Vector− purely mathematical concept, which is only used in physics or other applied sciences and which makes it possible to simplify the solution of some complex problems.
Vector− directed line segment.
I know elementary physics one has to operate with two categories of quantities − scalar and vector.
Scalar quantities (scalars) are quantities characterized by numerical value and sign. The scalars are the length − l, mass − m, path − s, time − t, temperature − T, electric charge − q, energy − W, coordinates, etc.
All apply to scalars. algebraic actions(addition, subtraction, multiplication, etc.).
Example 1.
Determine the total charge of the system, consisting of the charges included in it, if q 1 \u003d 2 nC, q 2 \u003d -7 nC, q 3 \u003d 3 nC.
Full system charge
q \u003d q 1 + q 2 + q 3 \u003d (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, for the definition of which it is necessary to specify, in addition to the numerical value, the direction as well. Vectors − speed v, force F, momentum p, tension electric field E, magnetic induction B and etc.
The numerical value of the vector (modulus) is denoted by a letter without a vector symbol or the vector is enclosed between vertical lines r = |r|.
Graphically, the vector is represented by an arrow (Fig. 1),
The length of which in a given scale is equal to its modulus, and the direction coincides with the direction of the vector.
Two vectors are equal if their moduli and directions are the same.
Vector quantities are added geometrically (according to the rule of vector algebra).
Finding a vector sum given component vectors is called vector addition.
The addition of two vectors is carried out according to the parallelogram or triangle rule. Total vector
c = a + b
equal to the diagonal of the parallelogram built on the vectors a and b. Module it
с = √(a 2 + b 2 − 2abcosα) (Fig. 2). 
For α = 90°, c = √(a 2 + b 2 ) is the Pythagorean theorem.
The same vector c can be obtained by the triangle rule if from the end of the vector a postpone vector b. Closing vector c (connecting the beginning of the vector a and the end of the vector b) is the vector sum of terms (components of vectors a and b).
The resulting vector is found as the closing one of the broken line, the links of which are the constituent vectors (Fig. 3). 
Example 3.
Add two forces F 1 \u003d 3 N and F 2 \u003d 4 N, vectors F1 and F2 make angles α 1 \u003d 10 ° and α 2 \u003d 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 F1 and F2, as sides, and modulo equal to its length.
Vector modulus F find by the law of cosines
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 a
(α 2 − α 1) = 90°, then F = √(F 1 2 + F 2 2 ).
Angle that vector F is with the Ox axis, we find by the formula
α \u003d arctg ((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 the vector a onto the axis Ox (Oy) is a scalar value depending on the angle α between the direction of the vector a and axes Ox (Oy). (Fig. 5) 
Vector projections a on the Ox and Oy axes rectangular system coordinates. (Fig. 6) 
In order to avoid mistakes when determining the sign of the vector projection onto the axis, it is useful to remember next rule: if the direction of the component coincides with the direction of the axis, then the projection of the vector on 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) 
Vector subtraction is an addition in which a vector is added to the first vector, numerically equal to the second, oppositely directed
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, it is necessary 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 towards 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 sum, and the other d− vector differences a and b(Fig. 9).
Vector product a per scalar k equals vector b= k a, whose modulus is k times more module vector a, and the direction is the same as the direction a for positive k and the opposite for negative k.
Example 4.
Determine the momentum of a body with a mass of 2 kg moving at a speed of 5 m/s. (Fig. 10) 
body momentum p= m v; p = 2 kg.m/s = 10 kg.m/s and is directed towards the speed v.
Example 5.
The charge q = −7.5 nC is placed in an electric field with intensity E = 400 V/m. Find the modulus and direction of the force acting on the charge.
Strength equals F= q E. Since the charge is negative, the force vector is directed to the side, vector opposite E. (Fig. 11) 
Division vector a by a scalar k is equivalent to multiplying a by 1/k.
Dot product vectors a and b call the scalar "c" equal to the product modules of these vectors by the cosine of the angle between them
(a.b) = (b.a) = c,
с = ab.cosα (Fig. 12) 
Example 6.
To find a job constant force F = 20 N if displacement S = 7.5 m and angle α between force and displacement α = 120°.
The work of a force is by definition dot product forces and movements
A = (F.S) = FScosα = 20 H × 7.5 m × cos120° = −150 × 1/2 = −75 J.
vector art vectors a and b call vector c, numerically equal to the product of the modules of the vectors a and b, multiplied by the sine of the angle between them:
c = a × b = ,
c = 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 right screw rule (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 in the conductor is 10 A and it forms an angle α = 30 ° with the direction of the field.
Amp 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 the same and equal to a, if the modulus of their sum is: a) 0; b) 2a; c) a; d) a√(2); e) a√(3)?
Decision.
a) Two vectors are directed along the same straight line in opposite sides. The sum of these vectors is equal to zero. 
b) Two vectors are directed along the same 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 equal to a. The resulting vector is found by 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
a 2 + a 2 + 2acosα = 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
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 = a1 + a2 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 \u003d a 1 2 + a 2 2; c) a 1 + a 2 \u003d a 1 - a 2?
Decision.
