Calculate the modulus of the geometric sum of vectors. Vectors
Many physical quantities are completely determined by specifying some number. These are, for example, volume, mass, density, body temperature, etc. Such quantities are called scalar. For this reason, numbers are sometimes called scalars. But there are also such quantities that are determined by setting not only a number, but also a certain direction. For example, when a body moves, one should indicate not only the speed with which the body moves, but also the direction of movement. In the same way, when studying the action of any force, it is necessary to indicate not only the value of this force, but also the direction of its action. Such quantities are called vector. To describe them, the concept of a vector was introduced, which turned out to be useful for mathematics.
Vector definition
Any ordered pair of points A to B in space defines directed segment, i.e. segment along with the direction given on it. If point A is the first, then it is called the beginning of the directed segment, and point B is called its end. The direction of the segment is the direction from the beginning to the end.
Definition
A directed segment is called a vector.
We will denote the vector by the symbol \(\overrightarrow(AB) \), where the first letter means the beginning of the vector, and the second - its end.
A vector whose beginning and end are the same is called zero and is denoted by \(\vec(0) \) or just 0.
The distance between the start and end of a vector is called its length and denoted by \(|\overrightarrow(AB)| \) or \(|\vec(a)| \).
The vectors \(\vec(a) \) and \(\vec(b) \) are called collinear if they lie on the same line or on parallel lines. Collinear vectors can be directed the same or opposite.
Now we can formulate the important concept of the equality of two vectors.
Definition
Vectors \(\vec(a) \) and \(\vec(b) \) are called equal (\(\vec(a) = \vec(b) \)) if they are collinear, have the same direction, and their lengths are equal .
On fig. 1, unequal vectors are shown on the left, and equal vectors \(\vec(a) \) and \(\vec(b) \) are shown on the right. It follows from the definition of vector equality that if given vector moved parallel to itself, you get a vector equal to the given one. In this regard, vectors in analytic geometry are called free.
Projection of a vector onto an axis
Let the axis \(u\) and some vector \(\overrightarrow(AB)\) be given in space. Let's draw through points A and B in the plane perpendicular to the axis \ (u \). Let us denote by A "and B" the points of intersection of these planes with the axis (see Figure 2).
The projection of the vector \(\overrightarrow(AB) \) onto the \(u\) axis is the value A"B" of the directed segment A"B" on the \(u\) axis. Recall that
\(A"B" = |\overrightarrow(A"B")| \) , if the direction \(\overrightarrow(A"B") \) is the same as the direction of the axis \(u \),
\(A"B" = -|\overrightarrow(A"B")| \) if the direction of \(\overrightarrow(A"B") \) is opposite to the direction of the \(u \) axis,
The projection of the vector \(\overrightarrow(AB) \) onto the axis \(u \) is denoted as follows: \(Pr_u \overrightarrow(AB) \).
Theorem
The projection of the vector \(\overrightarrow(AB) \) onto the axis \(u \) is equal to the length of the vector \(\overrightarrow(AB) \) times the cosine of the angle between the vector \(\overrightarrow(AB) \) and the axis \( u \) , i.e.
Comment
Let \(\overrightarrow(A_1B_1)=\overrightarrow(A_2B_2) \) and some axis \(u \) be given. Applying the formula of the theorem to each of these vectors, we obtain
Vector projections on coordinate axes
Let a rectangular coordinate system Oxyz and an arbitrary vector \(\overrightarrow(AB) \) be given in space. Let, further, \(X = Pr_u \overrightarrow(AB), \;\; Y = Pr_u \overrightarrow(AB), \;\; Z = Pr_u \overrightarrow(AB) \). The projections X, Y, Z of the vector \(\overrightarrow(AB) \) on the coordinate axes call it coordinates. At the same time they write
\(\overrightarrow(AB) = (X;Y;Z) \)
Theorem
Whatever two points A(x 1 ; y 1 ; z 1) and B(x 2 ; y 2 ; z 2) are, the coordinates of the vector \(\overrightarrow(AB) \) are defined by the following formulas:
X \u003d x 2 -x 1, Y \u003d y 2 -y 1, Z \u003d z 2 -z 1
Comment
If the vector \(\overrightarrow(AB) \) leaves the origin, i.e. x 2 = x, y 2 = y, z 2 = z, then the X, Y, Z coordinates of the vector \(\overrightarrow(AB) \) are equal to the coordinates of its end:
X=x, Y=y, Z=z.
Vector direction cosines
Let an arbitrary vector \(\vec(a) = (X;Y;Z) \); we assume that \(\vec(a) \) leaves the origin and does not lie in any coordinate plane. Let us draw through point A planes perpendicular to the axes. Together with coordinate planes they form a rectangular parallelepiped, the diagonal of which is the segment OA (see figure).
