Scalar value in physics examples. Between the hammer and the anvil

Vector quantity (vector) is a physical quantity that has two characteristics - the modulus and the 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 module 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 arbitrary point in space, the radius vector is the vector from the origin to that point.

The length of the radius vector, or its modulus, determines the distance at which the point is 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 abscissa axis in a counterclockwise direction.

the line along which the 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 the movement begins with the answer to the question: how did the position of the body in space change over a certain period of time? How is the change in the position of the body in space determined?

moving- directed segment (vector) connecting the initial and final positions of the body.

Speed(often denoted, from English. velocity or fr. vitesse) - vector physical quantity characterizing the speed of movement and direction of movement material point in space relative to the selected reference system (for example, angular velocity). The same word can be scalar, more precisely, the modulus of the derivative of the radius vector.

Science also uses speed in broad sense, as the rate of change of some quantity (not necessarily the radius vector) depending on another (more often changes in time, but also in space or any other). So, for example, they talk about the rate of temperature change, the rate chemical reaction, group velocity, connection velocity, angular velocity, etc. Mathematically characterized by the derivative of the function.

Acceleration(usually denoted , in theoretical mechanics), the time derivative of velocity is a vector quantity showing how much the velocity vector of a point (body) changes when it moves per unit of time (i.e., acceleration takes into account not only a change in the magnitude of the velocity, but also its direction).

For example, near the Earth, a body falling to the Earth, in the case where air resistance can be neglected, increases its speed by about 9.8 m / s every second, that is, its acceleration is 9.8 m / s².

A branch of mechanics that studies motion in three-dimensional Euclidean space, its recording, as well as the recording of velocities and accelerations in various systems reference is called kinematics.

The unit of acceleration is meters per second per second ( m/s 2, m/s 2), there is also an off-system unit Gal (Gal), used in gravimetry and equal to 1 cm/s 2 .

Derivative of acceleration with respect to time i.e. The value characterizing the rate of change of acceleration over time is called jerk.

The simplest movement of the body is one in which all points of the body move in the same way, describing the same trajectories. Such a movement is called progressive. We get this type of movement by moving the splinter so that it remains parallel to itself all the time. With translational motion, the trajectories can be both straight (Fig. 7, a) and curved (Fig. 7, b) lines.
It can be proved that during translational motion any straight line drawn in the body remains parallel to itself. This hallmark it is convenient to use to answer the question of whether a given movement of the body is translational. For example, when a cylinder rolls along a plane, the lines intersecting the axis do not remain parallel to themselves: rolling is not translational motion. When the T-square 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 the sewing machine moves forward, the piston in the cylinder of the steam engine or engine internal combustion, car body (but not wheels!) when driving on a straight road, etc.

Another simple type of movement is rotary motion body, or rotation. During rotational motion, all points of the body move along circles whose centers lie on a straight line. This 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 motionless. Rotation is not progressive movement: when the axis is rotated OO". The paths shown remain parallel only straight lines parallel to the axis of rotation.

Absolutely rigid body- the second reference 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 preserved in the process of any movements performed by this body. In other words, an absolutely rigid body not only does not change its shape, but also keeps 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, except for the kinetic energy of the translational or rotary motion.

3. Absolutely solid- a body (system), the mutual position of any points of which does not change, no matter what processes it participates in.

AT three-dimensional space and in the absence of bonds, 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.

End of work -

This topic belongs to:

An unproven and undisproved hypothesis is called an open problem.

Physics is closely related to mathematics, mathematics provides the apparatus through which physical laws can be precisely formulated.. theory gr.

