k and the opposite for negative k.
Example 4.
Determine the momentum of a body weighing 2 kg moving at a speed of 5 m/s. (Fig. 10) 
Body impulse p= m v; p = 2 kg.m/s = 10 kg.m/s and directed towards the speed v.
Example 5.
A charge q = −7.5 nC is placed in an electric field with a strength of E = 400 V/m. Find the magnitude and direction of the force acting on the charge.
The force is F= q E. Since the charge is negative, the force vector is directed in the direction opposite to the vector E. (Fig. 11) 
Division vector a by a scalar k is equivalent to multiplying a by 1/k.
Dot product vectors a And b called the scalar “c”, equal to the product of the moduli of these vectors and the cosine of the angle between them
(a.b) = (b.a) = c,
с = ab.cosα (Fig. 12) 
Example 6.
Find the work done by a constant force F = 20 N, if the displacement S = 7.5 m, and the angle α between the force and the displacement α = 120°.
The work done by a force is equal, by definition, to the scalar product of force and displacement
A = (F.S) = FScosα = 20 H × 7.5 m × cos120° = −150 × 1/2 = −75 J.
Vector artwork vectors a And b called a vector c, numerically equal to the product of the absolute values of vectors a and b multiplied by the sine of the angle between them:
c = a × b = ,
с = ab × sinα.
Vector c perpendicular to the plane in which the vectors lie a And b, and its direction is related to the direction of the vectors a And b the rule of the right screw (Fig. 13). 
Example 7.
Determine the force acting on a conductor 0.2 m long, placed in a magnetic field, the induction of which is 5 T, if the current strength in the conductor is 10 A and it forms an angle α = 30° with the direction of the field.
Ampere power
dF = I = Idl × B or F = I(l)∫(dl × B),
F = IlBsinα = 5 T × 10 A × 0.2 m × 1/2 = 5 N.
Consider problem solving.
1. How are two vectors directed, the moduli of which are identical and equal to a, if the modulus of their sum is equal to: a) 0; b) 2a; c) a; d) a√(2); e) a√(3)?
Solution.
a) Two vectors are directed along one straight line in opposite directions. The sum of these vectors is zero. 
b) Two vectors are directed along one straight line in the same direction. The sum of these vectors is 2a. 
c) Two vectors are directed at an angle of 120° to each other. The sum of the vectors is a. The resulting vector is found using the cosine theorem: 
a 2 + a 2 + 2aacosα = a 2 ,
cosα = −1/2 and α = 120°.
d) Two vectors are directed at an angle of 90° to each other. The modulus of the sum is equal to
a 2 + a 2 + 2aacosα = 2a 2 ,
cosα = 0 and α = 90°. 
e) Two vectors are directed at an angle of 60° to each other. The modulus of the sum is equal to
a 2 + a 2 + 2aacosα = 3a 2 ,
cosα = 1/2 and α = 60°.
Answer: The angle α between the vectors is equal to: a) 180°; b) 0; c) 120°; d) 90°; e) 60°.
2. If a = a 1 + a 2 orientation of vectors, what can be said about the mutual orientation of vectors a 1 And a 2, if: a) a = a 1 + a 2 ; b) a 2 = a 1 2 + a 2 2 ; c) a 1 + a 2 = a 1 − a 2?
Solution.
a) If the sum of vectors is found as the sum of the modules of these vectors, then the vectors are directed along one straight line, parallel to each other a 1 ||a 2.
b) If the vectors are directed at an angle to each other, then their sum is found using the cosine theorem for a parallelogram
a 1 2 + a 2 2 + 2a 1 a 2 cosα = a 2 ,
cosα = 0 and α = 90°.
vectors are perpendicular to each other a 1 ⊥ a 2.
c) Condition a 1 + a 2 = a 1 − a 2 can be executed if a 2− zero vector, then a 1 + a 2 = a 1 .
Answers. A) a 1 ||a 2; b) a 1 ⊥ a 2; V) a 2− zero vector.
