This is the vector sum of all forces acting on the body.
The cyclist leans towards the turn. The force of gravity and the reaction force of the support from the earth provide a resultant force that imparts the centripetal acceleration necessary for motion in a circle
Relationship with Newton's second law
Let's remember Newton's law: 
Equals effective force may be equal to zero in the case when one force is compensated by another, the same force, but opposite in direction. In this case, the body is at rest or moving uniformly.
If the resultant force is NOT zero, then the body moves with uniform acceleration. Actually, it is this force that is the reason Not uniform motion. Direction of resultant force Always coincides in direction with the acceleration vector.
When it is necessary to depict the forces acting on a body, while the body moves with uniform acceleration, it means that in the direction of acceleration the acting force is longer than the opposite one. If the body moves uniformly or is at rest, the length of the force vectors is the same.
Finding the resultant force
In order to find the resultant force, it is necessary: firstly, to correctly designate all the forces acting on the body; then draw coordinate axes, select their directions; in the third step it is necessary to determine the projections of the vectors on the axes; write down the equations. Briefly: 1) identify the forces; 2) select the axes and their directions; 3) find the projections of forces on the axis; 4) write down the equations.
How to write equations? If in a certain direction the body moves uniformly or is at rest, then the algebraic sum (taking into account signs) of the projections of forces is equal to zero. If a body moves uniformly accelerated in a certain direction, then the algebraic sum of the projections of forces is equal to the product of mass and acceleration, according to Newton’s second law.
Examples
A body moving uniformly on a horizontal surface is subject to the force of gravity, the reaction force of the support, the force of friction and the force under which the body moves.
Let's denote the forces, choose the coordinate axes
Let's find the projections
Writing down the equations
A body that is pressed against a vertical wall moves downward with uniform acceleration. The body is acted upon by the force of gravity, the force of friction, the reaction of the support and the force with which the body is pressed. The acceleration vector is directed vertically downwards. The resultant force is directed vertically downwards.
The body moves uniformly along a wedge whose slope is alpha. The body is acted upon by the force of gravity, the reaction force of the support, and the force of friction.
The main thing to remember
1) If the body is at rest or moving uniformly, then the resultant force is zero and the acceleration is zero;
2) If the body moves uniformly accelerated, then the resultant force is not zero;
3) The direction of the resultant force vector always coincides with the direction of acceleration;
4) Be able to write equations of projections of forces acting on a body
A block is a mechanical device, a wheel that rotates around its axis. Blocks can be mobile And motionless.
Fixed block used only to change the direction of force.
Bodies connected by an inextensible thread have equal accelerations.
Movable block designed to change the amount of effort applied. If the ends of the rope clasping the block make equal angles with the horizon, then lifting the load will require a force half as much as the weight of the load. The force acting on a load is related to its weight as the radius of a block is to the chord of an arc encircled by a rope.
The acceleration of body A is half the acceleration of body B.
In fact, any block is lever arm, in the case of a fixed block - equal arms, in the case of a movable one - with a ratio of shoulders of 1 to 2. As for any other lever, the following rule applies to the block: the number of times we win in effort, the same number of times we lose in distance
A system consisting of a combination of several movable and fixed blocks is also used. This system is called a polyspast.
According to Newton's first law, in inertial frames of reference, a body can change its speed only if other bodies act on it. The mutual action of bodies on each other is expressed quantitatively using a physical quantity such as force (). A force can change the speed of a body, both in magnitude and in direction. Force is a vector quantity; it has a modulus (magnitude) and a direction. The direction of the resultant force determines the direction of the acceleration vector of the body on which the force in question acts.
The basic law by which the direction and magnitude of the resultant force is determined is Newton’s second law:
where m is the mass of the body on which the force acts; - the acceleration that the force imparts to the body in question. The essence of Newton's second law is that the forces that act on a body determine the change in the speed of the body, and not just its speed. It must be remembered that Newton's second law works for inertial frames of reference.
