Construction according to the parallelogram rule. Addition of forces. Rule of parallelogram and polygon of forces. How to do addition using the parallelogram rule

Just as in Euclidean geometry, a point and a straight line are the main elements of the theory of planes, so a parallelogram is one of the key figures of convex quadrilaterals. From it, like threads from a ball, flow the concepts of “rectangle”, “square”, “rhombus” and other geometric quantities.

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Definition of parallelogram

convex quadrilateral, consisting of segments, each pair of which is parallel, is known in geometry as a parallelogram.

What a classic parallelogram looks like is depicted by a quadrilateral ABCD. The sides are called bases (AB, BC, CD and AD), the perpendicular drawn from any vertex to the side opposite to this vertex is called height (BE and BF), lines AC and BD are called diagonals.

Attention! Square, rhombus and rectangle are special cases of parallelogram.

Sides and angles: features of the relationship

Key properties, by and large, predetermined by the designation itself, they are proved by the theorem. These characteristics are as follows:

The sides that are opposite are identical in pairs.
Angles opposite each other are equal in pairs.

Proof: Consider ∆ABC and ∆ADC, which are obtained by dividing the quadrilateral ABCD with the straight line AC. ∠BCA=∠CAD and ∠BAC=∠ACD, since AC is common for them (vertical angles for BC||AD and AB||CD, respectively). It follows from this: ∆ABC = ∆ADC (the second sign of equality of triangles).

The segments AB and BC in ∆ABC correspond in pairs to the lines CD and AD in ∆ADC, which means that they are identical: AB = CD, BC = AD. Thus, ∠B corresponds to ∠D and they are equal. Since ∠A=∠BAC+∠CAD, ∠C=∠BCA+∠ACD, which are also pairwise identical, then ∠A = ∠C. The property has been proven.

Characteristics of the diagonals of a figure

Main feature of these lines of a parallelogram: the point of intersection divides them in half.

Proof: Let i.e. be the intersection point of diagonals AC and BD of figure ABCD. They form two commensurate triangles - ∆ABE and ∆CDE.

AB=CD since they are opposites. According to the lines and the secant, ∠ABE = ∠CDE and ∠BAE = ∠DCE.

By the second criterion of equality, ∆ABE = ∆CDE. This means that the elements ∆ABE and ∆CDE: AE = CE, BE = DE and at the same time they are proportional parts of AC and BD. The property has been proven.

Features of adjacent corners

Adjacent sides have a sum of angles equal to 180°, since they lie on the same side of parallel lines and a transversal. For quadrilateral ABCD:

∠A+∠B=∠C+∠D=∠A+∠D=∠B+∠C=180º

Properties of the bisector:

, lowered to one side, are perpendicular;
opposite vertices have parallel bisectors;
the triangle obtained by drawing a bisector will be isosceles.

Determination of the characteristic features of a parallelogram using the theorem

The characteristics of this figure follow from its main theorem, which states the following: a quadrilateral is considered a parallelogram in the event that its diagonals intersect, and this point divides them into equal segments.

Proof: let the lines AC and BD of the quadrilateral ABCD intersect in i.e. Since ∠AED = ∠BEC, and AE+CE=AC BE+DE=BD, then ∆AED = ∆BEC (by the first criterion for the equality of triangles). That is, ∠EAD = ∠ECB. They are also the internal cross angles of the secant AC for lines AD and BC. Thus, by definition of parallelism - AD || B.C. A similar property of lines BC and CD is also derived. The theorem is proven.

Calculating the area of a figure

Area of this figure found by several methods one of the simplest: multiplying the height and the base to which it is drawn.

Proof: draw perpendiculars BE and CF from vertices B and C. ∆ABE and ∆DCF are equal, since AB = CD and BE = CF. ABCD is equal in size to rectangle EBCF, since they consist of commensurate figures: S ABE and S EBCD, as well as S DCF and S EBCD. It follows from this that the area of this geometric figure is the same as that of a rectangle:

S ABCD = S EBCF = BE×BC=BE×AD.

To determine the general formula for the area of a parallelogram, let us denote the height as hb, and the side - b. Respectively:

Other ways to find area

Area calculations through the sides of the parallelogram and the angle, which they form, is the second known method.

Spr-ma - area;

a and b are its sides

α is the angle between segments a and b.

This method is practically based on the first, but in case it is unknown. always cuts off a right triangle whose parameters are found by trigonometric identities, that is. Transforming the relation, we get . In the equation of the first method, we replace the height with this product and obtain a proof of the validity of this formula.

Through the diagonals of a parallelogram and the angle, which they create when they intersect, you can also find the area.

Proof: AC and BD intersect to form four triangles: ABE, BEC, CDE and AED. Their sum is equal to the area of this quadrilateral.

The area of each of these ∆ can be found by the expression , where a=BE, b=AE, ∠γ =∠AEB. Since , the calculations use a single sine value. That is . Since AE+CE=AC= d 1 and BE+DE=BD= d 2, the area formula reduces to:

Application in vector algebra

The features of the constituent parts of this quadrilateral have found application in vector algebra, namely the addition of two vectors. The parallelogram rule states that if given vectorsAndNotare collinear, then their sum will be equal to the diagonal of this figure, the bases of which correspond to these vectors.

Proof: from an arbitrarily chosen beginning - i.e. - construct vectors and . Next, we construct a parallelogram OASV, where the segments OA and OB are sides. Thus, the OS lies on the vector or sum.

Formulas for calculating the parameters of a parallelogram

The identities are given under the following conditions:

a and b, α - sides and the angle between them;
d 1 and d 2, γ - diagonals and at the point of their intersection;
h a and h b - heights lowered to sides a and b;

Parameter	Formula
Finding the sides
along the diagonals and the cosine of the angle between them
along diagonals and sides
through the height and the opposite vertex
Finding the length of diagonals
on the sides and the size of the apex between them
along the sides and one of the diagonals	Conclusion The parallelogram, as one of the key figures of geometry, is used in life, for example, in construction when calculating the area of a site or other measurements. Therefore, knowledge about the distinctive features and methods of calculating its various parameters can be useful at any time in life.

Vector- a directed line segment, that is, a segment for which it is indicated which of its boundary points is the beginning and which is the end.

Vector starting at a point A (\displaystyle A) and end at a point B (\displaystyle B) usually denoted as . Vectors can also be denoted in small Latin letters with an arrow (sometimes a dash) above them, for example. Another common way of writing is to highlight the vector symbol in bold: a (\displaystyle \mathbf (a) ).

A vector in geometry is naturally compared to translation (parallel translation), which obviously clarifies the origin of its name (lat. vector, carrier). So, each directed segment uniquely defines some parallel transfer of plane or space: say, a vector A B → (\displaystyle (\overrightarrow (AB))) naturally determines the translation at which the point A (\displaystyle A) will go to point B (\displaystyle B), also vice versa, parallel transfer, in which A (\displaystyle A) goes into B (\displaystyle B), defines a single directed segment A B → (\displaystyle (\overrightarrow (AB)))(the only one is if we consider all directed segments of the same direction to be equal and - that is, consider them as; indeed, with parallel translation, all points are shifted in the same direction by the same distance, so in this understanding A 1 B 1 → = A 2 B 2 → = A 3 B 3 → = … (\displaystyle (\overrightarrow (A_(1)B_(1)))=(\overrightarrow (A_(2)B_(2)) )=(\overrightarrow (A_(3)B_(3)))=\dots )).

The interpretation of a vector as a transfer allows us to introduce an operation in a natural and intuitively obvious way - as a composition (sequential application) of two (or several) transfers; the same applies to the operation of multiplying a vector by a number.

Basic Concepts

A vector is a directed segment constructed from two points, one of which is considered the beginning and the other the end.

The coordinates of a vector are defined as the difference between the coordinates of its start and end points. For example, on a coordinate plane, if the start and end coordinates are given: T 1 = (x 1 , y 1) (\displaystyle T_(1)=(x_(1),y_(1))) And T 2 = (x 2 , y 2) (\displaystyle T_(2)=(x_(2),y_(2))), then the vector coordinates will be: V → = T 2 − T 1 = (x 2 , y 2) − (x 1 , y 1) = (x 2 − x 1 , y 2 − y 1) (\displaystyle (\overrightarrow (V))=T_ (2)-T_(1)=(x_(2),y_(2))-(x_(1),y_(1))=(x_(2)-x_(1),y_(2)-y_ (1))).

Vector length V → (\displaystyle (\overrightarrow (V))) is the distance between two points T 1 (\displaystyle T_(1)) And T 2 (\displaystyle T_(2)), it is usually denoted | V → | = | T 2 − T 1 | = | (x 2 − x 1 , y 2 − y 1) | = (x 2 − x 1) 2 + (y 2 − y 1) 2 (\displaystyle |(\overrightarrow (V))|=|T_(2)-T_(1)|=|(x_(2)- x_(1),y_(2)-y_(1))|=(\sqrt ((x_(2)-x_(1))^(2)+(y_(2)-y_(1))^( 2))))

The role of zero among vectors is played by the zero vector, whose beginning and end coincide T 1 = T 2 (\displaystyle T_(1)=T_(2)); it, unlike other vectors, is not assigned any direction.

For the coordinate representation of vectors, the concept is of great importance projection of the vector onto the axis(directional straight line, see figure). A projection is the length of a segment formed by the projections of the start and end points of a vector onto a given line, and the projection is assigned a plus sign if the direction of the projection corresponds to the direction of the axis, otherwise - a minus sign. The projection is equal to the length of the original vector multiplied by the cosine of the angle between the original vector and the axis; the projection of a vector onto the axis perpendicular to it is zero.

Applications

Vectors are widely used in geometry and applied sciences, where they are used to represent quantities that have a direction (forces, speeds, etc.). The use of vectors simplifies a number of operations - for example, determining angles between straight lines or segments, calculating the areas of figures. In computer graphics, normal vectors are used to create the correct lighting for the body. The use of vectors can be used as the basis for the coordinate method.

Types of vectors

Sometimes, instead of considering a set of vectors everyone directed segments (considering as distinct all directed segments whose beginnings and ends do not coincide), they take only some modification of this set (factor set), that is, some directed segments are considered equal if they have the same direction and length, although they may have different beginning (and end), that is, directed segments of the same length and direction are considered to represent the same vector; Thus, each vector turns out to have a corresponding whole class of directed segments, identical in length and direction, but differing in beginning (and end).

Yes, they talk about "free", "sliding" And "fixed" vectors. These types differ in the concept of equality of two vectors.

When talking about free vectors, they identify any vectors that have the same direction and length;
speaking about sliding vectors, they add that the origins of equal sliding vectors must coincide or lie on the same straight line on which the directed segments representing these vectors lie (so that one can be combined with another movement in the direction specified by it);
speaking about fixed vectors, they say that only vectors whose directions and origins coincide are considered equal (that is, in this case there is no factorization: there are no two fixed vectors with different origins that would be considered equal).

Formally:

They say that free vectors A B → (\displaystyle (\overrightarrow (AB))) and are equal if there are points E (\displaystyle E) And F (\displaystyle F) such that quadrilaterals A B F E (\displaystyle ABFE) And C D F E (\displaystyle CDFE)- parallelograms.

They say that sliding vectors A B → (\displaystyle (\overrightarrow (AB))) And C D → (\displaystyle \ (\overrightarrow (CD))) are equal if

Sliding vectors are especially used in mechanics. The simplest example of a sliding vector in mechanics is a force acting on a rigid body. Shifting the origin of the force vector along the straight line on which it lies does not change the moment of the force relative to any point; transferring it to another straight line, even if you do not change the magnitude and direction of the vector, can cause a change in its moment (even almost always will): therefore, when calculating the moment, the force cannot be considered as a free vector, that is, it cannot be considered applied to an arbitrary point of a rigid bodies.

They say that fixed vectors A B → (\displaystyle (\overrightarrow (AB))) And C D → (\displaystyle \ (\overrightarrow (CD))) are equal if the points coincide in pairs A (\displaystyle A) And C (\displaystyle C), B (\displaystyle B) And D (\displaystyle D).

In one case, a vector is a directed segment, and in other cases, different vectors are different equivalence classes of directed segments, determined by some specific equivalence relation. Moreover, the equivalence relation can be different, determining the type of vector (“free”, “fixed”, etc.). Simply put, within an equivalence class, all directed segments included in it are treated as completely equal, and each can equally represent the entire class.