If you need additional material on this topic, or you did not find what you were looking for, we recommend using the search in our database of works:

What will we do with the received material:

If this material turned out to be useful for you, you can save it to your page on social networks:

tweet

All topics in this section:

The principle of relativity in mechanics
Inertial reference systems and the principle of relativity. Galilean transformations. Transformation invariants. Absolute and relative speeds and accelerations. Postulates of special t

Rotational motion of a material point.
The rotational motion of a material point is the movement of a material point along a circle. Rotational motion - view mechanical movement. At

Connection between vectors of linear and angular velocities, linear and angular accelerations.
Measure of rotational motion: the angle φ by which the radius vector of a point rotates in a plane normal to the axis of rotation. Uniform rotational motion

Velocity and acceleration in curvilinear motion.
Curvilinear motion over complex view movement than rectilinear, since even if the movement occurs on a plane, then two coordinates that characterize the position of the body change. speed and

Acceleration during curvilinear motion.
Considering curvilinear motion 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 the body or mechanical system. The coordinates of the C. m. are determined by the formulas

The 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

Galilean principle of relativity
Inertial frame of reference Galileo's inertial frame of reference

Plastic deformation
Let's bend a little steel plate (for example, a hacksaw), and then let it go after a while. We will see that the hacksaw will completely (at least at a glance) 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
energy mechanical system, depending on the velocities of its points. K. e. T of a material point is measured by half the product of the mass m of this point and the square of its speed

Kinetic energy.
Kinetic energy - the energy of a moving body. (From Greek word kinema - movement). By definition, the kinetic energy of a reference frame at rest in a given frame

A value equal to half the product of the body's mass and the square of its speed.
=J. Kinetic energy is a relative value, depending on the choice of CO, because the speed of the body depends on the choice of CO. That.

Moment of power
· Moment of power. Rice. Moment of power. Rice. Moment of force, magnitudes

Kinetic energy of a rotating body
Kinetic energy is an additive quantity. Therefore, the kinetic energy of a body moving in an arbitrary way is equal to the sum kinetic energies 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 to work with

The basic equation of the dynamics of rotational motion
According to equation (5.8), Newton's second law for rotational motion P

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. Rotation about the axis is called 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 the course of physics, there are often such quantities, for the description of which it is enough to know only numerical values. For example, mass, time, length.

Quantities that are characterized only numerical value, are called scalar or scalars.

In addition to scalar quantities, quantities are used that have both a numerical value and a direction. For example, speed, acceleration, force.

Quantities that are characterized by a numerical value and direction are called vector or vectors.

Vector quantities are denoted by the corresponding letters with an arrow at the top or are highlighted in bold. For example, the force vector is denoted by \(\vec F\) or F . The numerical value of a vector quantity is called the modulus or length of the vector. The value of the force vector is denoted F or \(\left|\vec F\right|\).

Vector image

Vectors are represented by directed segments. The beginning of the vector is the point from which the directed segment begins (point BUT in fig. 1), the end of the vector is the point where the arrow ends (point B in fig. one).

Rice. one.

The two vectors are called equal if they have the same length and point in the same direction. Such vectors are represented by directed segments having same lengths and directions. For example, in fig. 2 shows the vectors \(\vec F_1 =\vec F_2\).

Rice. 2.

When depicting two or more vectors in one figure, the segments are built on a pre-selected scale. For example, in fig. Figure 3 shows vectors whose lengths \(\upsilon_1\) = 2 m/s, \(\upsilon_2\) = 3 m/s.

Rice. 3.

Vector specification method

On a plane, a vector can be specified in several ways:

1. Specify the coordinates of the beginning and end of the vector. For example, the vector \(\Delta\vec r\) in Fig. 4 is set by the coordinates of the beginning of the vector - (2, 4) (m), the end - (6, 8) (m).

Rice. 4.

2. Specify the module of the vector (its value) and the angle between the direction of the vector and some pre-selected direction on the plane. Often for such a direction in positive side axis 0 X. Angles measured counterclockwise from this direction are considered positive. On fig. 5 the vector \(\Delta\vec r\) is given by two numbers b and \(\alpha\) , indicating the length and direction of the vector.

Rice. 5.

Physics and mathematics cannot do without the concept of "vector quantity". It must be known and recognized, as well as be able to operate with it. You should definitely learn this so as not to get confused and not make stupid mistakes.