It is known from elementary geometry that the square of the length of the diagonal cuboid is equal to the sum the squares of the lengths of its three dimensions. Hence,
\(|OA|^2 = |OA_x|^2 + |OA_y|^2 + |OA_z|^2 \)
But \(|OA| = |\vec(a)|, \;\; |OA_x| = |X|, \;\; |OA_y| = |Y|, \;\;|OA_z| = |Z| \); thus we get
\(|\vec(a)|^2 = X^2 + Y^2 + Z^2 \)
or
\(|\vec(a)| = \sqrt(X^2 + Y^2 + Z^2) \)
This formula expresses the length arbitrary vector through its coordinates.
Denote by \(\alpha, \; \beta, \; \gamma \) the angles between the vector \(\vec(a) \) and the coordinate axes. From the formulas for the projection of the vector onto the axis and the length of the vector, we obtain
\(\cos \alpha = \frac(X)(\sqrt(X^2 + Y^2 + Z^2)) \)
\(\cos \beta = \frac(Y)(\sqrt(X^2 + Y^2 + Z^2)) \)
\(\cos \gamma = \frac(Z)(\sqrt(X^2 + Y^2 + Z^2)) \)
\(\cos \alpha, \;\; \cos \beta, \;\; \cos \gamma \) are called direction cosines of the vector \(\vec(a) \).
Squaring the left and right sides of each of the previous equalities and summing up the results, we have
\(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \)
those. the sum of the squared direction cosines of any vector is equal to one.
Linear operations on vectors and their main properties
Linear operations on vectors are the operations of adding and subtracting vectors and multiplying vectors by numbers.Addition of two vectors
Let two vectors \(\vec(a) \) and \(\vec(b) \) be given. The sum \(\vec(a) + \vec(b) \) is a vector that goes from the beginning of the vector \(\vec(a) \) to the end of the vector \(\vec(b) \) provided that the vector \(\vec(b) \) is attached to the end of the vector \(\vec(a) \) (see figure).Comment
The action of subtracting vectors is the opposite of the action of addition, i.e. the difference \(\vec(b) - \vec(a) \) of the vectors \(\vec(b) \) and \(\vec(a) \) is the vector which, together with the vector \(\vec(a) ) \) gives the vector \(\vec(b) \) (see figure).
Comment
Having determined the sum of two vectors, one can find the sum of any number of given vectors. Let, for example, given three vectors \(\vec(a),\;\; \vec(b), \;\; \vec(c) \). Adding \(\vec(a) \) and \(\vec(b) \), we get the vector \(\vec(a) + \vec(b) \). Now adding the vector \(\vec(c) \) to it, we get the vector \(\vec(a) + \vec(b) + \vec(c) \) 
The product of a vector by a number
Let a vector \(\vec(a) \neq \vec(0) \) and a number \(\lambda \neq 0 \) be given. The product \(\lambda \vec(a) \) is a vector that is collinear with the vector \(\vec(a) \), has a length equal to \(|\lambda| |\vec(a)| \), and a direction the same as the vector \(\vec(a) \) if \(\lambda > 0 \), and the opposite if \(\lambda geometric sense operations of multiplication of the vector \(\vec(a) \neq \vec(0) \) by the number \(\lambda \neq 0 \) can be expressed as follows: if \(|\lambda| >1 \), then when multiplying vector \(\vec(a) \) by the number \(\lambda \) the vector \(\vec(a) \) is "stretched" by \(\lambda \) times, and if \(|\lambda| 1 \ ).If \(\lambda =0 \) or \(\vec(a) = \vec(0) \), then the product \(\lambda \vec(a) \) is assumed to be equal to the zero vector.
Comment
Using the definition of multiplication of a vector by a number, it is easy to prove that if the vectors \(\vec(a) \) and \(\vec(b) \) are collinear and \(\vec(a) \neq \vec(0) \), then there exists (and only one) number \(\lambda \) such that \(\vec(b) = \lambda \vec(a) \)
Basic properties of linear operations
1. Commutative property of addition
\(\vec(a) + \vec(b) = \vec(b) + \vec(a) \)
2. Associative property of addition
\((\vec(a) + \vec(b))+ \vec(c) = \vec(a) + (\vec(b)+ \vec(c)) \)
3. Associative property of multiplication
\(\lambda (\mu \vec(a)) = (\lambda \mu) \vec(a) \)
4. Distributive property with respect to the sum of numbers
\((\lambda +\mu) \vec(a) = \lambda \vec(a) + \mu \vec(a) \)
5. Distributive property with respect to the sum of vectors
\(\lambda (\vec(a)+\vec(b)) = \lambda \vec(a) + \lambda \vec(b) \)
Comment
These properties line operations are of fundamental importance, since they make it possible to perform ordinary algebraic operations on vectors. For example, due to properties 4 and 5, it is possible to perform the multiplication of a scalar polynomial by a vector polynomial "term by term".