3. Two forces of 1.42 N each are applied to one point of the body at an angle of 60° to each other. At what angle should two forces of 1.75 N each be applied to the same point on the body so that their action balances the action of the first two forces?
Solution.
According to the conditions of the problem, two forces of 1.75 N each balance two forces of 1.42 N each. This is possible if the modules of the resulting vectors of force pairs are equal. We determine the resulting vector using the cosine theorem for a parallelogram. For the first pair of forces:
F 1 2 + F 1 2 + 2F 1 F 1 cosα = F 2 ,
for the second pair of forces, respectively
F 2 2 + F 2 2 + 2F 2 F 2 cosβ = F 2 .
Equating the left sides of the equations
F 1 2 + F 1 2 + 2F 1 F 1 cosα = F 2 2 + F 2 2 + 2F 2 F 2 cosβ.
Let's find the required angle β between the vectors
cosβ = (F 1 2 + F 1 2 + 2F 1 F 1 cosα − F 2 2 − F 2 2)/(2F 2 F 2).
After calculations,
cosβ = (2.1.422 + 2.1.422.cos60° − 2.1.752)/(2.1.752) = −0.0124,
β ≈ 90.7°.
Second solution.
Let's consider the projection of vectors onto the coordinate axis OX (Fig.). 
Using the relationship between the parties in right triangle, we get
2F 1 cos(α/2) = 2F 2 cos(β/2),
where
cos(β/2) = (F 1 /F 2)cos(α/2) = (1.42/1.75) × cos(60/2) and β ≈ 90.7°.
4. Vector a = 3i − 4j. What must be the scalar quantity c for |c a| = 7,5?
Solution.
c a= c( 3i − 4j) = 7,5
Vector module a will be equal
a 2 = 3 2 + 4 2 , and a = ±5,
then from
c.(±5) = 7.5,
let's find that
c = ±1.5.
5. Vectors a 1 And a 2 exit from the origin and have Cartesian end coordinates (6, 0) and (1, 4), respectively. Find the vector a 3 such that: a) a 1 + a 2 + a 3= 0; b) a 1 − a 2 + a 3 = 0.
Solution.
Let's depict the vectors in the Cartesian coordinate system (Fig.) 
a) The resulting vector along the Ox axis is
a x = 6 + 1 = 7.
The resulting vector along the Oy axis is
a y = 4 + 0 = 4.
For the sum of vectors to be equal to zero, it is necessary that the condition be satisfied
a 1 + a 2 = −a 3.
Vector a 3 modulo will be equal to the total vector a 1 + a 2, but directed in the opposite direction. Vector end coordinate a 3 is equal to (−7, −4), and the modulus
a 3 = √(7 2 + 4 2) = 8.1.
B) The resulting vector along the Ox axis is equal to
a x = 6 − 1 = 5,
and the resulting vector along the Oy axis
a y = 4 − 0 = 4.
When the condition is met
a 1 − a 2 = −a 3,
vector a 3 will have the coordinates of the end of the vector a x = –5 and a y = −4, and its modulus is equal to
a 3 = √(5 2 + 4 2) = 6.4.
6. A messenger walks 30 m to the north, 25 m to the east, 12 m to the south, and then takes an elevator to a height of 36 m in a building. What is the distance L traveled by him and the displacement S?
Solution.
Let us depict the situation described in the problem on a plane on an arbitrary scale (Fig.). 
End of vector O.A. has coordinates 25 m to the east, 18 m to the north and 36 up (25; 18; 36). The distance traveled by a person is equal to
L = 30 m + 25 m + 12 m +36 m = 103 m.
The magnitude of the displacement vector can be found using the formula
S = √((x − x o) 2 + (y − y o) 2 + (z − z o) 2 ),
where x o = 0, y o = 0, z o = 0.
S = √(25 2 + 18 2 + 36 2) = 47.4 (m).
Answer: L = 103 m, S = 47.4 m.
7. Angle α between two vectors a And b equals 60°. Determine the length of the vector c = a + b and angle β between vectors a And c. The magnitudes of the vectors are a = 3.0 and b = 2.0.
