A vector is usually understood as a quantity that has 2 main characteristics:
- module;
- direction.
Thus, two vectors are considered equal if the modules, as well as the directions of both, coincide. The value in question is most often written as a letter with an arrow drawn above it.
Among the most common quantities of the corresponding type are speed, force, and also, for example, acceleration.
From a geometric point of view, a vector can be a directed segment, the length of which correlates with its module.
If we consider a vector quantity separately from its direction, then it can in principle be measured. True, this will be, one way or another, a partial characteristic of the corresponding quantity. Full - is achieved only if it is supplemented with the parameters of the directional segment.
What is a scalar quantity?
By scalar we usually mean a quantity that has only one characteristic, namely a numerical value. In this case, the value under consideration can take a positive or negative value.
Common scalar quantities include mass, frequency, voltage, and temperature. With them it is possible to perform various mathematical operations - addition, subtraction, multiplication, division.
Direction (as a characteristic) is not typical for scalar quantities.
Comparison
The main difference between a vector quantity and a scalar quantity is that the first has key characteristics - magnitude and direction, while the second has a numerical value. It is worth noting that a vector quantity, like a scalar quantity, can in principle be measured, however, in this case its characteristics will only be partially determined, since there will be a lack of direction.
Having determined what the difference is between a vector and a scalar quantity, we will display the conclusions in a small table.
The two words that frighten schoolchildren - vector and scalar - are not actually 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. Every student probably noticed that in physics some quantities are denoted not only by a symbol, but also by an arrow on top. What do they mean? This will be discussed below. Let's try to figure out how it differs from scalar.
Examples of vectors. How are they designated?
What is meant by vector? That which characterizes movement. It doesn't matter whether in space or on a plane. What quantity is a vector quantity in general? For example, an airplane flies at a certain speed at a certain altitude, has a specific mass, and began moving from the airport with the required acceleration. What is the motion of an airplane? What made him fly? Of course, acceleration, speed. Vector quantities from the physics course are clear examples. To put it bluntly, a vector quantity is associated with motion, displacement.
Water also moves at a certain speed from the height of the mountain. Do you see? Movement is carried out not by volume or mass, but by speed. A tennis player allows the ball to move with the help of a racket. It sets the acceleration. By the way, the force applied in this case is also a vector quantity. Because it is obtained as a result of given speeds and accelerations. Power can also change and carry out specific actions. The wind that moves the leaves on the trees can also be considered an example. Because there is speed.
Positive and negative quantities
A vector quantity is a quantity that has a direction in the surrounding space and a magnitude. The scary word appeared again, this time module. Imagine that you need to solve a problem where a negative acceleration value will be recorded. In nature, negative meanings, it would seem, do not 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 playing tug of war. One team wears blue T-shirts, the other team wears yellow T-shirts. The latter turn out to be stronger. Let us assume that their force vector is directed positively. At the same time, the first ones cannot pull the rope, but they try. An opposing force arises.
Vector or scalar quantity?
Let's talk about how a vector quantity differs from a scalar quantity. Which parameter has no direction, but has its own meaning? Let's list some scalar quantities below:
Do they all have a direction? No. Which quantity is vector and which is scalar can only be shown with visual 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
Physics textbooks often contain formulas that have an arrow at the 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 the designation of these quantities. This was probably done to simplify things so that schoolchildren would not be misled. It is best to buy those books and reference books that indicate vectors in formulas.
The illustration will show which quantity is a vector one. It is recommended to pay attention to pictures and diagrams in physics lessons. Vector quantities have a direction. Where is it directed? Of course, down. This means that the arrow will be shown in the same direction.
Physics is studied in depth at technical universities. In many disciplines, teachers talk about what quantities are scalar and vector. Such knowledge is required in the following areas: construction, transport, natural sciences.
Vector- a purely mathematical concept that is only used in physics or other applied sciences and which allows one 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 a quadratic equation of the form
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, force F, impulse p, electric field strength E, magnetic induction B and 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) 
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 between 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 is equal to the 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 is S = 7.5 m, and the angle α between the force and the displacement is 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 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 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 sides in a 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.
We find the magnitude of the displacement vector 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.
The length of the vector equal to the sum of the 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; c = 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.
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 a geometric point of view. 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 the displacement of a 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 either a positive numerical value or 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 with a numerical value, so it can be displayed on a 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.