Vector projection theorems
Theorem
The projection of the sum of two vectors onto an axis is equal to the sum of their projections onto this axis, i.e.
\(Pr_u (\vec(a) + \vec(b)) = Pr_u \vec(a) + Pr_u \vec(b) \)
The theorem can be generalized to the case of any number of terms.
Theorem
When multiplying the vector \(\vec(a) \) by the number \(\lambda \), its projection onto the axis is also multiplied by this number, i.e. \(Ex_u \lambda \vec(a) = \lambda Ex_u \vec(a) \)
Consequence
If \(\vec(a) = (x_1;y_1;z_1) \) and \(\vec(b) = (x_2;y_2;z_2) \), then
\(\vec(a) + \vec(b) = (x_1+x_2; \; y_1+y_2; \; z_1+z_2) \)
Consequence
If \(\vec(a) = (x;y;z) \), then \(\lambda \vec(a) = (\lambda x; \; \lambda y; \; \lambda z) \) for any number \(\lambda \)
From here it is easy to deduce condition of collinearity of two vectors in coordinates.
Indeed, the equality \(\vec(b) = \lambda \vec(a) \) is equivalent to the equalities \(x_2 = \lambda x_1, \; y_2 = \lambda y_1, \; z_2 = \lambda z_1 \) or
\(\frac(x_2)(x_1) = \frac(y_2)(y_1) = \frac(z_2)(z_1) \) i.e. the vectors \(\vec(a) \) and \(\vec(b) \) are collinear if and only if their coordinates are proportional.
Decomposition of a vector in terms of a basis
Let the vectors \(\vec(i), \; \vec(j), \; \vec(k) \) - unit vectors coordinate axes, i.e. \(|\vec(i)| = |\vec(j)| = |\vec(k)| = 1 \), and each of them is equally directed with the corresponding coordinate axis (see figure). A triple of vectors \(\vec(i), \; \vec(j), \; \vec(k) \) is called basis.
The following theorem holds.
Theorem
Any vector \(\vec(a) \) can be expanded uniquely in the basis \(\vec(i), \; \vec(j), \; \vec(k)\; \), i.e. presented in the form
\(\vec(a) = \lambda \vec(i) + \mu \vec(j) + \nu \vec(k) \)
where \(\lambda, \;\; \mu, \;\; \nu \) are some numbers. 
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:
A bit 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
And the coordinates of the vectors look like:
Then c 1 \u003d a 1 + b 1 c 2 \u003d a 2 + b 2
If
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.
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 scalar example: 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 when performing linear 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.
Mathematical or physical quantities can be represented as scalars(numerical value), and vector quantities (magnitude and direction in space).
A vector is a directed line segment, for which it is indicated which of its boundary points is the beginning and which is the end. Thus, there are two components in the vector - this is its length and direction.
The image of the vector on the drawing.
When working with vectors, a certain Cartesian coordinate system is often introduced in which the coordinates of the vector are determined by decomposing it into basis vectors:
For a vector located in the coordinate space (x,y,z) and leaving the origin
The distance between the beginning and end of a vector is called its length, and to denote the length of a vector (its absolute value) use the modulo symbol.
Vectors located either on the same line or on parallel lines are called collinear. The null vector is considered collinear to any vector. Among collinear vectors distinguish between equally directed (co-directed) and oppositely directed vectors. Vectors are called coplanar if they lie either on the same plane or on straight lines parallel to the same plane.
1.Vector length (vector modulus)
The length of a vector defines it scalar value and depends on its coordinates, but does not depend on its direction. The length of a vector (or modulus of a vector) is calculated via arithmetic Square root from the sum of the squares of the coordinates (components) of the vector (the rule for calculating the hypotenuse in a right triangle is used, where the vector itself becomes the hypotenuse).
Through the coordinates, the module of the vector is calculated as follows:
For a vector located in the coordinate space (x,y) and coming out of the origin
For a vector located in coordinate space (x,y,z) and coming out of the origin, the formula will be similar to the formula for the diagonal of a rectangular parallelepiped, since the vector in space takes the same position relative to the coordinate axes.