Solution.
Vector length, equal to the amount vectors a And b Let's determine using the cosine theorem for a parallelogram (Fig.). 
с = √(a 2 + b 2 + 2abcosα).
After substitution
c = √(3 2 + 2 2 + 2.3.2.cos60°) = 4.4.
To determine the angle β, we use the sine theorem for triangle ABC:
b/sinβ = a/sin(α − β).
At the same time, you should know that
sin(α − β) = sinαcosβ − cosαsinβ.
Solving a simple trigonometric equation, we arrive at the expression
tgβ = bsinα/(a + bcosα),
hence,
β = arctan(bsinα/(a + bcosα)),
β = arctan(2.sin60/(3 + 2.cos60)) ≈ 23°.
Let's check using the cosine theorem for a triangle:
a 2 + c 2 − 2ac.cosβ = b 2 ,
where
cosβ = (a 2 + c 2 − b 2)/(2ac)
And
β = arccos((a 2 + c 2 − b 2)/(2ac)) = arccos((3 2 + 4.4 2 − 2 2)/(2.3.4.4)) = 23°.
Answer: c ≈ 4.4; β ≈ 23°.
Solve problems.
8. For vectors a And b defined in Example 7, find the length of the vector d = a − b corner γ
between a And d.
9. Find the projection of the vector a = 4.0i + 7.0j to a straight line, the direction of which makes an angle α = 30° with the Ox axis. Vector a and the straight line lie in the xOy plane.
10. Vector a makes an angle α = 30° with straight line AB, a = 3.0. At what angle β to straight line AB should the vector be directed? b(b = √(3)) so that the vector c = a + b was parallel to AB? Find the length of the vector c.
11. Three vectors are given: a = 3i + 2j − k; b = 2i − j + k; с = i + 3j. Find a) a+b; b) a+c; V) (a, b); G) (a, c)b − (a, b)c.
12. Angle between vectors a And b is equal to α = 60°, a = 2.0, b = 1.0. Find the lengths of the vectors c = (a, b)a + b And d = 2b − a/2.
13. Prove that the vectors a And b are perpendicular if a = (2, 1, −5) and b = (5, −5, 1).
14. Find the angle α between the vectors a And b, if a = (1, 2, 3), b = (3, 2, 1).
15. Vector a makes an angle α = 30° with the Ox axis, the projection of this vector onto the Oy axis is equal to a y = 2.0. Vector b perpendicular to the vector a and b = 3.0 (see figure). 
Vector c = a + b. Find: a) projections of the vector b on the Ox and Oy axis; b) the value of c and the angle β between the vector c and the Ox axis; c) (a, b); d) (a, c).
Answers:
9. a 1 = a x cosα + a y sinα ≈ 7.0.
10. β = 300°; c = 3.5.
11. a) 5i + j; b) i + 3j − 2k; c) 15i − 18j + 9 k.
12. c = 2.6; d = 1.7.
14. α = 44.4°.
15. a) b x = −1.5; b y = 2.6; b) c = 5; β ≈ 67°; c) 0; d) 16.0.
By studying physics, you have great opportunities to continue your education at a technical university. This will require a parallel deepening of knowledge in mathematics, chemistry, language, and less often other subjects. The winner of the Republican Olympiad, Savich Egor, graduates from one of the faculties of MIPT, where great demands are placed on knowledge in chemistry. If you need help at the State Academy of Sciences in chemistry, then contact the professionals; you will definitely receive qualified and timely assistance.
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 a 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, the axis of time, temperature, length (distance traveled), and others are often constructed.
In addition to scalar quantities, in various problems there are quantities for which, in addition to their numerical value, it is also necessary to know their direction in space. Such quantities are called vector. Physical examples of vector quantities include displacement 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. Let's look at a simple example from climatology. If we say that the wind is blowing at a speed of 10 m/s, then we will introduce a scalar value of 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 represented using vectors.
