How is the subtraction of vectors in physics. Rules for adding collinear vectors

standard definition: "A vector is a directed line segment." This is usually the limit of a graduate's knowledge of vectors. Who needs some kind of "directed segments"?

But in fact, what are vectors and why are they?
Weather forecast. "Wind northwest, speed 18 meters per second." Agree, both the direction of the wind (where it blows from) and the module (that is, absolute value) its speed.

Quantities that have no direction are called scalars. weight, work, electric charge not sent anywhere. They are characterized only numerical value- “how many kilograms” or “how many joules”.

Physical quantities that have not only absolute value, but also the direction, are called vector.

Speed, force, acceleration - vectors. For them, it is important "how much" and it is important "where". For example, acceleration free fall directed to the surface of the Earth, and its value is 9.8 m / s 2. momentum, tension electric field, induction magnetic field are also vector quantities.

Do you remember that physical quantities denoted by letters, Latin or Greek. The arrow above the letter indicates that the quantity is a vector:

Here is another example.
The car is moving from A to B. The end result is its movement from point A to point B, i.e. movement by a vector.

Now it is clear why a vector is a directed segment. Pay attention, the end of the vector is where the arrow is. Vector length is called the length of this segment. Designated: or

So far, we have been working with scalar quantities, according to the rules of arithmetic and elementary algebra. Vectors are a new concept. This is a different class mathematical objects. They have their own rules.

Once upon a time, we didn’t even know about numbers. Acquaintance with them began in lower grades. It turned out that numbers can be compared with each other, added, subtracted, multiplied and divided. We learned that there is a number one and a number zero.
Now we get to know vectors.

The concepts of "greater than" and "less than" do not exist for vectors - after all, their directions can be different. You can only compare the lengths of vectors.

But the concept of equality for vectors is.
Equal are called vectors having same lengths and the same direction. This means that the vector can be moved parallel to itself to any point in the plane.
single is called a vector whose length is 1 . Zero - a vector whose length is equal to zero, that is, its beginning coincides with the end.

It is most convenient to work with vectors in rectangular system coordinates - the same one in which we draw graphs of functions. Each point in the coordinate system corresponds to two numbers - its x and y coordinates, abscissa and ordinate.
The vector is also given by two coordinates:

Here, the coordinates of the vector are written in brackets - in x and in y.
They are easy to find: the coordinate of the end of the vector minus the coordinate of its beginning.

If the vector coordinates are given, its length is found by the formula

Vector addition

There are two ways to add vectors.

one . parallelogram rule. To add the vectors and , we place the origins of both at the same point. We complete the parallelogram and draw the diagonal of the parallelogram from the same point. This will be the sum of the vectors and .

Remember the fable about the swan, cancer and pike? They tried very hard, but they never moved the cart. After all, the vector sum of the forces applied by them to the cart was equal to zero.

2. The second way to add vectors is the triangle rule. Let's take the same vectors and . We add the beginning of the second to the end of the first vector. Now let's connect the beginning of the first and the end of the second. This is the sum of the vectors and .

By the same rule, you can add several vectors. We attach them one by one, and then connect the beginning of the first to the end of the last.

Imagine that you are going from point A to point B, from B to C, from C to D, then to E and then to F. The end result of these actions is a move from A to F.

When adding vectors and we get:

Vector subtraction

The vector is directed opposite to the vector . The lengths of the vectors and are equal.

Now it is clear what subtraction of vectors is. The difference of the vectors and is the sum of the vector and the vector .

Multiply a vector by a number

Multiplying a vector by a number k results in a vector whose length is k times different from the length . It is codirectional with the vector if k Above zero, and is directed opposite if k is less than zero.

Dot product of vectors

Vectors can be multiplied not only by numbers, but also by each other.

The scalar product of vectors is the product of the lengths of vectors and the cosine of the angle between them.