If several forces act on a body, then their combined action is characterized by the resultant force. Let us assume that several forces act on the body simultaneously, and the body moves with an acceleration equal to the vector sum of the accelerations that would appear under the influence of each of the forces separately. The forces acting on the body and applied to one point must be added according to the rule of vector addition. The vector sum of all forces acting on a body at one moment in time is called the resultant force ():
When several forces act on a body, Newton's second law is written as:
The resultant of all forces acting on the body can be equal to zero if there is mutual compensation of the forces applied to the body. In this case, the body moves at a constant speed or is at rest.
When depicting forces acting on a body in a drawing, in the case of uniformly accelerated movement of the body, the resultant force directed along the acceleration should be depicted longer than the oppositely directed force (sum of forces). In the case of uniform motion (or rest), the magnitude of the force vectors directed in opposite directions is the same.
To find the resultant force, you should depict in the drawing all the forces that must be taken into account in the problem acting on the body. Forces should be added according to the rules of vector addition.
Examples of solving problems on the topic “Resultant force”
EXAMPLE 1
| Exercise | A small ball hangs on a thread, it is at rest. What forces act on this ball, depict them in the drawing. What is the resultant force applied to the body? |
| Solution | Let's make a drawing. Let's consider the reference system associated with the Earth. In our case, this reference system can be considered inertial. A ball suspended on a thread is acted upon by two forces: the force of gravity directed vertically downward () and the reaction force of the thread (tension force of the thread): . Since the ball is at rest, the force of gravity is balanced by the tension force of the thread: Expression (1.1) corresponds to Newton’s first law: the resultant force applied to a body at rest in an inertial frame of reference is zero. |
| Answer | The resultant force applied to the ball is zero. |
EXAMPLE 2
| Exercise | Two forces act on the body and and , where are constant quantities. . What is the resultant force applied to the body? |
| Solution | Let's make a drawing. Since the vectors of force and are perpendicular to each other, therefore, we find the length of the resultant as: |
When several forces are simultaneously applied to one body, the body begins to move with acceleration, which is the vector sum of the accelerations that would arise under the influence of each force separately. The rule of vector addition is applied to forces acting on a body and applied to one point.
Definition 1
The vector sum of all forces simultaneously acting on a body is the force resultant, which is determined by the rule of vector addition of forces:
R → = F 1 → + F 2 → + F 3 → + . . . + F n → = ∑ i = 1 n F i → .
The resultant force acts on a body in the same way as the sum of all forces acting on it.
Definition 2To add 2 forces use rule parallelogram(picture 1).
Picture 1 . Addition of 2 forces according to the parallelogram rule
Let us derive the formula for the modulus of the resultant force using the cosine theorem:
R → = F 1 → 2 + F 2 → 2 + 2 F 1 → 2 F 2 → 2 cos α
Definition 3
If it is necessary to add more than 2 forces, use polygon rule: from the end
The 1st force must draw a vector equal and parallel to the 2nd force; from the end of the 2nd force it is necessary to draw a vector equal and parallel to the 3rd force, etc.
Figure 2. Addition of forces using the polygon rule
The final vector drawn from the point of application of forces to the end of the last force is equal in magnitude and direction to the resultant force. Figure 2 clearly illustrates an example of finding the resultant forces from 4 forces: F 1 →, F 2 →, F 3 →, F 4 →. Moreover, the summed vectors do not necessarily have to be in the same plane.
The result of the force acting on a material point will depend only on its module and direction. U solid There are certain sizes. Therefore, forces with the same magnitudes and directions cause different movements of a rigid body depending on the point of application.
Definition 4
Line of action of force called a straight line passing through the force vector.
Figure 3. Addition of forces applied to different points of the body
If forces are applied to different points of the body and do not act parallel to each other, then the resultant is applied to the point of intersection of the lines of action of the forces (Figure 3 ). A point will be in equilibrium if the vector sum of all forces acting on it is equal to 0: ∑ i = 1 n F i → = 0 → . In this case, the sum of the projections of these forces onto any coordinate axis is also equal to 0.
Definition 5Decomposition of forces into two components- this is the replacement of one force by 2, applied at the same point and producing the same effect on the body as this one force. The decomposition of forces is carried out, like addition, by the parallelogram rule.
The problem of decomposing one force (the modulus and direction of which are given) into 2, applied at one point and acting at an angle to each other, has a unique solution in the following cases when the following are known:
- directions of 2 component forces;
- module and direction of one of the component forces;
- modules of 2 component forces.
It is necessary to decompose the force F into 2 components located in the same plane with F and directed along straight lines a and b (Figure 4 ). Then it is enough to draw 2 straight lines from the end of the vector F, parallel to straight lines a and b. The segment F A and the segment F B represent the required forces.
Figure 4. Decomposition of the force vector in directions
Example 2
The second version of this problem is to find one of the projections of the force vector using the given force vectors and the 2nd projection (Figure 5 a).
Figure 5. Finding the projection of the force vector from given vectors
In the second version of the problem, it is necessary to construct a parallelogram along the diagonal and one of the sides, as in planimetry. Figure 5 b shows such a parallelogram and indicates the desired component F 2 → force F → .
So, the 2nd solution: add to the force a force equal to - F 1 → (Figure 5 c). As a result, we obtain the desired force F →.
Example 3
Three forces F 1 → = 1 N; F 2 → = 2 N; F 3 → = 3 N are applied to one point, are in the same plane (Figure 6 a) and make angles with the horizontal α = 0 °; β = 60°; γ = 30° respectively. It is necessary to find the resultant force.
Solution
Figure 6. Finding the resultant force from given vectors
Let's draw mutually perpendicular axes O X and O Y so that the O X axis coincides with the horizontal along which the force F 1 → is directed. Let's make a projection of these forces onto the coordinate axes (Figure 6 b). The projections F 2 y and F 2 x are negative. The sum of the projections of forces onto the coordinate axis O X is equal to the projection onto this axis of the resultant: F 1 + F 2 cos β - F 3 cos γ = F x = 4 - 3 3 2 ≈ - 0.6 N.
Similarly, for projections onto the O Y axis: - F 2 sin β + F 3 sin γ = F y = 3 - 2 3 2 ≈ - 0.2 N.
We determine the modulus of the resultant using the Pythagorean theorem:
F = F x 2 + F y 2 = 0.36 + 0.04 ≈ 0.64 N.
We find the direction of the resultant using the angle between the resultant and the axis (Figure 6 c):
t g φ = F y F x = 3 - 2 3 4 - 3 3 ≈ 0.4.
Example 4
A force F = 1 kN is applied at point B of the bracket and is directed vertically downward (Figure 7 a). It is necessary to find the components of this force in the directions of the bracket rods. All necessary data is shown in the figure.
Solution
Figure 7. Finding the components of force F in the directions of the bracket rods
Given:
F = 1 k N = 1000 N
Let the rods be screwed to the wall at points A and C. Figure 7 b shows the decomposition of the force F → into components along the directions A B and B C. From here it is clear that
F 1 → = F t g β ≈ 577 N;
F 2 → = F cos β ≈ 1155 N.
Answer: F 1 → = 557 N; F 2 → = 1155 N.
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Section 1. "STATICS"
Newtons
The arm of a force is the shortest distance from a point to the line of action of the force
The product of the force on the arm is equal to the moment of the force.
8. Formulate a “rule” right hand» to determine the direction of the moment of force.
9. How is the main moment of a system of forces relative to a point determined?
Main point relative to the center – the vector sum of the moments of all forces applied to the body relative to the same center.
10. What is called a pair of forces? Why equals the moment couples of forces? Does it depend on the choice of point? What is the direction and magnitude of the moment of a pair of forces?
A force pair is a system of forces in which the forces are equal, parallel and opposite to each other. The moment is equal to the product of one of the forces on the shoulder, does not depend on the choice of point, and is directed perpendicular to the plane in which the pair lies.
11. State Poinsot’s theorem.
Any system of forces acting on an absolutely rigid body can be replaced by one force and one pair of forces. In this case, the force will be the main vector, and the moment of the couple will be the main moment of this system of forces.12. Formulate the necessary and sufficient conditions balance of the system of forces.
For the equilibrium of a plane system of forces, it is necessary and sufficient that the algebraic sums of the projections of all forces onto two coordinate axes and the algebraic sum of the moments of all forces relative to an arbitrary point are equal to zero. The second form of the equilibrium equation is the equality to zero of the algebraic sums of the moments of all forces relative to any three points that do not lie on the same straight line
14. What systems of forces are called equivalent?
If, without disturbing the state of the body, one system of forces (F 1, F 2, ..., F n) can be replaced by another system (P 1, P 2, ..., P n) and vice versa, then such systems of forces are called equivalent
15. What force is called the resultant of this system of forces?
When a system of forces (F 1, F 2, ..., F n) is equivalent to one force R, then R is called. resultant. The resultant force can replace the action of all given forces. But not every system of forces has a resultant.
16. It is known that the sum of the projections of all forces applied to the body onto a given axis is equal to zero. What is the direction of the resultant of such a system?
17. Formulate the axiom of inertia (Galileo’s principle of inertia).
Under the influence of mutually balancing forces material point(body) is at rest or moving in a straight line and uniformly
28. Formulate the axiom of equilibrium between two forces.
Two forces applied to an absolutely rigid body will be balanced if and only if they are equal in magnitude, act in the same straight line and are directed in opposite directions
19. Is it possible to transfer a force along its line of action without changing the kinematic state of an absolutely rigid body?
Without changing the kinematic state of an absolutely rigid body, the force can be transferred along the line of its action, keeping its modulus and direction unchanged.
20. Formulate the axiom of the parallelogram of forces.
Without changing the state of the body, two forces applied to one point can be replaced by one resultant force applied at the same point and equal to them geometric sum
21. How is Newton's third law formulated?
Every action has an equal and opposite reaction
22. Which solid body is called non-free?
The forces acting between the bodies of the system are called internal.
Articulated and movable support. This type of connection is structurally made in the form of a cylindrical hinge that can move freely along the surface. The reaction of the articulated movable support is always directed perpendicular to the supporting surface
Hinged-fixed support. The reaction of a hinged-fixed support is represented in the form of unknown components and , the lines of action of which are parallel or coincide with the coordinate axes
29. Which support is called rigid embedding (pinching)?
This is an unusual type of connection, since in addition to preventing movement in the plane, the rigid seal prevents the rotation of the rod (beam) relative to the point. Therefore, the coupling reaction is reduced not only to the reaction (,), but also to the reactive torque
30. What support is called a thrust bearing?
Thrust bearing and spherical hinge This type of connection can be represented in the form of a rod having a spherical surface at the end, which is attached to a support, which is part of a spherical cavity. A spherical hinge prevents movement in any direction in space, so its reaction is represented in the form of three components , , , parallel to the corresponding coordinate axes
31. Which support is called a spherical joint?
32. What system of forces is called convergent? How are the equilibrium conditions for a system of converging forces formulated?
If a (absolutely rigid) body is in equilibrium under the action of a plane system of three not parallel forces(i.e. forces, at least two of which are non-parallel), then their lines of action intersect at one point.
34. What is the sum of two parallel forces directed in the same direction? IN different sides?
the resultant of two parallel forces F 1 and F 2 of the same direction has the same direction, its module is equal to the sum of the modules of the forces added, and the point of application divides the segment between the points of application of forces into parts inversely proportional to the modules of the forces: R = F 1 + F 2 ; AC/BC=F 2 /F 1. The resultant of two oppositely directed parallel forces has a direction of force that is larger in magnitude and a magnitude equal to the difference in the magnitudes of the forces.
37. How is Varignon’s theorem formulated?
If the subject flat system forces are reduced to a resultant, then the moment of this resultant relative to any point is equal to the algebraic sum of the moments of all forces of a given system relative to that same point.
40. How is the center of parallel forces determined?
According to Varignon's theorem
41. How is the center of gravity of a solid body determined?
45. Where is the center of gravity of the triangle?
Median intersection point
46. Where is the center of gravity of the pyramid and cone?
Section 2. “KINEMATICS”
1. What is called the trajectory of a point? What motion of a point is called rectilinear? Curvilinear?
The line along which material moves dot , called trajectory .
If the trajectory is a straight line, then the movement of the point is called rectilinear; if the trajectory is a curved line, then the movement is called curvilinear
2. How is Cartesian determined? rectangular system coordinates?
3. How is the absolute speed of a point in a stationary (inertial) coordinate system determined? What is the direction of the velocity vector in relation to its trajectory? What are the projections of the velocity of a point on the Cartesian coordinate axis?
For a point, these dependencies are as follows: the absolute speed of a point is equal to the geometric sum of the relative and portable speeds, that is:
.
3. How is the absolute acceleration of a point in a stationary (inertial) coordinate system determined? What are the projections of the acceleration of a point on the Cartesian coordinate axis?
5. How is the angular velocity vector of a rigid body determined when it rotates around fixed axis? What is the direction of the angular velocity vector?
Angular velocity- vector physical quantity, characterizing the speed of rotation of the body. The angular velocity vector is equal in magnitude to the angle of rotation of the body per unit time:
a is directed along the axis of rotation according to the gimlet rule, that is, in the direction into which a gimlet with a right-hand thread would be screwed if it rotated in the same direction.
6. How a vector is determined angular acceleration of a rigid body when it rotates around a fixed axis? What is the direction of the angular acceleration vector?
When a body rotates around a fixed axis, the angular acceleration in magnitude is equal to:
The angular acceleration vector α is directed along the axis of rotation (to the side during accelerated rotation and in the opposite direction during slow rotation).
When rotating around a fixed point, the angular acceleration vector is defined as the first derivative of the angular velocity vector ω with respect to time, that is
8. What are the absolute, portable and relative speeds of a point during its complex motion?
9. How are portable and relative accelerations determined during complex motion of a point?
10. How is Coriolis acceleration determined for complex motion of a point?
11. State the Coriolis theorem.
Theorem on the addition of accelerations (Coriolis theorem): 
, Where 
– Coriolis acceleration (Coriolis acceleration) – in the case of non-translational portable movement absolute acceleration = the geometric sum of the translational, relative and Coriolis accelerations.
12. At what movements are the points equal to zero:
b) normal acceleration?
14. What body movement is called translational? What are the velocities and accelerations of the points of the body during such movement?
16. What body movement is called rotational? What are the velocities and accelerations of the points of the body during such movement?
17. How are the tangential and centripetal accelerations of a point on a rigid body rotating around a fixed axis expressed?
18. What does it feel like locus points of a rigid body rotating around a fixed axis, whose velocities at a given moment have the same magnitude and the same direction?
19. What motion of a body is called plane-parallel? What are the velocities and accelerations of the points of the body during such movement?
20. How is the instantaneous center of velocities determined? flat figure, moving in its own plane?
21. How can you graphically find the position of the instantaneous center of velocities if the velocities of two points of a plane figure are known?
22. What will be the velocities of the points of a flat figure in the case when the instantaneous center of rotation of this figure is infinitely distant?
23. How are the projections of the velocities of two points of a plane figure onto a straight line connecting these points related to each other?
24. Given two points ( A And IN) of a moving flat figure, and it is known that the speed of the point A perpendicular to AB. How is the speed of the point directed? IN?
Section 1. "STATICS"
1. What factors determine the force acting on a solid?
2. In what units is force measured in the SI system?
Newtons
3. What is the main vector of the force system? How to construct a force polygon for a given system of forces?
The main vector is the vector sum of all forces applied to the body
5. What is called the moment of force relative to a given point? What is the direction of the moment of force relative to the force vector and the radius vector of the point of application of the force?
The moment of a force relative to a point (center) is a vector that is numerically equal to the product of the modulus of the force by the arm, i.e., by the shortest distance from the specified point to the line of action of the force. It is directed perpendicular to the plane of propagation of force and r.v. points.
6. In what case is the moment of a force relative to a point equal to zero?
When the arm is equal to 0 (The center of the moments is located on the line of action of the force)
7. How is the leverage of a force relative to a point determined? What is the product of force and arm?
How vector addition occurs is not always clear to students. Children have no idea what is hidden behind them. You just have to remember the rules, and not think about the essence. Therefore, it is precisely about the principles of addition and subtraction of vector quantities that a lot of knowledge is required.
The addition of two or more vectors always results in one more. Moreover, it will always be the same, regardless of how it is found.
Most often in school course geometry considers the addition of two vectors. It can be performed according to the triangle or parallelogram rule. These drawings look different, but the result of the action is the same.
How does addition occur using the triangle rule?
It is used when the vectors are non-collinear. That is, they do not lie on the same straight line or on parallel ones.
In this case, the first vector must be plotted from some arbitrary point. From its end it is required to draw parallel and equal to the second. The result will be a vector starting from the beginning of the first and ending at the end of the second. The pattern resembles a triangle. Hence the name of the rule.
If the vectors are collinear, then this rule can also be applied. Only the drawing will be located along one line.
How is addition performed using the parallelogram rule?
Yet again? applies only to non-collinear vectors. The construction is carried out according to a different principle. Although the beginning is the same. We need to set aside the first vector. And from its beginning - the second. Based on them, complete the parallelogram and draw a diagonal from the beginning of both vectors. This will be the result. This is how vector addition is performed according to the parallelogram rule.
So far there have been two. But what if there are 3 or 10 of them? Use the following technique.
How and when does the polygon rule apply?
If you need to perform addition of vectors, the number of which is more than two, do not be afraid. It is enough to put them all aside sequentially and connect the beginning of the chain with its end. This vector will be the required sum.
What properties are valid for operations with vectors?
About the zero vector. Which states that when added to it, the original is obtained.
About the opposite vector. That is, about one that has the opposite direction and equal magnitude. Their sum will be zero.
On the commutativity of addition. What has been known since primary school. Changing the positions of the terms does not change the result. In other words, it doesn't matter which vector to put off first. The answer will still be correct and unique.
On the associativity of addition. This law allows you to add any vectors from a triple in pairs and add a third to them. If you write this using symbols, you get the following:
first + (second + third) = second + (first + third) = third + (first + second).
What is known about vector difference?
There is no separate subtraction operation. This is due to the fact that it is essentially addition. Only the second of them is given the opposite direction. And then everything is done as if adding vectors were considered. Therefore, there is practically no talk about their difference.
In order to simplify the work with their subtraction, the triangle rule is modified. Now (when subtracting) the second vector must be set aside from the beginning of the first. The answer will be the one that connects the end point of the minuend with the same one as the subtrahend. Although you can postpone it as described earlier, simply by changing the direction of the second.
How to find the sum and difference of vectors in coordinates?
The problem gives the coordinates of the vectors and requires finding out their values for the final result. In this case, there is no need to perform constructions. That is, you can use simple formulas that describe the rule for adding vectors. They look like this:
a (x, y, z) + b (k, l, m) = c (x + k, y + l, z + m);
a (x, y, z) -b (k, l, m) = c (x-k, y-l, z-m).
It is easy to see that the coordinates just need to be added or subtracted depending on the specific task.
First example with solution
Condition. Given a rectangle ABCD. Its sides are equal to 6 and 8 cm. The intersection point of the diagonals is designated by the letter O. It is required to calculate the difference between the vectors AO and VO.
Solution. First you need to draw these vectors. They are directed from the vertices of the rectangle to the point of intersection of the diagonals.
If you look closely at the drawing, you can see that the vectors are already combined so that the second of them is in contact with the end of the first. It's just that his direction is wrong. It should start from this point. This is if the vectors are added, but the problem involves subtraction. Stop. This action means that you need to add the oppositely directed vector. This means that VO needs to be replaced with OV. And it turns out that the two vectors have already formed a pair of sides from the triangle rule. Therefore, the result of their addition, that is, the desired difference, is the vector AB.
And it coincides with the side of the rectangle. To write down your numerical answer, you will need the following. Draw a rectangle lengthwise so that the larger side is horizontal. Start numbering the vertices from the bottom left and go counterclockwise. Then the length of vector AB will be equal to 8 cm.
Answer. The difference between AO and VO is 8 cm.
Second example and its detailed solution
Condition. The diagonals of the rhombus ABCD are 12 and 16 cm. The point of their intersection is indicated by the letter O. Calculate the length of the vector formed by the difference between the vectors AO and VO.
Solution. Let the designation of the vertices of the rhombus be the same as in the previous problem. Similar to the solution of the first example, it turns out that the desired difference is equal to the vector AB. And its length is unknown. Solving the problem came down to calculating one of the sides of the rhombus.
For this purpose, you will need to consider the triangle ABO. It is rectangular because the diagonals of a rhombus intersect at an angle of 90 degrees. And its legs are equal to half the diagonals. That is, 6 and 8 cm. The side sought in the problem coincides with the hypotenuse in this triangle.
To find it you will need the Pythagorean theorem. The square of the hypotenuse will be equal to the sum of the numbers 6 2 and 8 2. After squaring, the values obtained are: 36 and 64. Their sum is 100. It follows that the hypotenuse is equal to 10 cm.
Answer. The difference between the vectors AO and VO is 10 cm.
Third example with detailed solution
Condition. Calculate the difference and sum of two vectors. Their coordinates are known: the first one has 1 and 2, the second one has 4 and 8.
Solution. To find the sum you will need to add the first and second coordinates in pairs. The result will be the numbers 5 and 10. The answer will be a vector with coordinates (5; 10).
For the difference, you need to subtract the coordinates. After performing this action, the numbers -3 and -6 will be obtained. They will be the coordinates of the desired vector.
Answer. The sum of the vectors is (5; 10), their difference is (-3; -6).
Fourth example
Condition. The length of the vector AB is 6 cm, BC is 8 cm. The second is laid off from the end of the first at an angle of 90 degrees. Calculate: a) the difference between the modules of the vectors VA and BC and the module of the difference between VA and BC; b) the sum of the same modules and the module of the sum.
Solution: a) The lengths of the vectors are already given in the problem. Therefore, calculating their difference is not difficult. 6 - 8 = -2. The situation with the difference module is somewhat more complicated. First you need to find out which vector will be the result of the subtraction. For this purpose, one should set aside the vector VA, which is directed towards the opposite side AB. Then draw the vector BC from its end, directing it in the direction opposite to the original one. The result of subtraction is the vector CA. Its modulus can be calculated using the Pythagorean theorem. Simple calculations lead to a value of 10 cm.
b) The sum of the moduli of the vectors is equal to 14 cm. To find the second answer, some transformation will be required. Vector BA is oppositely directed to that given - AB. Both vectors are directed from the same point. In this situation, you can use the parallelogram rule. The result of the addition will be a diagonal, and not just a parallelogram, but a rectangle. Its diagonals are equal, which means that the modulus of the sum is the same as in the previous paragraph.
Answer: a) -2 and 10 cm; b) 14 and 10 cm.