For the geometric representation of vector quantities, directed segments are used, that is, segments that have a fixed direction in space. In this case, the length of the segment is equal to the numerical value vector quantity, and its direction coincides with the direction of the vector quantity. The directed segment characterizing a given vector quantity is called geometric vector or just a 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 contributes to the generalization and simplification of formulas and results. Often vector quantities and the vectors representing them are 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) unbranched AC circuits; 5) applied mechanics; 6) theoretical mechanics; 7) physics; 8) hydraulics: 9) machine parts; 10) strength of materials; 11) management; 12) chemistry; 13) kinematics; 14) statics, etc.
2. Definition of a vector. A straight line segment is defined by two equal points - its ends. But we can consider a directed segment defined by an ordered pair of points. It is known about these points 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 - 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 vector (“free”, “fixed”, etc.). Simply put, within an equivalence class, all directed segments included in it are treated as completely equal, and each can equally represent the entire class.
Vectors play an important role in the study of infinitesimal transformations of space.
Definition 1. We will call a directed segment (or, what is the same, an ordered pair of points) vector. The direction on the segment is usually marked with an arrow. Over letter designation when writing a vector, an arrow is placed, for example: (in this case, the letter corresponding to the beginning of the vector must be placed in front). In books, letters denoting a vector are often typed in bold, for example: A.
We will also include the so-called zero vector, whose beginning and end coincide, as vectors.
A vector whose beginning coincides with its end is called zero. The zero vector is denoted simply as 0.
The distance between the start and end of a vector is called its length(and also module and absolute value). The length of the vector is denoted by | | or | |. The length of a vector, or the modulus of a vector, is the length of the corresponding directed segment: | | = .
The vectors are called collinear, if they are located on the same 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. The null vector is considered collinear to any vector, since it has no specific 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 not be collinear. By virtue of the condition adopted above, the zero vector is collinear with any vector and coplanar with any pair of vectors, i.e. if among three vectors at least one is zero, then they are coplanar.
2) The word “coplanar” essentially means: “having a common plane,” i.e., “located in the same plane.” But since we are talking here about free vectors that can be transferred (without changing length and direction) in an arbitrary way, we must call vectors parallel to the same plane coplanar, 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 completely defined if these two vectors are not parallel to each other. We will simply call any plane to which these coplanar vectors are parallel the plane of these vectors.
Definition 2. The two vectors are called equal, if they are collinear, have the same direction and have equal lengths.
You must always remember that the equality of the lengths of two vectors does not mean that these vectors are equal.
By the very meaning of the definition, two vectors that are separately equal to the third are equal to each other. Obviously, all zero vectors are equal to each other.
From this definition it immediately follows that by choosing any point A", we can construct (and, moreover, only one) vector A" B", equal to some given vector, or, as they say, move the vector to point A."
Comment. For vectors there are no concepts 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. A unit vector whose direction coincides with the direction of vector a is called ortom vector and is denoted a.
3. About 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 as the same number. With vectors, as we 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 the vector , and not in another, equal 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 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 for numbers, equality means coincidence: two Vectors are equal if and only if they are the same Vector.
In parallel transfer of space, 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, parallel transfer of space can be identified with a Vector composed of all these directed segments.
It is well known from the initial physics course that a force can be represented by a directed segment. But it cannot be represented by a Vector, since forces represented by equal directed segments produce, generally speaking, different actions. (If a 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 only one of the reasons why, along with Vectors, i.e. sets (or, as they say, classes) of equal directed segments, it is necessary to consider individual representatives of these classes. Under these circumstances, the application of Definition 3 becomes more difficult a large number reservations We will adhere to Definition 1, and according to general sense it will always be clear whether we are talking about a well-defined vector, or whether anyone equal to it can be substituted in its place.
In connection with the definition of a vector, it is worth explaining the meaning of some words found in the literature.
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.
Vector quantity (vector) is a physical quantity that has two characteristics - modulus and direction in space.
Examples of vector quantities: speed (), force (), acceleration (), etc.
Geometrically, a vector is depicted as a directed segment of a straight line, the length of which on a scale is the absolute value of the vector.