Pay attention - we multiplied two vectors, and we got a scalar, that is, a number. For example, in physics mechanical work is equal to the scalar product of two vectors - force and displacement:

If the vectors are perpendicular, their dot product is zero.
And this is how the scalar product is expressed in terms of the coordinates of the vectors and:

From the formula for dot product you can find the angle between the vectors:

This formula is especially convenient in stereometry. For example, in problem 14 profile exam in mathematics, you need to find the angle between intersecting lines or between a line and a plane. Problem 14 is often solved several times faster by the vector method than by the classical one.

AT school curriculum in mathematics, only the scalar product of vectors is studied.
It turns out that, in addition to the scalar, there is also a vector product, when a vector is obtained as a result of multiplying two vectors. Who passes the exam in physics, knows what the Lorentz force and the Ampère force are. The formulas for finding these forces include exactly vector products.

Vectors are a very useful mathematical tool. You will be convinced of this in the first course.

Definition

The addition of vectors and is carried out according to triangle rule.

sum two vectors and such a third vector is called, the beginning of which coincides with the beginning, and the end with the end, provided that the end of the vector and the beginning of the vector coincide (Fig. 1).

For addition vectors The parallelogram rule also applies.

Definition

parallelogram rule- if two non-collinear vectors u lead to a common origin, then the vector coincides with the diagonal of the parallelogram built on the vectors u (Fig. 2). Moreover, the beginning of the vector coincides with the beginning of the given vectors.

Definition

The vector is called opposite vector to the vector if it collinear vector , equal to it in length, but directed in the opposite direction to the vector.

The vector addition operation has the following properties:

Definition

difference vectors and a vector is called such that the condition is satisfied: (Fig. 3).

Multiply a vector by a number

Definition

work vector per number is called a vector that satisfies the conditions:

Properties of multiplication of a vector by a number:

Here u are arbitrary vectors, and are arbitrary numbers.

Euclidean space(also Euclidean space) - in the original sense, the space whose properties are described axioms euclidean geometry. In this case, it is assumed that the space has dimension equal to 3.

In the modern sense, in a more general sense, it can denote one of the similar and closely related objects: finite-dimensional real vector space with a positive definite scalar product, or metric space corresponding to such a vector space. In this article, the first definition will be taken as the initial one.

Dimensional Euclidean space is also often used (if it is clear from the context that the space has a Euclidean structure).

To define the Euclidean space, it is easiest to take as the main concept dot product. The Euclidean vector space is defined as finite-dimensional vector space above field real numbers, on whose vectors real-valued function with the following three properties:

affine space, corresponding to such a vector space, is called the Euclidean affine space, or simply the Euclidean space .

An example of a Euclidean space is a coordinate space consisting of all possible n-ok real numbers scalar product in which is determined by the formula

Basis and vector coordinates

Basis (other Greekβασις, basis) - the set of such vectors in vector space that any vector of this space can be uniquely represented as linear combination vectors from this set - basis vectors.

In the case when the basis is infinite, the concept of "linear combination" needs to be clarified. This leads to two main types of definition:

Hamel basis, whose definition considers only finite linear combinations. The Hamel basis is used mainly in abstract algebra (in particular, in linear algebra).

Schauder basis, whose definition also considers infinite linear combinations, namely, expansion in ranks. This definition is used mainly in functional analysis, in particular for Hilbert space,

In finite-dimensional spaces, both types of basis coincide.

Vector coordinates are the coefficients of the only possible linear combination basic vectors in the selected coordinate system equal to the given vector.

where are the coordinates of the vector.

Scalar product.

operation on two vectors, the result of which is number[when vectors are considered, numbers are often called scalars], which does not depend on the coordinate system and characterizes the lengths of the factor vectors and corner between them. This operation corresponds to the multiplication length vector x on the projection vector y per vector x. This operation is usually considered as commutative and linear for each factor.

