Useful Tips

Vector module


Let's analyze the vector. The first coordinate is $ a_x = 4 $, and the second coordinate is $ a_y = -3 $. Since two coordinates are given, we conclude that the problem is flat. You must apply the first formula. We substitute the values ​​from the condition of the problem into it:

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Example 1
Find the length of a vector by its coordinates $ overline = (4, -3) $
Vector Length $ | overline | = 5 $

We immediately notice that a spatial problem is given. Namely, $ a_x = 4, a_y = 2, a_z = 4 $. To find the length of the vector, we use the second formula. Substitute the unknowns into it:

Example 2
Find the length of the vector by the coordinates $ overline = (4,2,4) $
Vector length $ | overline | = 6 $

The problem is given flat judging by the presence of only two coordinates of the vectors. But this time given the beginning and end of the vector. Therefore, first we find the coordinates of the vector $ overline $, and only then its length according to the coordinate formula:

Now that the coordinates of the vector $ overline $ became known, you can use the familiar formula:

Example 3
Find the length of the vector if the coordinates of its beginning and end are known. $ A = (2,1), B = (- 1,3) $
$ | overline| = sqrt <13> $

In the article, we answered the question: "How to find the length of a vector?" using formulas. And also considered practical examples of solving problems on the plane and in space. It should be noted that there are similar formulas for spaces more than three-dimensional.

The formula for the length of an n-dimensional vector

In the case of n-dimensional space, the module of the vector a = <a 1 , a 2,. , an > can be found using the following formula:

| a | = (nai 2 ) 1/2
i = 1

Examples of calculating the length of a vector for spaces with dimension greater than 3

Decision: | a | = √ 1 2 + (-3) 2 + 3 2 + (-1) 2 = √ 1 + 9 + 9 + 1 = √ 20 = 2√ 5

Decision: | a | = √ 2 2 + 4 2 + 4 2 + 6 2 + 2 2 = √ 4 + 16 + 16 + 36 + 4 = √ 76 = 2√ 19.

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In order to emphasize that this is a vector (and not a scalar), use the line above, the arrow above, bold or gothic:

Addition of vectors is almost always indicated by a plus sign:

Multiplication by a number - simply by writing side by side, without a special sign, for example:

and the number is usually written on the left.

Multiplication by a matrix is ​​also indicated by writing next to it, without a special sign, but here the permutation of the factors in the general case affects the result. The action of a linear operator on a vector is also indicated by writing the operator on the left, without a special sign.

Intuitively, a vector is understood as an object having a magnitude, direction, and (optionally) an application point. The rudiments of vector calculus appeared along with the geometric model of complex numbers (Gauss, 1831). Hamilton published the developed operations with vectors as part of his quaternion calculus (the imaginary components of the quaternion formed the vector). Hamilton proposed the term itself vector (lat.Vector, carrier) and described some operations of vector analysis. This formalism was used by Maxwell in his works on electromagnetism, thereby drawing the attention of scientists to the new calculus. Gibbs (Elements of Vector Analysis) (1880s) soon came out, and then Heaviside (1903) gave vector analysis a modern look. There are no generally accepted vector notations; bold type, a line or arrow above a letter, the Gothic alphabet, etc. are used.

In geometry, vectors mean directional segments. This interpretation is often used in computer graphics, building lighting maps, using normals to surfaces. Also, using vectors, you can find the areas of various shapes, such as triangles and parallelograms, as well as the volumes of bodies: a tetrahedron and a parallelepiped.
Sometimes a direction is identified with a vector.

A vector in geometry is naturally associated with a transfer (parallel transfer), which, obviously, clarifies the origin of its name (Latin vector, carrier) Indeed, any directed segment unambiguously defines a parallel translation of a plane or space, and vice versa, parallel translation uniquely defines a single directed segment (unambiguously - if all directed segments of the same direction and length are considered equal - that is, they are considered as free vectors) .

Interpretation of a vector as a transfer allows a natural and intuitive way to introduce the operation of adding vectors - as a composition (sequential application) of two (or several) transfers, the same applies to the operation of multiplying the vector by a number.

In linear algebra, a vector is an element of a linear space, which corresponds to the general definition below. Vectors can have a different nature: directed segments, matrices, numbers, functions, and others, however, all linear spaces of the same dimension are isomorphic to each other.
This concept of a vector is most often used in solving systems of linear algebraic equations, as well as when working with linear operators (an example of a linear operator is the rotation operator). Often this definition is expanded by defining a norm or a scalar product (possibly both), after which they operate on normalized and Euclidean spaces, connect the notion of the angle between the vectors with the scalar product, and the notion of the length of the vector with the norm. Many mathematical objects (for example, matrices, tensors, etc.), including those having a structure more general than a finite (and sometimes even countable) ordered list, satisfy the axioms of a vector space, that is, from the point of view of algebra, they are vectors .