2. Angle between vectors
The angle between two vectors plotted from one point is the shortest angle by which one of the vectors must be rotated around its origin to the position of the second vector. The angle between vectors is determined using an expression to determine the scalar product of vectors
Thus, the cosine of the angle between vectors is equal to the ratio of the scalar product to the product of the lengths or modules of the vectors. This formula can be used if the lengths of vectors and their scalar product, or the vectors are given by coordinates in rectangular system coordinates on a plane or in space in the form: and .
If vectors A and B are given in three-dimensional space and the coordinates of each of them are given in the form: and , then the angle between the vectors is determined by the following expression:
It should be noted that the angle between the vectors and can also be determined by applying the cosine theorem for a triangle: the square of any side of the triangle is equal to the sum of the squares of the other two sides minus double product these sides by the cosine of the angle between them.
where AB, OA, OB is the corresponding side of the triangle.
Cosine theorem for a triangle
Applied to vector calculus given formula will be rewritten as follows:
Thus, the angle between the vectors and is determined by the following expression:
where and is the module (length) of the vector, and is the module (length) of the vector, which is determined from the difference of two vectors. The unknowns entering the equation are determined by the coordinates of the vectors and .
3. Vector addition
The addition of two vectors and (the sum of two vectors) is the operation of calculating the vector , all elements of which are equal to the pairwise sum of the corresponding elements of the vectors and . If the vectors are given in a rectangular coordinate system sum of vectors
IN graphical form, With position of two free vectors can be carried out both according to the rule of a triangle, and according to the rule of a parallelogram.
Addition of two vectors
The addition of two sliding vectors is defined only in the case when the lines on which they are located intersect. The addition of two fixed vectors is defined only if they have a common origin.
triangle rule.
To add two vectors and 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 formed triangle, and its beginning coincides with the beginning of the first vector, and the end with the end of the second vector.
where is the angle between the vectors when the beginning of one coincides with the end of the other.
parallelogram rule.
To add two vectors and according to the parallelogram rule, both of these vectors are transferred parallel to themselves so that their beginnings coincide. Then the sum vector is given by the diagonal of the parallelogram built on them, coming from their common origin.
The module (length) of the sum vector is determined by the cosine theorem:
where is the angle between the vectors coming out of the same point.
Note:
As you can see, depending on which angle is chosen, the sign in front of the cosine of the angle changes in the formula for determining the module (length) of the sum vector.
4. Difference of vectors
The difference of vectors and (vector subtraction) is the operation of calculating a vector , all elements of which are equal to the pairwise difference of the corresponding elements of the vectors and . If the vectors are given in a rectangular coordinate system vector difference and can be found using the following formula:
In graphical form, the difference of vectors and is the sum of the vector and the vector opposite to the vector, i.e. 
Difference of two free vectors
The difference of two free vectors in graphical form can be determined both by the triangle rule and by the parallelogram rule. The modulus (length) of the difference vector is determined by the cosine theorem. Depending on the angle used in the formula, the sign in front of the cosine changes (discussed earlier).
5. Dot product of vectors
The scalar product of two vectors is called real number, equal to the product lengths of multiplied vectors by the cosine of the angle between them. The scalar product of vectors and is denoted by one of the following notation or or and is defined by the formula:
where are the lengths of the vectors and, respectively, and is the cosine of the angle between the vectors.
Dot product of two vectors
The scalar product can also be calculated through the coordinates of vectors in a rectangular coordinate system in a plane or in space.
The scalar product of two vectors on a plane or in three-dimensional space in a rectangular coordinate system is the sum of the products of the corresponding coordinates of the vectors and .
Thus, for vectors and on a plane in a rectangular Cartesian system coordinates, the formula for calculating the scalar product is as follows:
For three-dimensional space the formula for calculating the scalar product of vectors and has the following form:
Properties of the scalar product.
1. The commutativity property of the scalar product
2. The distributivity property of the scalar product
3. Associative property of the scalar product (associativity)
where is an arbitrary real number.
It should be noted that in the case of:
If the dot product is positive, then the angle between the vectors is acute (less than 90 degrees);
If the dot product is negative, then the angle between the vectors is obtuse (greater than 90 degrees);
If the dot product is 0, then the vectors are orthogonal (which lie perpendicular to each other);
If the scalar product is equal to the product of the lengths of the vectors, then these vectors are collinear with each other (parallel).
6. Vector product of vectors
A vector product of two vectors is a vector for which the following conditions are met:
1. the vector is orthogonal (perpendicular) to the plane of the vectors and ;
The meaning of the name Maria for a girl and her fate
What is the difference between the name Natalya, Natasha and Natalia?
What is the difference between the name Natalya, Natasha and Natalia?