Scalar product two vectors is equal to the sum of the products of their respective coordinates:

vector product

this is pseudovector, perpendicular plane constructed by two factors, which is the result of binary operation"vector multiplication" over vectors in 3D euclidean space. Vector product does not have properties commutativity and associativity(is anticommutative) and, in contrast to dot product of vectors, is a vector. Widely used in many technical and physical applications. For example, angular momentum and Lorentz force mathematically written as a vector product. The cross product is useful for "measuring" the perpendicularity of vectors - the modulus of the cross product of two vectors is equal to the product of their moduli if they are perpendicular, and decreases to zero if the vectors are parallel or anti-parallel.

vector product two vectors can be calculated using determinant matrices

mixed product

Mixed product vectors -scalar product vector on the vector product vectors and:

Sometimes it is called triple scalar product vectors, apparently due to the fact that the result is scalar(more precisely - pseudoscalar).

Geometric sense: The modulus of the mixed product is numerically equal to the volume parallelepiped educated vectors .mixed product three vectors can be found through the determinant

Plane in space

Plane - algebraic surface first order: in Cartesian coordinate system plane can be set equation first degree.

Some characteristic properties of a plane

Plane - surface, containing completely each direct, connecting any points;

Two planes are either parallel or intersect in a straight line.

The line is either parallel to the plane, or intersects it at one point, or is on the plane.

Two lines perpendicular to the same plane are parallel to each other.

Two planes perpendicular to the same line are parallel to each other.

Similarly segment and interval, a plane that does not include extreme points can be called an interval plane, or an open plane.

General equation (complete) of the plane

where and are constants, and at the same time they are not equal to zero; in vector form:

where is the radius vector of the point, the vector perpendicular to the plane (normal vector). Guidescosines vector :

The vector \(\overrightarrow(AB)\) can be viewed as moving a point from position \(A\) (start of movement) to position \(B\) (end of movement). That is, the trajectory of movement in this case is not important, only the beginning and end are important!

\(\blacktriangleright\) Two vectors are collinear if they lie on the same line or on two parallel lines.
Otherwise, the vectors are called non-collinear.

\(\blacktriangleright\) Two collinear vectors are said to be codirectional if their directions are the same.
If their directions are opposite, then they are called oppositely directed.

Addition rules collinear vectors:

co-directional end first. Then their sum is a vector, the beginning of which coincides with the beginning of the first vector, and the end coincides with the end of the second one (Fig. 1).

\(\blacktriangleright\) To add two opposite directions vector, you can postpone the second vector from start first. Then their sum is a vector, the beginning of which coincides with the beginning of both vectors, the length is equal to the difference in the lengths of the vectors, the direction coincides with the direction of the longer vector (Fig. 2).

Rules for adding non-collinear vectors \(\overrightarrow (a)\) and \(\overrightarrow(b)\) :

\(\blacktriangleright\) Triangle rule (Fig. 3).

It is necessary to postpone the vector \(\overrightarrow (b)\) from the end of the vector \(\overrightarrow (a)\) . Then the sum is a vector whose beginning coincides with the beginning of the vector \(\overrightarrow (a)\) , and whose end coincides with the end of the vector \(\overrightarrow (b)\) .

\(\blacktriangleright\) Parallelogram rule (Fig. 4).

It is necessary to postpone the vector \(\overrightarrow (b)\) from the beginning of the vector \(\overrightarrow (a)\) . Then the sum \(\overrightarrow (a)+\overrightarrow (b)\) is a vector coinciding with the diagonal of the parallelogram built on the vectors \(\overrightarrow (a)\) and \(\overrightarrow (b)\) (the beginning of which coincides with the beginning of both vectors).

\(\blacktriangleright\) To find the difference of two vectors \(\overrightarrow(a)-\overrightarrow(b)\), you need to find the sum of the vectors \(\overrightarrow (a)\) and \(-\overrightarrow(b)\) : \(\overrightarrow(a)-\overrightarrow(b)=\overrightarrow(a)+(-\overrightarrow(b))\)(Fig. 5).

Task 1 #2638

Task level: More difficult than the exam

Given a right triangle \(ABC\) with a right angle \(A\) , point \(O\) is the center of the circumscribed circle around the given triangle. Vector coordinates \(\overrightarrow(AB)=\(1;1\)\), \(\overrightarrow(AC)=\(-1;1\)\). Find the sum of the coordinates of the vector \(\overrightarrow(OC)\) .