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 by the law of cosines 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 performed if a 2− zero vector, then a 1 + a 2 = a 1 .
Answers. a) a 1 ||a 2; b) a 1 ⊥ a 2; in) 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 of the body so that their action balances the action of the first two forces?
Decision.
According to the condition 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. The resulting vector is determined by the cosine theorem for a parallelogram. For the first pair of forces:
F 1 2 + F 1 2 + 2F 1 F 1 cosα \u003d 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 parts 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.
Find the desired 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°.
The second way to solve.
Consider the projection of vectors onto the coordinate axis OX (Fig.). 
Using the ratio between the sides 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 value c so that |c a| = 7,5?
Decision.
c a= c( 3i − 4j) = 7,5
Vector modulus a will be equal to
a 2 = 3 2 + 4 2 , and a = ±5,
then from
c.(±5) = 7.5,
find that
c = ±1.5.
5. Vectors a 1 and a 2 come out of the origin and have Cartesian coordinates ends (6, 0) and (1, 4), respectively. Find a vector a 3 such that: a) a 1 + a 2 + a 3= 0; b) a 1 − a 2 + a 3 = 0.
Decision.
Let's draw the vectors in Cartesian system coordinates (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
a 1 + a 2 = −a 3.
Vector a 3 modulo will be equal to the total vector a1 + a2 but directed in the opposite direction. End vector coordinate a 3 is equal to (−7, −4), and the modulus
a 3 \u003d √ (7 2 + 4 2 ) \u003d 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
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
a 3 \u003d √ (5 2 + 4 2) \u003d 6.4.
6. The messenger travels 30 m to the north, 25 m to the east, 12 m to the south, and then in the building rises in an elevator to a height of 36 m. What is the distance traveled by him L and the displacement S?
Decision.
Let us depict the situation described in the problem on a plane on an arbitrary scale (Fig.). 
End of vector OA has coordinates 25 m to the east, 18 m to the north and 36 up (25; 18; 36). The path traveled by a person is
L = 30 m + 25 m + 12 m +36 m = 103 m.
The module of the displacement vector is found by 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 \u003d √ (25 2 + 18 2 + 36 2 ) \u003d 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 the angle β between the vectors a and c. The magnitudes of the vectors are a = 3.0 and b = 2.0.
Decision.
The length of the vector equal to the sum vectors a and b we 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 the simple trigonometric equation, we arrive at the expression
tgβ = bsinα/(a + bcosα),
hence,
β = arctg(bsinα/(a + bcosα)),
β = arctg(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 \u003d arccos ((a 2 + c 2 - b 2) / (2ac)) \u003d arccos ((3 2 + 4.4 2 - 2 2) / (2.3.4.4)) \u003d 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 injection γ
between a and d.
9. Find the projection of the vector a = 4.0i + 7.0j to a straight line whose direction makes an angle α = 30° with the Ox axis. Vector a and the line lie in the xOy plane.
10. Vector a makes an angle α = 30° with the straight line AB, a = 3.0. At what angle β to the 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; c = i + 3j. find a) a+b; b) a+c; in) (a,b); G) (a, c)b − (a, b)c.
12. Angle between vectors a and b equals 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 a y = 2.0. Vector b perpendicular to the vector a and b = 3.0 (see figure). 
Vector c = a + b. Find: a) vector projections b on the Ox and Oy axes; b) the value c and the angle β between the vector c and axis Ox; c) (a, b); d) (a, c).
Answers:
9. a 1 \u003d 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 + 9k.
12. c = 2.6; d = 1.7.
14. α = 44.4°.
15. a) b x \u003d -1.5; b y = 2.6; b) c = 5; β ≈ 67°; c) 0; d) 16.0.
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By vector it is customary to understand a quantity that has 2 main characteristics:
- module;
- direction.
So, two vectors are recognized as equal if the modules, as well as the directions of both, coincide. The value under consideration is most often written as a letter, over which an arrow is drawn.
Among the most common quantities of the corresponding type are speed, force, and also, for example, acceleration.
With geometric point of view, a vector can be a directed segment, the length of which is related to its modulus.
If we consider vector quantity apart from the direction, it can in principle be measured. True, this will be, one way or another, a partial characteristic of the corresponding value. Full - is achieved only if it is supplemented with the parameters of the directed segment.
What is a scalar value?
By scalar it is customary to understand a value that has only 1 characteristic, namely - numerical value. In this case, the considered value can take a positive or negative value.
Common scalar quantities include mass, frequency, voltage, temperature. With them it is possible to produce various mathematical operations- addition, subtraction, multiplication, division.
Direction (as a characteristic) is not characteristic of scalar quantities.
Comparison
The main difference between a vector quantity and a scalar quantity is that the first key features- module and direction, the second - a numerical value. It is worth noting that a vector quantity, like a scalar one, can in principle be measured, however, in this case, its characteristics will be determined only partially, since there will be a lack of direction.
Having determined what is the difference between a vector and a scalar quantity, we will reflect the conclusions in a small table.
Two words that frighten a schoolboy - 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, let's 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 stand for? This will be discussed below. Let's try to figure out how it differs from scalar.