Functional analysis considers functional spaces - infinite-dimensional linear spaces. Their elements may be functions. Based on this representation of a function, the theory of Fourier series is constructed. Similarly, with linear algebra, a norm, a scalar product, or a metric on a function space is often introduced. Some methods for solving differential equations, for example, the finite element method, are based on the concept of a function as an element of a Hilbert space.

The most general definition of a vector is given by means of general algebra:

Many results of linear algebra are generalized to unitary modules over non-commutative bodies and even arbitrary modules over rings, so in the most general case, in some contexts, any element of a module over a ring can be called a vector.

Vector, as a structure having both magnitude (module) and direction, is considered in physics as a mathematical model of speed, force, and related quantities, kinematic or dynamic. The mathematical model of many physical fields (for example, electromagnetic fields or fluid velocity fields) are vector fields.

Abstract multidimensional and infinite-dimensional (in the spirit of functional analysis) vector spaces are used in the Lagrangian and Hamiltonian formalism as applied to mechanical and other dynamical systems, as well as in quantum mechanics (see State Vector).

Vector - (sequence, tuple) of homogeneous elements. This is the most general definition in the sense that ordinary vector operations may not be specified at all, they may be less, or they may not satisfy the usual axioms of linear space. It is in this form that the vector is understood in programming, where, as a rule, it is denoted by an identifier name with square brackets (for example, object) The list of properties models the definition of the class and state of an object accepted in system theory. So the types of vector elements determine the class of the object, and the values ​​of the elements determine its state. However, it is likely that this use of the term already goes beyond what is usually accepted in algebra, and in mathematics in general.

Addition of vectors. Vector amount. Rules for adding vectors. Geometric sum. Online calculator

In mechanics, there are two types of quantities:

  • scalar quantities specifying some numerical value - time, temperature, mass, etc.
  • vector quantities which, together with some numerical value, determine the direction - speed, force, etc.

We first consider the algebraic approach to the addition of vectors.

Coordinate addition of vectors.

Let two vectors given coordinate-wise be given (in order to calculate the coordinates of a vector, it is necessary to subtract the corresponding coordinates of its beginning from the corresponding coordinates of its end, i.e. from the first coordinate - the first, from the second - the second, etc.):

Then the coordinates of the vector obtained by adding these two vectors are calculated by the formula:

In the two-dimensional case, everything is absolutely analogous, just discard the third coordinate.

Now let's move on to the geometric meaning of adding two vectors:.

When adding vectors, their numerical values ​​and directions must be taken into account. There are several commonly used addition methods:

  • parallelogram rule
  • triangle rule
  • trigonometric method

Parallelogram rule.

The procedure for adding vectors according to the parallelogram rule is as follows:

  • draw the first vector, given its magnitude and direction
  • from the beginning of the first vector draw the second vector, also using its magnitude and its direction
  • supplement the drawing to a parallelogram, assuming that two drawn vectors are its sides
  • the resulting vector will be the diagonal of the parallelogram, and its beginning will coincide with the beginning of the first (and, therefore, the second) vector.

Triangle rule

The addition of vectors according to the triangle rule is as follows:

  • draw the first vector using data on its length (numerical value) and direction
  • draw the second vector from the end of the first vector, also taking into account its size and its direction
  • the resulting vector will be a vector whose beginning coincides with the beginning of the first vector, and the end with the end of the second.

Trigonometric method

The resulting addition vector of two coplanar vectors can be calculated using the cosine theorem:

F = numerical value of the vector

α = angle between vectors 1 and 2

The angle between the resulting vector and one of the original vectors can be calculated by the sine theorem:

α = angle between the original vectors

An example is the addition of vectors.

Force 1 is 5 kN and acts on the body in a direction 80 o different from the direction of action of the second force, equal to 8 kN.

The resulting force is calculated as follows:

Fres = 1/2

The angle between the resulting force and the first force is equal to:

And the angle between the second and the resulting force can be calculated as follows: as

α = arcsin

Online vector addition calculator.

The calculator below can be used for any vector quantities (force, speed, etc.) The point of origin of the vector coincides with the beginnings of both source vectors.

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