Because the triangle \(ABC\) is right-angled, then the center of the circumscribed circle lies in the middle of the hypotenuse, i.e. \(O\) is the middle of \(BC\) .

notice, that \(\overrightarrow(BC)=\overrightarrow(AC)-\overrightarrow(AB)\), Consequently, \(\overrightarrow(BC)=\(-1-1;1-1\)=\(-2;0\)\).

Because \(\overrightarrow(OC)=\dfrac12 \overrightarrow(BC)\), then \(\overrightarrow(OC)=\(-1;0\)\).

Hence, the sum of the coordinates of the vector \(\overrightarrow(OC)\) is equal to \(-1+0=-1\) .

Answer: -1

Task 2 #674

Task level: More difficult than the exam

\(ABCD\) is a quadrilateral whose sides contain the vectors \(\overrightarrow(AB)\) , \(\overrightarrow(BC)\) , \(\overrightarrow(CD)\) , \(\overrightarrow(DA) \) . Find the length of the vector \(\overrightarrow(AB) + \overrightarrow(BC) + \overrightarrow(CD) + \overrightarrow(DA)\).

\(\overrightarrow(AB) + \overrightarrow(BC) = \overrightarrow(AC)\), \(\overrightarrow(AC) + \overrightarrow(CD) = \overrightarrow(AD)\), then
\(\overrightarrow(AB) + \overrightarrow(BC) + \overrightarrow(CD) + \overrightarrow(DA) = \overrightarrow(AC) + \overrightarrow(CD) + \overrightarrow(DA)= \overrightarrow(AD) + \overrightarrow(DA) = \overrightarrow(AD) - \overrightarrow(AD) = \vec(0)\).
The null vector has length equal to \(0\) .

A vector can be thought of as a displacement, then \(\overrightarrow(AB) + \overrightarrow(BC)\)- move from \(A\) to \(B\) , and then from \(B\) to \(C\) - in the end it is a move from \(A\) to \(C\) .

With this interpretation, it becomes clear that \(\overrightarrow(AB) + \overrightarrow(BC) + \overrightarrow(CD) + \overrightarrow(DA) = \vec(0)\), because as a result, here we moved from the point \(A\) to the point \(A\) , that is, the length of such a movement is equal to \(0\) , which means that the vector of such a movement itself is \(\vec(0)\) .

Answer: 0

Task 3 #1805

Task level: More difficult than the exam

Given a parallelogram \(ABCD\) . The diagonals \(AC\) and \(BD\) intersect at the point \(O\) . Let, then \(\overrightarrow(OA) = x\cdot\vec(a) + y\cdot\vec(b)\)

\[\overrightarrow(OA) = \frac(1)(2)\overrightarrow(CA) = \frac(1)(2)(\overrightarrow(CB) + \overrightarrow(BA)) = \frac(1)( 2)(\overrightarrow(DA) + \overrightarrow(BA)) = \frac(1)(2)(-\vec(b) - \vec(a)) = - \frac(1)(2)\vec (a) - \frac(1)(2)\vec(b)\]\(\Rightarrow\) \(x = - \frac(1)(2)\) , \(y = - \frac(1)(2)\) \(\Rightarrow\) \(x + y = - one\) .

Answer: -1

Task 4 #1806

Task level: More difficult than the exam

Given a parallelogram \(ABCD\) . The points \(K\) and \(L\) lie on the sides \(BC\) and \(CD\), respectively, and \(BK:KC = 3:1\) , and \(L\) is the midpoint \ (CD\) . Let \(\overrightarrow(AB) = \vec(a)\), \(\overrightarrow(AD) = \vec(b)\), then \(\overrightarrow(KL) = x\cdot\vec(a) + y\cdot\vec(b)\), where \(x\) and \(y\) are some numbers. Find the number equal to \(x + y\) .

\[\overrightarrow(KL) = \overrightarrow(KC) + \overrightarrow(CL) = \frac(1)(4)\overrightarrow(BC) + \frac(1)(2)\overrightarrow(CD) = \frac (1)(4)\overrightarrow(AD) + \frac(1)(2)\overrightarrow(BA) = \frac(1)(4)\vec(b) - \frac(1)(2)\vec (a)\]\(\Rightarrow\) \(x = -\frac(1)(2)\) , \(y = \frac(1)(4)\) \(\Rightarrow\) \(x + y = -0 ,25\) .

Answer: -0.25

Task 5 #1807

Task level: More difficult than the exam

Given a parallelogram \(ABCD\) . The points \(M\) and \(N\) lie on the sides \(AD\) and \(BC\) respectively, where \(AM:MD = 2:3\) and \(BN:NC = 3): one\) . Let \(\overrightarrow(AB) = \vec(a)\), \(\overrightarrow(AD) = \vec(b)\), then \(\overrightarrow(MN) = x\cdot\vec(a) + y\cdot\vec(b)\)

\[\overrightarrow(MN) = \overrightarrow(MA) + \overrightarrow(AB) + \overrightarrow(BN) = \frac(2)(5)\overrightarrow(DA) + \overrightarrow(AB) + \frac(3 )(4)\overrightarrow(BC) = - \frac(2)(5)\overrightarrow(AD) + \overrightarrow(AB) + \frac(3)(4)\overrightarrow(BC) = -\frac(2 )(5)\vec(b) + \vec(a) + \frac(3)(4)\vec(b) = \vec(a) + \frac(7)(20)\vec(b)\ ]\(\Rightarrow\) \(x = 1\) , \(y = \frac(7)(20)\) \(\Rightarrow\) \(x\cdot y = 0.35\) .

Answer: 0.35

Task 6 #1808

Task level: More difficult than the exam

Given a parallelogram \(ABCD\) . The point \(P\) lies on the diagonal \(BD\) , the point \(Q\) lies on the side \(CD\) , where \(BP:PD = 4:1\) , and \(CQ:QD = 1:9 \) . Let \(\overrightarrow(AB) = \vec(a)\), \(\overrightarrow(AD) = \vec(b)\), then \(\overrightarrow(PQ) = x\cdot\vec(a) + y\cdot\vec(b)\), where \(x\) and \(y\) are some numbers. Find the number equal to \(x\cdot y\) .

\[\begin(gathered) \overrightarrow(PQ) = \overrightarrow(PD) + \overrightarrow(DQ) = \frac(1)(5)\overrightarrow(BD) + \frac(9)(10)\overrightarrow( DC) = \frac(1)(5)(\overrightarrow(BC) + \overrightarrow(CD)) + \frac(9)(10)\overrightarrow(AB) =\\ = \frac(1)(5) (\overrightarrow(AD) + \overrightarrow(BA)) + \frac(9)(10)\overrightarrow(AB) = \frac(1)(5)(\overrightarrow(AD) - \overrightarrow(AB)) + \frac(9)(10)\overrightarrow(AB) = \frac(1)(5)\overrightarrow(AD) + \frac(7)(10)\overrightarrow(AB) = \frac(1)(5) \vec(b) + \frac(7)(10)\vec(a)\end(gathered)\]

\(\Rightarrow\) \(x = \frac(7)(10)\) , \(y = \frac(1)(5)\) \(\Rightarrow\) \(x\cdot y = 0, fourteen\) . and \(ABCO\) is a parallelogram; \(AF \parallel BE\) and \(ABOF\) – parallelogram \(\Rightarrow\) \[\overrightarrow(BC) = \overrightarrow(AO) = \overrightarrow(AB) + \overrightarrow(BO) = \overrightarrow(AB) + \overrightarrow(AF) = \vec(a) + \vec(b)\ ]\(\Rightarrow\) \(x = 1\) , \(y = 1\) \(\Rightarrow\) \(x + y = 2\) .