How to distinguish a scalar value from a vector one?

The first always has only one characteristic. This is its numerical value. Most scalars can take both positive and negative values. Examples are electrical charge, work, or temperature. But there are some scalars that cannot be negative, such as length and mass.

Vector quantity, except numerical value, which is always taken modulo, is also characterized by direction. Therefore, it can be depicted graphically, that is, in the form of an arrow, the length of which is equal to the modulus of the value directed in a certain direction.

When writing, each vector quantity is indicated by an arrow sign on the letter. If a in question about a numerical value, then the arrow is not written or it is taken modulo.

What actions are most often performed with vectors?

First, a comparison. They may or may not be equal. In the first case, their modules are the same. But this is not the only condition. They must also have the same or opposite directions. In the first case, they should be called equal vectors. In the second, they are opposite. If at least one of these conditions is not met, then the vectors are not equal.

Then comes the addition. It can be done according to two rules: a triangle or a parallelogram. The first prescribes to postpone first one vector, then from its end the second. The result of the addition will be the one that needs to be drawn from the beginning of the first to the end of the second.

The parallelogram rule can be used when you need to add vector quantities in physics. Unlike the first rule, here they should be postponed from one point. Then build them to a parallelogram. The result of the action should be considered the diagonal of the parallelogram drawn from the same point.

If a vector quantity is subtracted from another, then they are again plotted from one point. Only the result will be a vector that matches the one drawn from the end of the second to the end of the first.

What vectors are studied in physics?

There are as many of them as there are scalars. You can simply remember what vector quantities exist in physics. Or know the signs by which they can be calculated. For those who prefer the first option, such a table will come in handy. It contains the main vector

Now a little more about some of these quantities.

The first value is speed

It is worth starting to give examples of vector quantities from it. This is due to the fact that it is studied among the first.

Velocity is defined as a characteristic of the motion of a body in space. It specifies a numerical value and a direction. Therefore, speed is a vector quantity. In addition, it is customary to divide it into types. The first one is linear speed. It is introduced when considering rectilinear uniform motion. In this case, it turns out to be equal to the ratio of the path traveled by the body to the time of movement.

The same formula can be used for uneven movement. Only then will it be average. Moreover, the time interval to be chosen must necessarily be as short as possible. When the time interval tends to zero, the velocity value is already instantaneous.

If arbitrary motion is considered, then here speed is always a vector quantity. After all, it has to be decomposed into components directed along each vector directing the coordinate lines. In addition, it is defined as the derivative of the radius vector, taken with respect to time.

The second value is strength

It determines the measure of the intensity of the impact that is exerted on the body by other bodies or fields. Since force is a vector quantity, it necessarily has its own modulo value and direction. Since it acts on the body, the point to which the force is applied is also important. To obtain visual representation about force vectors, you can refer to the following table.

Also, the resultant force is also a vector quantity. It is defined as the sum of all acting on the body mechanical forces. To determine it, it is necessary to perform addition according to the principle of the triangle rule. Only you need to postpone the vectors in turn from the end of the previous one. The result will be the one that connects the beginning of the first to the end of the last.

The third quantity is displacement

While moving, the body describes a certain line. It's called a trajectory. This line can be completely different. More important is not her appearance, and the start and end points of the movement. They are connected by a segment called displacement. This is also a vector quantity. Moreover, it is always directed from the beginning of the movement to the point where the movement was stopped. It is accepted to designate it Latin letter r.

Here the following question may arise: “Is the path a vector quantity?”. AT general case this statement is not true. Way equal to length trajectory and has no definite direction. An exception is the situation when it is considered in one direction. Then the modulus of the displacement vector coincides in value with the path, and their direction turns out to be the same. Therefore, when considering movement along a straight line without changing the direction of movement, the path can be included in the examples of vector quantities.