Vector examples. How are they labeled
What is meant by vector? That which characterizes movement. It doesn't matter if it's in space or on a plane. What is a vector quantity? For example, an airplane flies at a certain speed at a certain height, has a specific mass, and starts moving from the airport with the required acceleration. What is the movement of an aircraft? What made him fly? Of course, acceleration, speed. The vector quantities from the physics course are good 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? Movement is carried out due to not volume or mass, namely speed. The tennis player allows the ball to move with the help of a racket. It sets the acceleration. By the way, attached to this case force is also a vector quantity. Because it is obtained as a result of given speeds and accelerations. Force is also capable of changing, specific actions. The wind that shakes the leaves on the trees can also be considered an example. Because there is speed.
Positive and negative values
A vector quantity is a quantity that has a direction in the surrounding space and a module. The frightening word appeared again, this time module. Imagine that you need to solve a problem where the negative value of acceleration will be fixed. In nature negative values does not seem to exist. How can speed be negative?
A 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 is equal to reaction. The guys are pulling the rope. One team is in blue jerseys, the other is in yellow jerseys. The second ones are stronger. Assume that the vector of their force is directed positively. At the same time, the former fail to pull the rope, but they try. There is an opposing force.
Vector or scalar quantity?
Let's talk about the difference between a vector quantity and a scalar quantity. Which parameter has no direction, but has its own meaning? Let's list some scalars below:
Do they all have direction? No. Which quantity is vector and which is scalar can only be shown 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 also a vector quantity.
Vector and scalar in formulas
In physics textbooks, there are often formulas in which there is an arrow on top. Remember Newton's second law. Force ("F" with an arrow above) equals the product of mass ("m") and acceleration ("a" with an arrow above). As mentioned above, force and acceleration are vector quantities, but mass is scalar.
Unfortunately, not all publications have the designation of these quantities. Probably, this was done to simplify, so as not to mislead schoolchildren. It is best to buy those books and reference books that indicate vectors in formulas.
The illustration will show which quantity is a vector. It is recommended to pay attention to pictures and diagrams in physics lessons. Vector quantities have a direction. Where it is directed Of course, down. So the arrow will be shown in the same direction.
AT technical universities study physics in depth. Within many disciplines, teachers talk about which quantities are scalar and vector. Such knowledge is required in the areas: construction, transport, natural sciences.
Quantities are called scalar (scalars) if, after choosing a unit of measure, they are completely characterized by one number. Examples of scalar quantities are angle, surface, volume, mass, density, electric charge, resistance, temperature.
Two types of scalars should be distinguished: 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. Temperature and mass are examples of pure scalars.
3.1.2. Pseudoscalars.
Like pure scalars, pseudoscalars are defined with a single number, absolute value 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, cuboid, the projections of the edges of which onto 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, then the determinant will change sign. Volume is a pseudoscalar. Pseudoscalars are also angle, area, surface. 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.
The axis is an infinite straight line on which the positive direction is chosen. Let such a straight line, and the direction from
considered positive. Consider a segment on this straight line and assume that the number measuring the length is 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 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. We will call rotation about the 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 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.
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 modulo value and direction are the same. To designate a vector quantity, letters are most often used, over which an arrow is displayed. 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 a geometric point of view. A vector is a line segment that has a direction. The length of such a segment corresponds to the value of its module. physical example vector quantity is the displacement material point moving in space. Parameters such as the acceleration of this point, the speed and the forces acting on it, 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 result will display only partial characteristics of the value. For her full measurement the value should be supplemented with other parameters of the directed segment.
In vector algebra, there is a concept zero vector . Under this concept is meant a point. As for the direction of the zero vector, it is considered indefinite. The zero vector is denoted by the arithmetic zero 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 values. Equality means a complete match in all respects.
What is a scalar value?
Unlike a vector, a scalar quantity has only one parameter - it is its numerical value. It should be noted that the analyzed value can have both a positive numerical value and a negative one.
Examples include mass, voltage, frequency, or temperature. With these values, you can perform various arithmetic operations: addition, division, subtraction, multiplication. For a scalar quantity, such a characteristic as direction is not characteristic.
A scalar quantity is measured by a numeric value, so it can be displayed on coordinate axis. For example, very often they build the axis of the distance traveled, temperature or time.
Main differences between scalar and vector quantities
From the descriptions given above, it can be seen that the main difference between vector quantities and scalar quantities lies in their characteristics. A vector quantity has a direction and a modulus, while a scalar quantity has only a numerical value. Of course, a vector quantity, like a scalar one, can be measured, but such a characteristic will not be complete, since there is no direction.
In order to more clearly present the difference between a scalar quantity and a vector quantity, an example should be given. To do this, we take such a field of knowledge as climatology. If we say that the wind is blowing at a speed of 8 meters per second, then a scalar value will be introduced. But, if we say that the north wind blows at a speed of 8 meters per second, then we will talk about the vector value.
Vectors play huge role in modern mathematics, as well as in many areas of mechanics and physics. Majority physical quantities can be represented as vectors. This makes it possible to generalize and substantially simplify the formulas and results used. Often vector values and vectors are identified with each other. For example, in physics one hears that speed or force is a vector.