Answer: 2

High school students preparing for passing the exam in mathematics and at the same time expect to receive decent points, they must definitely repeat the topic “Rules for adding and subtracting several vectors”. As can be seen from many years of practice, such tasks are included in the certification test every year. If a graduate has difficulties with tasks from the “Geometry on a Plane” section, for example, in which it is required to apply the rules of addition and subtraction of vectors, he should definitely repeat or re-understand the material in order to successfully pass the exam.

The educational project "Shkolkovo" offers new approach in preparation for the certification test. Our resource is built in such a way that students can identify the most difficult sections for themselves and fill in knowledge gaps. Shkolkovo specialists have prepared and systematized all the necessary material to prepare for the certification test.

To USE tasks, in which it is necessary to apply the rules of addition and subtraction of two vectors, did not cause difficulties, we recommend that you first of all refresh your memory basic concepts. Students can find this material in the "Theoretical Reference" section.

If you have already remembered the vector subtraction rule and the basic definitions on this topic, we suggest that you consolidate your knowledge by completing the appropriate exercises that were selected by experts educational portal"Shkolkovo". For each problem, the site presents a solution algorithm and gives the correct answer. The topic "Rules of vector addition" presents various exercises; after completing two or three relatively easy tasks, students can successively move on to more difficult ones.

To hone their own skills in such tasks, for example, as schoolchildren have the opportunity online, being in Moscow or any other city in Russia. If necessary, the task can be saved in the "Favorites" section. Thanks to this, you can quickly find examples of interest and discuss the algorithms for finding the correct answer with the teacher.

In mathematics and physics, students and schoolchildren often come across tasks for vector quantities and for performing various operations on them. What is the difference between vector quantities and scalar quantities familiar to us, the only characteristic of which is a numerical value? Because they have direction.

The use of vector quantities is most clearly explained in physics. by the most simple examples are forces (friction force, elastic force, weight), velocity and acceleration, since in addition to numerical values they also have a direction of action. For comparison, let's take example scalars : this can be the distance between two points or the mass of the body. Why is it necessary to perform operations on vector quantities such as addition or subtraction? This is necessary in order to be able to determine the result of the action of a vector system consisting of 2 or more elements.

Definitions of vector mathematics

Let us introduce the main definitions used in the implementation line operations.

A vector is a directed (having a start point and an end point) segment.
The length (modulus) is the length of the directed segment.
Collinear vectors are two vectors that are either parallel to the same line or lie simultaneously on it.
Oppositely directed vectors are called collinear and, at the same time, directed in different sides. If their direction coincides, then they are co-directional.
Vectors are equal when they are codirectional and have the same absolute value.
The sum of two vectors a and b is such a vector c, the beginning of which coincides with the beginning of the first, and the end - with the end of the second, provided that b starts at the same point it ends a.
Vector difference a and b call the amount a and ( - b ), where ( - b ) - opposite to the vector b. Also, the definition of the difference of two vectors can be given as follows: by the difference c couple vectors a and b call this c, which, when added to the subtrahend b forms a reduced a.

Analytical method

The analytical method involves obtaining the coordinates of the difference according to the formula without construction. It is possible to calculate for flat (2D), volume (3D), or n-dimensional space.

For two-dimensional space and vector quantities a {a₁;a₂) and b {b₁;b₂} calculations will be next view: c {c₁; c₂} = {a₁ – b₁; a₂ – b₂}.

In the case of adding a third coordinate, the calculation will be carried out in a similar way, and for a {a₁;a₂; a₃) and b {b₁;b₂; b₃) the coordinates of the difference will also be obtained by pairwise subtraction: c {c₁; c₂; c₃} = {a₁ – b₁; a₂ – b₂; a₃–b₃}.

Computing the difference graphically

In order to construct the difference graphically, use the triangle rule. To do this, you must perform the following sequence of actions:

By given coordinates construct the vectors for which you need to find the difference.
Combine their ends (i.e., construct two directed segments equal to the given ones, which will end at the same point).
Connect the beginnings of both directed segments and indicate the direction; the resulting one will start at the same point where the vector being minuend started and end at the start point of the vector being subtracted.

The result of the subtraction operation is shown in the figure below..

There is also a method for constructing a difference, slightly different from the previous one. Its essence lies in the application of the theorem on the difference of vectors, which is formulated as follows: in order to find the difference of a pair of directed segments, it is enough to find the sum of the first of them with the segment opposite to the second. The construction algorithm will look like this:

Construct initial directed segments.
The one that is subtrahend must be reflected, i.e., construct an oppositely directed and equal segment; then combine its beginning with the reduced one.
Construct the sum: connect the beginning of the first segment with the end of the second.

The result of this decision is shown in the figure:

Problem solving

To consolidate the skill, we will analyze several tasks in which it is required to calculate the difference analytically or graphically.

Task 1. There are 4 points on the plane: A (1; -3), B (0; 4), C (5; 8), D (-3; 2). Determine the coordinates of the vector q = AB - CD, and also calculate its length.

Solution. First you need to find the coordinates AB and CD. To do this, subtract the coordinates of the initial points from the coordinates of the end points. For AB the beginning is A(1; -3), and the end - B(0; 4). Calculate the coordinates of the directed segment:

AB {0 - 1; 4 - (- 3)} = {- 1; 7}

A similar calculation is performed for CD:

CD {- 3 - 5; 2 - 8} = {- 8; - 6}

Now, knowing the coordinates, you can find the difference of the vectors. Formula for analytical solution flat tasks has been discussed previously: c = a- b coordinates look like ( c₁; c₂} = {a₁ – b₁; a₂ – b₂). For a specific case, you can write:

q = {- 1 - 8; 7 - (- 6)} = { - 9; - 1}

To find the length q, we use the formula | q| = √(q₁² + q₂²) = √((- 9)² + (- 1)²) = √(81 + 1) = √82 ≈ 9.06.

Task 2. The figure shows the vectors m, n and p.

It is necessary to construct differences for them: p- n; m- n; m-n- p. Find out which one has the smallest modulus.

Solution. The task requires three constructions. Let's look at each part of the task in more detail.

Part 1. In order to portray p-n, Let's use the triangle rule. To do this, using parallel translation, we connect the segments so that their end point coincides. Now let's connect the starting points and define the direction. In our case, the difference vector starts in the same place as the subtracted one. n.

Part 2. Let's portray m-n. Now for the solution we use the theorem on the difference of vectors. To do this, construct a vector opposite n, and then find its sum with m. The result will look like this:

Part 3 In order to find the difference m-n-p, split the expression into two steps. Because in vector algebra there are laws similar to the laws of arithmetic, then the following options are possible:

m-(n+p): in this case, the sum is built first n+p, which is then subtracted from m;
(m-n)-p: here first you need to find m-n, and then subtract from this difference p;
(m-p)-n: the first action is determined m-p, after which from the result you need to subtract n.

Since in the previous part of the problem we have already found the difference m-n, we can only subtract from it p. Let us construct the difference of two given vectors using the difference theorem. The answer is shown in the image below (red color indicates intermediate result, and green - final).

It remains to determine which of the segments has the smallest modulus. Recall that the concepts of length and modulus in vector mathematics are identical. Estimate visually the lengths p- n, m-n and m-n-p. Obviously, the answer in the last part of the problem is the shortest and has the smallest modulus, namely m-n-p.

Sum of vectors. The length of the vector. Dear friends, there is a group of tasks with vectors in the back exam types. Quite a wide range of tasks (important to know theoretical basis). Most are resolved orally. Questions are related to finding the length of a vector, the sum (difference) of vectors, the scalar product. There are also many tasks, in the solution of which it is necessary to carry out actions with the coordinates of the vectors.

The theory behind vectors is simple and should be well understood. In this article, we will analyze the tasks associated with finding the length of a vector, as well as the sum (difference) of vectors. Some theoretical points:

Vector concept

A vector is a directed line segment.

All vectors that have the same direction and are equal in length are equal.

*All four vectors above are equal!

That is, if we use parallel translation to move the vector given to us, we will always get a vector equal to the original one. Thus, there can be an infinite number of equal vectors.

Vector notation

Vector can be denoted by latin capital letters, for example:

With this form of notation, the letter denoting the beginning of the vector is first written, then the letter denoting the end of the vector.

Another vector is denoted by one letter Latin alphabet(uppercase):

A designation without arrows is also possible:

The sum of the two vectors AB and BC will be the vector AC.
It is written as AB + BC \u003d AC.

This rule is called - triangle rule.

That is, if we have two vectors - let's call them conditionally (1) and (2), and the end of the vector (1) coincides with the beginning of the vector (2), then the sum of these vectors will be a vector whose beginning coincides with the beginning of the vector (1) , and the end coincides with the end of the vector (2).

Conclusion: if we have two vectors on the plane, we can always find their sum. Using parallel translation, you can move any of these vectors and connect its beginning to the end of another. For example:

Let's move the vector b, or in another way - we will construct equal to it:

How is the sum of several vectors found? By the same principle:

* * *

parallelogram rule

This rule is a consequence of the above.

For vectors with common beginning their sum is represented by the diagonal of the parallelogram built on these vectors.

Let's build a vector equal to vector b so that its beginning coincides with the end of the vector a, and we can build a vector that will be their sum:

Some more important information needed to solve problems.

A vector equal in length to the original one, but oppositely directed, is also denoted but has the opposite sign:

This information is extremely useful for solving problems in which there is a question of finding the difference of vectors. As you can see, the difference of vectors is the same sum in a modified form.

Let two vectors be given, find their difference:

We built a vector opposite to the vector b, and found the difference.

Vector coordinates

To find the vector coordinates, you need to subtract the corresponding start coordinates from the end coordinates:

That is, the coordinates of the vector are a pair of numbers.

If a

And the coordinates of the vectors look like:

Then c 1 \u003d a 1 + b 1 c 2 \u003d a 2 + b 2

If a

Then c 1 \u003d a 1 - b 1 c 2 \u003d a 2 - b 2

Vector modulus

The module of a vector is its length, determined by the formula:

The formula for determining the length of a vector if the coordinates of its beginning and end are known:

Consider the tasks:

The two sides of the rectangle ABCD are 6 and 8. The diagonals intersect at point O. Find the length of the difference between the vectors AO and BO.

Let's find a vector that will be the result of AO - VO:

AO -VO \u003d AO + (-VO) \u003d AB

That is, the difference between the vectors AO and VO will be a vector AB. And its length is eight.

Rhombus diagonals ABCD are 12 and 16. Find the length of the vector AB +AD.

Let's find a vector that will be the sum of vectors AD and AB BC equal to the vector AD. So AB+AD=AB+BC=AC

AC is the length of the diagonal of the rhombus AC, it is equal to 16.

The diagonals of the rhombus ABCD intersect at a point O and are equal to 12 and 16. Find the length of the vector AO + BO.

Let's find a vector that will be the sum of the vectors AO and BO BO is equal to the vector OD,

AD is the length of the side of the rhombus. The problem is to find the hypotenuse in right triangle AOD. Let's calculate the legs:

According to the Pythagorean theorem:

The diagonals of the rhombus ABCD intersect at the point O and are equal to 12 and 16. Find the length of the vector AO –BO.

Let's find a vector that will be the result of AO - VO:

AB is the length of the side of the rhombus. The problem is reduced to finding the hypotenuse AB in a right triangle AOB. calculate the legs:

According to the Pythagorean theorem:

Parties right triangle ABCs are 3.

Find the length of the vector AB -AC.

Let's find the result of the difference of vectors:

CB is equal to three, because the condition says that the triangle is equilateral and its sides are equal to 3.

27663. Find the length of the vector a (6; 8).

27664. Find the square of the length of the vector AB.