The fourth quantity is acceleration

It is a characteristic of the rate of change of speed. Moreover, the acceleration can be both positive and negative meaning. At rectilinear motion it is directed in the direction of higher speed. If the movement is by curvilinear trajectory, then its acceleration vector is decomposed into two components, one of which is directed to the center of curvature along the radius.

Allocate the average and instantaneous value of acceleration. The first should be calculated as the ratio of the change in speed over a certain period of time to this time. When the considered time interval tends to zero, one speaks of instantaneous acceleration.

The fifth quantity is momentum

In another way, it is also called the amount of motion. Momentum is a vector quantity due to the fact that it is directly related to the speed and force applied to the body. Both of them have a direction and give it to the impulse.

By definition, the last is equal to the product body weight for speed. Using the concept of the momentum of a body, one can write the well-known Newton's law in a different way. It turns out that the change in momentum is equal to the product of the force and the time interval.

In physics important role has the law of conservation of momentum, which states that in a closed system of bodies its total momentum is constant.

We have very briefly listed what quantities (vector) are studied in the course of physics.

The problem of inelastic impact

Condition. There is a fixed platform on the rails. A car is approaching it at a speed of 4 m/s. and wagon - 10 and 40 tons, respectively. The car hits the platform, an automatic coupler occurs. It is necessary to calculate the speed of the wagon-platform system after the impact.

Decision. First, you need to enter the notation: the speed of the car before the impact - v 1, the car with the platform after the coupling - v, the mass of the car m 1, the platform - m 2. According to the condition of the problem, it is necessary to find out the value of the speed v.

The rules for solving such tasks require a schematic representation of the system before and after the interaction. It is reasonable to direct the OX axis along the rails in the direction where the car is moving.

Under these conditions, the wagon system can be considered closed. This is determined by the fact that external forces can be neglected. Gravity and are balanced, and friction on the rails is not taken into account.

According to the law of conservation of momentum, their vector sum before the interaction of the car and the platform is equal to the total for the coupler after the impact. At first, the platform did not move, so its momentum was zero. Only the car moved, its momentum is the product of m 1 and v 1 .

Since the impact was inelastic, i.e., the wagon clung to the platform, and then it began to roll together in the same direction, the impulse of the system did not change direction. But its meaning has changed. Namely, the product of the sum of the mass of the wagon with the platform and the desired speed.

You can write the following equality: m 1 * v 1 \u003d (m 1 + m 2) * v. It will be true for the projection of momentum vectors on the selected axis. From it it is easy to derive the equality that will be required to calculate the desired speed: v \u003d m 1 * v 1 / (m 1 + m 2).

According to the rules, you should convert the values ​​\u200b\u200bfor mass from tons to kilograms. Therefore, when substituting them into the formula, you should first multiply the known values ​​​​by a thousand. Simple calculations give a number of 0.75 m/s.

Answer. The speed of the wagon with the platform is 0.75 m/s.

Dividing the body into parts

Condition. The speed of a flying grenade is 20 m/s. It breaks into two pieces. The mass of the first is 1.8 kg. It continues to move in the direction in which the grenade was flying at a speed of 50 m/s. The second fragment has a mass of 1.2 kg. What is its speed?

Decision. Let the fragment masses be denoted by the letters m 1 and m 2 . Their speeds will respectively be v 1 and v 2 . starting speed grenades v. In the problem, you need to calculate the value v 2 .

In order for the larger fragment to continue moving in the same direction as the entire grenade, the second must fly in reverse side. If we choose for the direction of the axis the one that had initial impulse, then after the break, the large fragment flies along the axis, and the small fragment flies against the axis.

In this problem, it is allowed to use the law of conservation of momentum due to the fact that the explosion of a grenade occurs instantly. Therefore, despite the fact that gravity acts on the grenade and its parts, it does not have time to act and change the direction of the momentum vector with its modulus value.