All operations on vectors (addition, multiplication by a number, scalar and vector products, calculation of modulus or length, angle between vectors, etc.) are, in principle, defined identically for all types of vectors; the difference in types is reduced in this regard only to that for moving and fixed ones, a restriction is imposed on the possibility of performing operations between two vectors that have different beginnings (for example, for two fixed vectors, addition is prohibited - or makes no sense - if their beginnings are different; however, for all cases when this operation is allowed - or has meaning - it is the same as for free vectors). Therefore, often the vector type is not explicitly stated at all; it is assumed that it is obvious from the context. Moreover, depending on the context of the problem, the same vector can be considered as fixed, sliding or free; for example, in mechanics, vectors of forces applied to a body can be summed up regardless of the point of application when finding the resultant (both in statics and dynamics when studying the movement of the center of mass, changes in momentum, etc.), but cannot be added to each other without taking into account the points of application when calculating the torque (also in statics and dynamics).

Relationships between vectors

Coordinate representation

When working with vectors, a certain Cartesian coordinate system is often introduced and the coordinates of the vector are determined in it, decomposing it into basis vectors. Basis expansion can be represented geometrically using vector projections onto coordinate axes. If the coordinates of the beginning and end of the vector are known, the coordinates of the vector itself are obtained by subtracting the coordinates of its beginning from the coordinates of the end of the vector.

A B → = (A B x , A B y , A B z) = (B x − A x , B y − A y , B z − A z) (\displaystyle (\overrightarrow (AB))=(AB_(x), AB_(y),AB_(z))=(B_(x)-A_(x),B_(y)-A_(y),B_(z)-A_(z)))

Coordinate unit vectors, denoted by i → , j → , k → (\displaystyle (\vec (i)),(\vec (j)),(\vec (k))), corresponding to the axes x , y , z (\displaystyle x,y,z). Then the vector a → (\displaystyle (\vec (a))) can be written as

a → = a x i → + a y j → + a z k → (\displaystyle (\vec (a))=a_(x)(\vec (i))+a_(y)(\vec (j))+a_(z) (\vec (k)))

Any geometric property can be written in coordinates, after which the study from geometric becomes algebraic and is often simplified. The opposite, generally speaking, is not entirely true: it is usually customary to say that only those relations that hold in any Cartesian coordinate system have a “geometric interpretation” invariant).

Operations on vectors

Vector module

Vector module A B → (\displaystyle (\overrightarrow (AB))) is a number equal to the length of the segment A B (\displaystyle AB). Denoted as | A B → | (\displaystyle |(\overrightarrow (AB))|). Through coordinates it is calculated as:

| a → | = a x 2 + a y 2 + a z 2 (\displaystyle |(\vec (a))|=(\sqrt (a_(x)^(2)+a_(y)^(2)+a_(z)^( 2))))

Vector addition

In coordinate representation, the sum vector is obtained by summing the corresponding coordinates of the terms:

a → + b → = (a x + b x , a y + b y , a z + b z) (\displaystyle (\vec (a))+(\vec (b))=(a_(x)+b_(x),a_ (y)+b_(y),a_(z)+b_(z)))

To geometrically construct the sum vector c → = a → + b → (\displaystyle (\vec (c))=(\vec (a))+(\vec (b))) use different rules (methods), but they all give the same result. The use of one or another rule is justified by the problem being solved.

Triangle rule

The triangle rule follows most naturally from the understanding of a vector as a transfer. It is clear that the result of sequential application of two transfers a → (\displaystyle (\vec (a))) and some point will be the same as applying one transfer at once corresponding to this rule. To add two vectors a → (\displaystyle (\vec (a))) And b → (\displaystyle (\vec (b))) according to the triangle rule, both of these vectors are transferred parallel to themselves so that the beginning of one of them coincides with the end of the other. Then the sum vector is given by the third side of the resulting triangle, and its beginning coincides with the beginning of the first vector, and its end with the end of the second vector.

This rule can be directly and naturally generalized to the addition of any number of vectors, turning into broken line rule:

Three point rule

If the segment A B → (\displaystyle (\overrightarrow (AB))) depicts vector a → (\displaystyle (\vec (a))), and the segment B C → (\displaystyle (\overrightarrow (BC))) depicts vector b → (\displaystyle (\vec (b))), then the segment A C → (\displaystyle (\overrightarrow (AC))) depicts vector a → + b → (\displaystyle (\vec (a))+(\vec (b))) .

Polygon rule

The beginning of the second vector coincides with the end of the first, the beginning of the third with the end of the second, and so on, the sum n (\displaystyle n) vectors is a vector, with the beginning coinciding with the beginning of the first one, and the end coinciding with the end n (\displaystyle n)-th (that is, depicted by a directed segment closing the polyline). Also called the broken line rule.

Parallelogram rule

To add two vectors a → (\displaystyle (\vec (a))) And b → (\displaystyle (\vec (b))) According to the parallelogram rule, both of these vectors are transferred parallel to themselves so that their origins coincide. Then the sum vector is given by the diagonal of the parallelogram constructed on them, starting from their common origin. (It is easy to see that this diagonal coincides with the third side of the triangle when using the triangle rule).

The parallelogram rule is especially convenient when there is a need to depict the sum vector as immediately applied to the same point to which both terms are applied - that is, to depict all three vectors as having a common origin.

Vector sum modulus

Modulus of the sum of two vectors can be calculated using the cosine theorem:

| a → + b → | 2 = | a → | 2 + | b → | 2 + 2 | a → | | b → | cos ⁡ (a → , b →) (\displaystyle |(\vec (a))+(\vec (b))|^(2)=|(\vec (a))|^(2)+|( \vec (b))|^(2)+2|(\vec (a))||(\vec (b))|\cos((\vec (a)),(\vec (b))) ), Where a → (\displaystyle (\vec (a))) And b → (\displaystyle (\vec (b))).

If the vectors are depicted in accordance with the triangle rule and the angle is taken according to the drawing - between the sides of the triangle - which does not coincide with the usual definition of the angle between vectors, and therefore with the angle in the above formula, then the last term acquires a minus sign, which corresponds to the cosine theorem in its direct formulation.

For the sum of an arbitrary number of vectors a similar formula is applicable, in which there are more terms with cosine: one such term exists for each pair of vectors from the summed set. For example, for three vectors the formula looks like this:

| a → + b → + c → | 2 = | a → | 2 + | b → | 2 + | c → | 2 + 2 | a → | | b → | cos ⁡ (a → , b →) + 2 | a → | | c → | cos ⁡ (a → , c →) + 2 | b → | | c → | cos ⁡ (b → , c →) . (\displaystyle |(\vec (a))+(\vec (b))+(\vec (c))|^(2)=|(\vec (a))|^(2)+|(\ vec (b))|^(2)+|(\vec (c))|^(2)+2|(\vec (a))||(\vec (b))|\cos((\vec (a)),(\vec (b)))+2|(\vec (a))||(\vec (c))|\cos((\vec (a)),(\vec (c) ))+2|(\vec (b))||(\vec (c))|\cos((\vec (b)),(\vec (c))).)

Vector subtraction

Two vectors a → , b → (\displaystyle (\vec (a)),(\vec (b))) and the vector of their difference

To obtain the difference in coordinate form, you need to subtract the corresponding coordinates of the vectors:

a → − b → = (a x − b x , a y − b y , a z − b z) (\displaystyle (\vec (a))-(\vec (b))=(a_(x)-b_(x),a_ (y)-b_(y),a_(z)-b_(z)))

To obtain the difference vector c → = a → − b → (\displaystyle (\vec (c))=(\vec (a))-(\vec (b))) the beginnings of the vectors are connected by the beginning of the vector c → (\displaystyle (\vec (c))) there will be an end b → (\displaystyle (\vec (b))) and the end is the end a → (\displaystyle (\vec (a))). If we write using vector points, then A C → − A B → = B C → (\displaystyle (\overrightarrow (AC))-(\overrightarrow (AB))=(\overrightarrow (BC))).

Vector difference module

Three vectors a → , b → , a → − b → (\displaystyle (\vec (a)),(\vec (b)),(\vec (a))-(\vec (b))), as with addition, form a triangle, and the expression for the difference module is similar:

| a → − b → | 2 = | a → | 2 + | b → | 2 − 2 | a → | | b → | cos ⁡ (a → , b →) , (\displaystyle |(\vec (a))-(\vec (b))|^(2)=|(\vec (a))|^(2)+| (\vec (b))|^(2)-2|(\vec (a))||(\vec (b))|\cos((\vec (a)),(\vec (b)) ),)

Where cos ⁡ (a → , b →) (\displaystyle \cos((\vec (a)),(\vec (b))))- cosine of the angle between vectors a → (\displaystyle (\vec (a))) And b → . (\displaystyle (\vec (b)).)

The difference from the formula for the modulus of the sum is in the sign in front of the cosine; in this case, you need to carefully monitor which angle is taken (the version of the formula for the modulus of the sum with the angle between the sides of a triangle when summing according to the triangle rule does not differ in form from this formula for the modulus of the difference, but you need to have Note that different angles are taken here: in the case of a sum, the angle is taken when the vector b → (\displaystyle (\vec (b))) is carried to the end of the vector a → (\displaystyle (\vec (a))), when the modulus of the difference is sought, the angle between the vectors applied to one point is taken; expression for the modulus of the sum using the same angle as in this expression for the modulus of the difference, differs in the sign in front of the cosine).

In order to perform the operation of adding vectors, there are several methods that, depending on the situation and the type of vectors in question, may be more convenient to use. Let's look at the rules for adding vectors:

Triangle rule

The triangle rule is as follows: in order to add two vectors x, y, you need to construct vector x so that its beginning coincides with the end of vector y. Then their sum will be the value of vector z, and the beginning of vector z will coincide with the beginning of vector x, and the end with the end of vector y.

The triangle rule helps if the number of vectors that need to be summed is no more than two.

Polygon rule

The polygon rule is the simplest and most convenient for adding any number of vectors on a plane or in space. The essence of the rule is as follows: when adding vectors, you need to sequentially add them one after another, so that the beginning of the next vector coincides with the end of the previous one, while the vector that closes the resulting curve is the sum of the added vectors. This is clearly shown by the equality w= x + y + z, where the vector w is the sum of these vectors. In addition, it should be noted that changing the places of the terms of the vectors does not change the sum, that is, (x + y) + z = x + (y + z).

Parallelogram rule

The parallelogram rule is used to add vectors that originate from the same point. This rule states that the sum of vectors x and y, originating at one point, will be a third vector z, also emanating from this point, and the vectors x and y are the sides of the parallelogram, and the vector z is its diagonal. In this case, it also does not matter in what order the vectors will be added.

Thus, the polygon rule, the triangle rule and the parallelogram rule help solve vector addition problems of absolutely any complexity, both on the plane and in space.

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 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 a school geometry course, the addition of two vectors is considered. 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, one that has the opposite direction and equal magnitude. Their sum will be zero.

On the commutativity of addition. Something that has been known since elementary 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 one 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 adding, 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 BO.

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, the vector BA, which is directed in the opposite direction AB, should be set aside. 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.

Vector- a directed line segment, that is, a segment for which it is indicated which of its boundary points is the beginning and which is the end.

Basic Concepts[ | ]

A vector is a directed segment constructed from two points, one of which is considered the beginning and the other the end.

The role of zero among vectors is played by the zero vector, whose beginning and end coincide T 1 = T 2 (\displaystyle T_(1)=T_(2)); it, unlike other vectors, is not assigned any direction.

Applications [ | ]

Types of vectors [ | ]

Yes, they talk about "free", "sliding" And "fixed" vectors. These types differ in the concept of equality of two vectors.

When talking about free vectors, they identify any vectors that have the same direction and length;
speaking about sliding vectors, they add that the origins of equal sliding vectors must coincide or lie on the same straight line on which the directed segments representing these vectors lie (so that one can be combined with another movement in the direction specified by it);
speaking about fixed vectors, they say that only vectors whose directions and origins coincide are considered equal (that is, in this case there is no factorization: there are no two fixed vectors with different origins that would be considered equal).

Formally:

They say that sliding vectors A B → (\displaystyle (\overrightarrow (AB))) And C D → (\displaystyle \ (\overrightarrow (CD))) are equal if

Relationships between vectors[ | ]

Coordinate representation[ | ]

A B → = (A B x , A B y , A B z) = (B x − A x , B y − A y , B z − A z) (\displaystyle (\overrightarrow (AB))=(AB_(x), AB_(y),AB_(z))=(B_(x)-A_(x),B_(y)-A_(y),B_(z)-A_(z)))

a → = a x i → + a y j → + a z k → (\displaystyle (\vec (a))=a_(x)(\vec (i))+a_(y)(\vec (j))+a_(z) (\vec (k)))

Operations on vectors[ | ]

Vector module [ | ]

| a → | = a x 2 + a y 2 + a z 2 (\displaystyle |(\vec (a))|=(\sqrt (a_(x)^(2)+a_(y)^(2)+a_(z)^( 2))))

Vector addition[ | ]

In coordinate representation, the sum vector is obtained by summing the corresponding coordinates of the terms:

a → + b → = (a x + b x , a y + b y , a z + b z) (\displaystyle (\vec (a))+(\vec (b))=(a_(x)+b_(x),a_ (y)+b_(y),a_(z)+b_(z)))

Triangle rule[ | ]

This rule can be directly and naturally generalized to the addition of any number of vectors, turning into broken line rule:

Three point rule[ | ]

Polygon rule[ | ]

Parallelogram rule[ | ]

Vector sum modulus[ | ]

Modulus of the sum of two vectors can be calculated using the cosine theorem:

Vector subtraction[ | ]

Two vectors a → , b → (\displaystyle (\vec (a)),(\vec (b))) and the vector of their difference

To obtain the difference in coordinate form, you need to subtract the corresponding coordinates of the vectors:

a → − b → = (a x − b x , a y − b y , a z − b z) (\displaystyle (\vec (a))-(\vec (b))=(a_(x)-b_(x),a_ (y)-b_(y),a_(z)-b_(z)))

Vector difference module[ | ]

Three vectors a → , b → , a → − b → (\displaystyle (\vec (a)),(\vec (b)),(\vec (a))-(\vec (b))), as with addition, form a triangle, and the expression for the difference module is similar:

Where cos ⁡ (a → , b →) (\displaystyle \cos((\vec (a)),(\vec (b))))- cosine of the angle between vectors a → (\displaystyle (\vec (a))) And b → . (\displaystyle (\vec (b)).)

Multiplying a vector by a number[ | ]

Vector multiplication a → (\displaystyle (\vec (a))) per number α > 0 (\displaystyle \alpha >0), gives a codirectional vector with a length of α (\displaystyle \alpha ) times more.
Vector multiplication a → (\displaystyle (\vec (a))) per number α < 0 {\displaystyle \alpha <0} , gives an oppositely directed vector with a length of | α | (\displaystyle |\alpha |) times more. Multiplying a vector by a number in coordinate form is done by multiplying all coordinates by this number.

Construction according to the parallelogram rule. Addition of forces. Rule of parallelogram and polygon of forces. How to do addition using the parallelogram rule

Definition of parallelogram

Sides and angles: features of the relationship

Characteristics of the diagonals of a figure

Features of adjacent corners

Determination of the characteristic features of a parallelogram using the theorem

Calculating the area of ​​a figure

Other ways to find area

Application in vector algebra

Formulas for calculating the parameters of a parallelogram

Conclusion

Basic Concepts

Applications

Types of vectors

Relationships between vectors

Coordinate representation

Operations on vectors

Vector module

Vector addition

Triangle rule

Three point rule

Polygon rule

Parallelogram rule

Vector sum modulus

Vector subtraction

Vector difference module

Triangle rule

Polygon rule

Parallelogram rule

How does addition occur using the triangle rule?

How is addition performed using the parallelogram rule?

How and when does the polygon rule apply?

What properties are valid for operations with vectors?

What is known about vector difference?

How to find the sum and difference of vectors in coordinates?

First example with solution

Second example and its detailed solution

Third example with detailed solution

Fourth example

Basic Concepts[ | ]

Applications [ | ]

Types of vectors [ | ]

Relationships between vectors[ | ]

Coordinate representation[ | ]

Operations on vectors[ | ]

Vector module [ | ]

Vector addition[ | ]

Triangle rule[ | ]

Three point rule[ | ]

Polygon rule[ | ]

Parallelogram rule[ | ]

Vector sum modulus[ | ]

Vector subtraction[ | ]

Vector difference module[ | ]

Multiplying a vector by a number[ | ]

Calculating the area of a figure