Radius vector(usually denoted or simply) - a vector that specifies the position of a point in space relative to some pre-fixed point, called the origin.
For an arbitrary point in space, the radius vector is the vector going from the origin to that point.
The length of the radius vector, or its modulus, determines the distance at which the point is located from the origin, and the arrow indicates the direction to this point in space.
On a plane, the angle of the radius vector is the angle by which the radius vector is rotated relative to the x-axis in a counterclockwise direction.
the line along which a body moves is called trajectory of movement. Depending on the shape of the trajectory, all movements can be divided into rectilinear and curvilinear.
The description of movement begins with an answer to the question: how has the position of the body in space changed over a certain period of time? How is a change in the position of a body in space determined?
Moving- a directed segment (vector) connecting the initial and final position of the body.
Speed(often denoted , from English. velocity or fr. vitesse) is a vector physical quantity that characterizes the speed of movement and direction of movement of a material point in space relative to the selected reference system (for example, angular velocity). The same word can be used to refer to a scalar quantity, or more precisely, the modulus of the derivative of the radius vector.
Science also uses speed in in a broad sense, as the speed of change of some quantity (not necessarily the radius vector) depending on another (usually changes in time, but also in space or any other). For example, they talk about the rate of temperature change, the rate chemical reaction, group speed, connection speed, angular speed, etc. Mathematically characterized by the derivative of the function.
Acceleration(usually denoted in theoretical mechanics), the derivative of speed with respect to time is a vector quantity showing how much the speed vector of a point (body) changes as it moves per unit time (i.e. acceleration takes into account not only the change in the magnitude of the speed, but also its direction ).
For example, near the Earth, a body falling on the Earth, in the case where air resistance can be neglected, increases its speed by approximately 9.8 m/s every second, that is, its acceleration is equal to 9.8 m/s².
The branch of mechanics that studies motion in three-dimensional Euclidean space, its recording, as well as the recording of velocities and accelerations in various reference systems, is called kinematics.
The unit of acceleration is meters per second per second ( m/s 2, m/s 2), there is also a non-system unit Gal (Gal), used in gravimetry and equal to 1 cm/s 2.
Derivative of acceleration with respect to time i.e. the quantity characterizing the rate of change of acceleration over time is called jerk.
The simplest movement of a body is one in which all points of the body move equally, describing the same trajectories. This movement is called progressive. We obtain this type of motion by moving the splinter so that it remains parallel to itself at all times. During forward motion, trajectories can be either straight (Fig. 7, a) or curved (Fig. 7, b) lines.
It can be proven that during translational motion, any straight line drawn in the body remains parallel to itself. It is convenient to use this characteristic feature to answer the question of whether a given body movement is translational. For example, when a cylinder rolls along a plane, straight lines intersecting the axis do not remain parallel to themselves: rolling is not a translational motion. When the crossbar and square move along the drawing board, any straight line drawn in them remains parallel to itself, which means they move forward (Fig. 8). The needle of a sewing machine, the piston in the cylinder of a steam engine or engine moves progressively internal combustion, car body (but not wheels!) when driving on a straight road, etc.
Another simple type of movement is rotational movement body, or rotation. During rotational motion, all points of the body move in circles whose centers lie on a straight line. This straight line is called the axis of rotation (straight line 00" in Fig. 9). The circles lie in parallel planes perpendicular to the axis of rotation. The points of the body lying on the axis of rotation remain stationary. Rotation is not a translational movement: when the axis rotates OO" . The trajectories shown remain parallel only straight lines parallel to the axis of rotation.
Absolutely solid body- the second supporting object of mechanics along with the material point.
There are several definitions:
1. An absolutely rigid body is a model concept of classical mechanics, denoting a set of material points, the distances between which are maintained during any movements performed by this body. In other words, an absolutely solid body not only does not change its shape, but also maintains the distribution of mass inside unchanged.
2. An absolutely rigid body is a mechanical system that has only translational and rotational degrees of freedom. “Hardness” means that the body cannot be deformed, that is, no other energy can be transferred to the body other than the kinetic energy of translational or rotational motion.
3. Absolutely solid- a body (system), the relative position of any points of which does not change, no matter what processes it participates in.
In three-dimensional space and in the absence of connections, an absolutely rigid body has 6 degrees of freedom: three translational and three rotational. The exception is a diatomic molecule or, in the language of classical mechanics, a solid rod of zero thickness. Such a system has only two rotational degrees of freedom.
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An unproven and unrefuted hypothesis is called an open problem.
Physics is closely related to mathematics; mathematics provides an apparatus with the help of which physical laws can be precisely formulated.. theory Greek consideration.. standard method of testing theories direct experimental verification experiment criterion of truth however often..
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The principle of relativity in mechanics
Inertial reference systems and the principle of relativity. Galileo's transformations. Transformation invariants. Absolute and relative speeds and accelerations. Postulates of special technology
Rotational motion of a material point.
Rotational motion of a material point is the movement of a material point in a circle. Rotational movement is a type of mechanical movement. At
Relationship between the vectors of linear and angular velocities, linear and angular accelerations.
A measure of rotational motion: the angle φ through which the radius vector of a point rotates in a plane normal to the axis of rotation. Uniform rotational motion
Speed and acceleration during curved motion.
Curvilinear movement more complex look movement than a rectilinear one, since even if the movement occurs on a plane, two coordinates that characterize the position of the body change. Speed and
Acceleration during curved motion.
Considering curvilinear movement body, we see that its speed is different at different moments. Even in the case when the magnitude of the speed does not change, there is still a change in the direction of the speed
Newton's equation of motion
(1) where the force F in the general case
Center of mass
center of inertia, geometric point, the position of which characterizes the distribution of masses in a body or mechanical system. The coordinates of the central mass are determined by the formulas
Law of motion of the center of mass.
Using the law of momentum change, we obtain the law of motion of the center of mass: dP/dt = M∙dVc/dt = ΣFi The center of mass of the system moves in the same way as
Galileo's principle of relativity
· Inertial reference system Galileo's inertial reference system
Plastic deformation
Bend the steel plate (for example, a hacksaw) a little, and then release it after a while. We will see that the hacksaw will completely (at least in appearance) restore its shape. If we take
EXTERNAL AND INTERNAL FORCES
. In mechanics external forces in relation to a given system of material points (i.e., such a set of material points in which the movement of each point depends on the positions or movements of all axes
Kinetic energy
the energy of a mechanical system, depending on the speed of movement of its points. K. e. T of a material point is measured by half the product of the mass m of this point by the square of its speed
Kinetic energy.
Kinetic energy is the energy of a moving body. (From the Greek word kinema - movement). By definition, the kinetic energy of something at rest in a given frame of reference
A value equal to half the product of a body's mass and the square of its speed.
=J. Kinetic energy is a relative quantity, depending on the choice of CO, because the speed of the body depends on the choice of CO. That.
moment of force
· Moment of force. Rice. Moment of power. Rice. Moment of force, quantities
Kinetic energy of a rotating body
Kinetic energy is an additive quantity. Therefore, the kinetic energy of a body moving in an arbitrary manner is equal to the sum of the kinetic energies of all n materials
Work and power during rotation of a rigid body.
Work and power during rotation of a rigid body. Let's find an expression for work at temp
Basic equation for the dynamics of rotational motion
According to equation (5.8), Newton’s second law for rotational motion P
In mathematics, a vector is a directed segment of a certain length. In physics, a vector quantity is understood as full description some physical quantity that has a modulus and direction of action. Let's consider the basic properties of vectors, as well as examples of physical quantities that are vector.
Scalars and vectors
Scalar quantities in physics are parameters that can be measured and represented by a single number. For example, temperature, mass, and volume are scalars because they are measured in degrees, kilograms, and cubic meters, respectively.
In most cases, it turns out that the number defining a scalar quantity does not contain comprehensive information. For example, considering this physical characteristics, as acceleration, it will not be enough to say that it is equal to 5 m/s 2, since you need to know where it is directed, against the speed of the body, at some angle to this speed or otherwise. Besides acceleration, an example of a vector quantity in physics is velocity. Also included in this category are force, electric field strength, and much more.
According to the definition of a vector quantity as a segment directed in space, it can be represented as a set of numbers (vector components) if it is considered in a certain coordinate system. Most often in physics and mathematics problems arise that, to describe a vector, require knowledge of its two (problems on a plane) or three (problems in space) components.
Definition of a vector in n-dimensional space
In n-dimensional space, where n is an integer, a vector will be uniquely defined if its n components are known. Each component represents the coordinate of the end of the vector along the corresponding coordinate axis, provided that the beginning of the vector is at the origin of the coordinate system of n-dimensional space. As a result, the vector can be represented as follows: v = (a 1, a 2, a 3, ..., a n), where a 1 - scalar value 1st component of vector v. Accordingly, in 3-dimensional space the vector will be written as v = (a 1, a 2, a 3), and in 2-dimensional space - v = (a 1, a 2).
How is a vector quantity denoted? Any vector in 1-dimensional, 2-dimensional and 3-dimensional spaces can be represented as a directed segment lying between points A and B. In this case, it is denoted as AB →, where the arrow indicates that we are talking about a vector quantity. The sequence of letters is usually indicated from the beginning of the vector to its end. This means that if the coordinates of points A and B, for example, in 3-dimensional space, are equal to (x 1, y 1, z 1) and (x 2, y 2, z 2), respectively, then the components of the vector AB → will be equal (x 2 -x 1, y 2 -y 1, z 2 -z 1).
Graphical representation of the vector
In drawings, it is customary to depict a vector quantity as a segment; at its end there is an arrow indicating the direction of action of the physical quantity of which it is a representation. This segment is usually signed, for example, v → or F →, so that it is clear what characteristic we are talking about.
A graphical representation of a vector helps to understand where the physical quantity is applied and in what direction it acts. In addition, it is convenient to perform many mathematical operations on vectors using their images.
Mathematical operations on vectors
Vector quantities, just like regular numbers, can be added, subtracted, and multiplied with each other and with other numbers.
The sum of two vectors is understood as the third vector, which is obtained if the summed parameters are arranged so that the end of the first coincides with the beginning of the second vector, and then connect the beginning of the first and the end of the second. To perform this mathematical operation, three main methods have been developed:
- The parallelogram method consists of constructing geometric figure on two vectors that originate from the same point in space. The diagonal of this parallelogram, which extends from the common point of origin of the vectors, will be their sum.
- The polygon method, the essence of which is that the beginning of each subsequent vector should be located at the end of the previous one, then the total vector will connect the beginning of the first and the end of the last.
- An analytical method that consists of pairwise addition of the corresponding components of known vectors.
As for the difference in vector quantities, it can be replaced by adding the first parameter with the one that is opposite in direction to the second.
Multiplication of a vector by a certain number A is performed by simple rule: Each component of the vector should be multiplied by this number. The result is also a vector whose modulus is A times greater than the original one, and the direction is either the same or opposite to the original one, it all depends on the sign of the number A.
You cannot divide a vector or a number by it, but dividing a vector by the number A is similar to multiplying by the number 1/A.
Dot and cross product
Vector multiplication can be performed using two in various ways: scalar and vector.
The scalar product of vector quantities is a method of multiplying them, the result of which is one number, that is, a scalar. IN matrix form dot product is written as the row component of the 1st vector to the column component of the 2nd. As a result, in n-dimensional space we get the formula: (A → *B →) = a 1 *b 1 +a 2 *b 2 +...+a n *b n .
In 3-dimensional space, the dot product can be defined differently. To do this, you need to multiply the modules of the corresponding vectors by the cosine of the angle between them, that is, (A → *B →) = |A → |*|B → |*cos(θ AB). From this formula it follows that if the vectors are directed in the same direction, then the scalar product is equal to the multiplication of their modules, and if the vectors are perpendicular to each other, then it turns out to be zero. Note that the modulus of a vector in a rectangular coordinate system is defined as square root from the sum of the squares of the components of this vector.
The vector product is understood as the multiplication of a vector by a vector, the result of which is also a vector. Its direction turns out to be perpendicular to each of the multiplied parameters, and the length is equal to the product of the moduli of the vectors and the sine of the angle between them, that is, A → x B → = |A → |*|B → |*sin(θ AB), where the sign "x" denotes the vector product. In matrix form, this type of product is represented as a determinant, the rows of which are the elementary vectors of a given coordinate system and the components of each vector.
Both scalar and vector artwork used in mathematics and physics to determine many quantities, for example, the area and volume of figures.
Speed and acceleration
In physics, speed is understood as the rate of change in the location of a given material point. Speed is measured in SI units in meters per second (m/s), and is denoted by the symbol v → . Acceleration refers to the rate at which speed changes. Acceleration is measured in meters per square second (m/s2), and is usually denoted by the symbol a →. The value of 1 m/s2 means that for every second the body increases its speed by 1 m/s.
Velocity and acceleration are vector quantities that participate in the formulas of Newton's second law and the displacement of a body as a material point. Velocity is always directed along the direction of motion, but acceleration can be directed in any way relative to the moving body.
Physical quantity force
Force is a vector physical quantity that reflects the intensity of interaction between bodies. It is designated by the symbol F → and measured in newtons (N). By definition, 1 N is a force capable of changing the speed of a body with a mass of 1 kg by 1 m/s for every second of time.
This physical quantity is widely used in physics, since the energy characteristics of interaction processes are associated with it. The nature of the force can be very different, for example, the gravitational forces of planets, the force that makes a car move, the elastic forces of solid media, electrical forces that describe behavior electric charges, magnetic, nuclear forces that determine the stability of atomic nuclei, and so on.
Vector quantity pressure
Another quantity closely related to the concept of force is pressure. In physics, it is understood as the normal projection of force onto the area on which it acts. Since force is a vector, then, according to the rule of multiplying a number by a vector, pressure will also be a vector quantity: P → = F → /S, where S is the area. Pressure is measured in pascals (Pa), 1 Pa is the parameter at which a perpendicular force of 1 N acts on a surface of 1 m2. Based on the definition, the pressure vector is directed in the same direction as the force vector.
In physics, the concept of pressure is often used in the study of phenomena in liquids and gases (for example, Pascal's law or the ideal gas equation of state). Pressure is closely related to the temperature of a body, since the kinetic energy of atoms and molecules, the representation of which is temperature, explains the nature of the existence of pressure itself.
Electric field strength
There is an electric field around any charged body, the force characteristic of which is its intensity. This intensity is defined as the force acting at a given point in the electric field on a unit charge placed at this point. The electric field strength is denoted by the letter E → and is measured in newtons per coulomb (N/C). The intensity vector is directed along the electric field line in its direction if the charge is positive, and against it if the charge is negative.
The electric field strength created by a point charge can be determined at any point using Coulomb's law.
Magnetic induction
The magnetic field, as scientists Maxwell and Faraday showed in the 19th century, is closely related to the electric field. Thus, a changing electric field generates a magnetic field, and vice versa. Therefore, both types of fields are described in terms of electromagnetic physical phenomena.
Magnetic induction describes the force properties of a magnetic field. Is magnetic induction a scalar or vector quantity? This can be understood by knowing that it is determined through the force F → acting on a charge q, which flies at a speed v → in a magnetic field, according to the following formula: F → = q*|v → x B → |, where B → - magnetic induction. Thus, answering the question whether magnetic induction is a scalar or vector quantity, we can say that it is a vector that is directed from the north magnetic pole to the south. B is measured → in tesla (T).
Physical quantity candela
Another example of a vector quantity is the candela, which is introduced into physics as the luminous flux, measured in lumens, passing through a surface bounded by an angle of 1 steradian. Candela reflects the brightness of light because it indicates the density of luminous flux.