The sum of the vector values ​​of the momentum after the grenade burst is equal to the one before it. If we write down the conservation law in projection onto the OX axis, then it will look like this: (m 1 + m 2) * v = m 1 * v 1 - m 2 * v 2 . It is easy to express the desired speed from it. It is determined by the formula: v 2 \u003d ((m 1 + m 2) * v - m 1 * v 1) / m 2. After substitution of numerical values ​​and calculations, 25 m / s is obtained.

Answer. The speed of a small fragment is 25 m/s.

Problem about shooting at an angle

Condition. A tool is mounted on a platform of mass M. A projectile of mass m is fired from it. It takes off at an angle α to the horizon with a speed v (given relative to the ground). It is required to find out the speed of the platform after the shot.

Decision. In this problem, you can use the momentum conservation law in projection onto the OX axis. But only in the case when the projection of the external resultant forces is equal to zero.

For the direction of the OX axis, you need to choose the side where the projectile will fly, and parallel to the horizontal line. In this case, the projections of the forces of gravity and the reaction of the support on OX will be equal to zero.

The problem will be solved in general view, since there are no specific data for known quantities. The formula is the answer.

The momentum of the system before the shot was equal to zero, since the platform and the projectile were stationary. Let the desired speed of the platform be denoted by the Latin letter u. Then its momentum after the shot is determined as the product of the mass and the projection of the velocity. Since the platform rolls back (against the direction of the OX axis), the momentum value will be with a minus sign.

The momentum of the projectile is the product of its mass and the projection of the velocity on the OX axis. Due to the fact that the velocity is directed at an angle to the horizon, its projection is equal to the velocity multiplied by the cosine of the angle. In literal equality, it will look like this: 0 = - Mu + mv * cos α. From it, by simple transformations, the answer formula is obtained: u = (mv * cos α) / M.

Answer. The speed of the platform is determined by the formula u = (mv * cos α) / M.

River crossing problem

Condition. The width of the river along its entire length is the same and equal to l, its banks are parallel. The speed of the flow of water in the river v 1 and the own speed of the boat v 2 are known. one). When crossing, the bow of the boat is directed strictly to the opposite shore. How far s will it be carried downstream? 2). At what angle α should the bow of the boat be directed so that it reaches the opposite bank strictly perpendicular to the point of departure? How much time t will it take for such a crossing?

Decision. one). The full speed of the boat is the vector sum of the two quantities. The first of these is the course of the river, which is directed along the banks. The second is the boat's own speed, perpendicular to the shores. The drawing shows two similar triangles. The first is formed by the width of the river and the distance that the boat carries. The second is the velocity vectors.

The following entry follows from them: s / l = v 1 / v 2. After the transformation, a formula for the desired value is obtained: s \u003d l * (v 1 / v 2).

2). In this version of the problem, the total velocity vector is perpendicular to the banks. It is equal to the vector sum of v 1 and v 2 . The sine of the angle by which the own velocity vector must deviate is equal to the ratio of the modules v 1 and v 2 . To calculate the travel time, you will need to divide the width of the river by the calculated total speed. The value of the latter is calculated by the Pythagorean theorem.

v = √(v 2 2 - v 1 2), then t = l / (√(v 2 2 - v 1 2)).

Answer. one). s \u003d l * (v 1 / v 2), 2). sin α \u003d v 1 / v 2, t \u003d l / (√ (v 2 2 - v 1 2)).

 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 that are characterized by a numerical value and a sign. The scalars are the length − l, mass − m, path − s, time − t, temperature − T, electric charge − q, energy − W, coordinates, etc.
All algebraic operations (addition, subtraction, multiplication, etc.) are applied to scalar values.

 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 indicate, 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 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)


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 greater than the modulus of the 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 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 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.
Find the work of 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 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.
By studying physics, you have great opportunities continue your education in 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, Egor Savich, is graduating from one of the departments of the Moscow Institute of Physics and Technology, where great demands are made on knowledge of chemistry. If you need help in the GIA in chemistry, then contact the professionals, you will definitely be provided with qualified and timely assistance.

 See also: