Vector

Vectors have a special and a more general meaning in mathematics. In most general form is a vector in Element one Vector space, D.h. an object, with its-same to be added can and with numbers (so-called Scalar one) to be multiplied can. A multiplication of vectors is not defined generally.

In that Geometry a vector is one Class of Arrows more directly Length (Amount), more directly Direction and more directly Orientation.

In order to shift for example the triangle ABC in the figure to the position A'B'C ', each point must be shifted 7 units upward to the right and 3. It moves thereby along an arrow <math> \vec v</math>. There these arrows in length, Direction and orientation everything agree, one combines it into a class (vector class), which one with < likewise briefly;math> \vec v</math> designated. Each arrow is in Representative this class. One describes the class by the shift, those their arrows cause, in the example:

<math> \vec v = \begin{pmatrix} 7 \ \ 3 \end{pmatrix}</math>,

in the three-dimensional area accordingly with 3 coordinates, see further below.

The definition of the vector in that linear algebra as Element one Vector space is a much more more comprehensive, beside the "conventional", geometrical vectors most diverse mathematical objects (Numbers, Consequences, Functions and Transformations) contains. Therefore is also each vector in Tensor, although one by convention two-dimensional vectors as Matrix and multidimensional vectors tensors calls. Also all designated sizes - like length specifications in m, Money etc.. usf. - are in this sense vectors.

In that Differentialgeometrie, that Physics and that Technology the expression refers Vector normally on a geometrical vector of the Euclidean area, by an amount, a direction and an orientation are given. Examples are Speed, Impulse, Strength, Moment and Acceleration. According to this definition is a vector in Tensor first stage. All following views refer to such vectors, general characteristics are under Vector space.

One knows vectors scalar Sizes how Distance, Energy, Time, Temperature, Charge, Achievement, Work and Mass confront, those an amount, but no direction and no orientation have.

Vectors are normally free, that is, they do not have a fixed starting point. A vector can therefore as those Quantity all "arrows", those kollinear (Dh. are parallel, thus possess the same direction), equivalent long and equivalent oriented are, are regarded. They serve generally for it, one Direction to indicate - see also Direction vector.

Have in contrast to it bound vectors one Origin (starting point). They can for example, as so mentioned Radius vectors, the position one Point indicate in the area. Forces, those on rigid bodies work, are partly bound. They work along a certain straight line. It is no matter, attack it at which point of the straight lines. One calls it "line-volatile" vectors.

A vector with the length 1 is called Unit vector. One can make each vector a unit vector, by one it standardized, that is called all coordinates by the amount (the length) of the vector divides.

A vector with same amount, same direction however opposite orientation of another vector is its Gegenvektor.

Table of contents 1 Representational forms 2 Arithmetic operations 2.1 Addition and subtraction 2.2 Multiplication with a scalar 2.3 Dot product 2.4 Angle between two vectors 2.5 Cross product 3 See also 4 History 5 Literature 6 Web on the left of

Representational forms

Variable one, for vectors stand, frequently by an arrow are marked (<math> \vec{a}</math> and/or. <math> \overrightarrow{AB}</math>) or fat written (A, OFF). In the English-language area the way of writing is <math> \underline a</math> and/or. <math> \underline{AB}</math> more common.(Note: In this article constantly the arrow way of writing is used, in addition, in other Wikipedia articles the bold print occurs.) is the amount, thus those Length, the vector meant, becomes the vector with two senkrechten Amount lines put in parentheses: <math>|\vec{a}|</math>

Vectors become graphic normally as Arrows represented:

A becomes in this case as output or starting point and B point or terminator point of the vector calls. The situation of the head of the arrow gives the orientation of the vector, the length its amount and the arrow shank its direction on. This vector can also as <math> \overrightarrow{AB}</math> designated and its amount as are <math>|\overrightarrow{AB}|</math> and/or. <math> \overline{AB}</math>. Is to be considered, that the vector not to the points A and B is bound, separate, that these it only define.

In order to be able to count on vectors meaningfully, the graphic notation is naturally unpractical. In one n-dimensional Vectors know Euclidean area as Linear combination of n Basis vectors this area to be represented. In cartesian coordinate system one takes for it n one on the other in pairs normally standing Unit vectors.

As example of this article is always the three-dimensional vector space <math> \mathbb{R}^3</math> with a cartesian coordinate system serve. Are <math> \vec{i}</math>, <math> \vec{j}</math> and <math> \vec{k}</math> the unit vectors toward the x -, y and/or. zAxle, each vector can as

<math> \vec{a} = a_1 \vec{i} + a_2 \vec{j} + a_3 \vec{k}</math>

are written down. Those real numbers A₁, A₂ and A₃ are clearly through <math> \vec{a}</math> specified. Often one writes vectors also briefly as 3×1 or 1×3Matrix:

<math>

 \vec{a} = \begin{pmatrix}a_1 \ \ a_2 \ \ a_3 \end{pmatrix} \quad \mbox{or} \quad \vec{a} = \begin{pmatrix}a_1 & a_2 & a_3 \ \ \end{pmatrix}

</math>

With this way of writing is the choice of the Coordinate system not held, if however always the cartesian coordinate system is not indicated meant different one, since it is simplest for many calculations.

One can then the coordinates vector also in such a way represent:

<math> \vec{a} = a_1 \begin{pmatrix}1 \ \ 0 \ \ 0 \ \ \end{pmatrix} + a_2 \begin{pmatrix}0 \ \ 1 \ \ 0 \ \ \end{pmatrix} + a_3 \begin{pmatrix}0 \ \ 0 \ \ 1 \ \ \end{pmatrix}</math>

From that Sentence of Pythagoras follows, that the amount of the vector can be computed as follows:

<math>|\vec{a}| = \sqrt{{a_1}^2 + {a_2}^2 + {a_3}^2}</math>

Arithmetic operations

Addition and Subtraction

Those Sum the two vectors

<math> \vec A = a_1 \vec i + a_2 \vec j + a_3 \vec k \ \mathrm{und} \ \vec b = b_1 \vec i + b_2 \vec j + b_3 \vec k</math>

computes itself as:

<math>

 \vec{a}+ \vec{b} = (a_1+b_1) \vec{i} + (a_2+b_2) \vec{j} + (a_3+b_3) \vec{k} = \begin{pmatrix}a_1+b_1 \ \ a_2+b_2 \ \ a_3+b_3 \end{pmatrix}

</math>.

The vectoraddition being able graphically to interpret by one the starting point of the second vector to the terminator point of the first vector attaches. The arrow of the starting point of the first vector up to the terminator point of the second vector represents the result vector:

The vectors <math> \vec a</math> and <math> \vec b</math> can here as Sides one Parallelogram are understood and the result vector as (longer) Diagonal one. To the addition of vectors that applies Associative law and that Commutative law.

Those Difference these two vectors is:

<math>

 \vec{a} - \vec{b} = (a_1-b_1) \vec{i} + (a_2-b_2) \vec{j} + (a_3-b_3) \vec{k} = \begin{pmatrix}a_1-b_1 \ \ a_2-b_2 \ \ a_3-b_3 \end{pmatrix}

</math>.

The geometrical interpretation that Subtraction from two vectors is: Two vectors are subtracted, by attaching the starting point of the Gegenvektors of the second vector to the terminator point of the first vector. The arrow of the result vector begins in the starting point of the first vector and ends in the terminator point of the Gegenvektors of the second vector. Geometrically the connecting vector between the terminator points second and the first vector corresponds to that.

Multiplication with a scalar

Vectors can also real numbers, often Scalar one called, in order to be able to differentiate it from vectors, are multiplied:

<math>

 r \vec{a} = (ra_1) \vec{i} + (ra_2) \vec{j} + (ra_3) \vec{k} = \begin{pmatrix}ra_1 \ \ ra_2 \ \ ra_3 \end{pmatrix}

</math>

The length of the resulting vector is < therefore;math>|r| \cdot| \vec{a}|</math>. If the scalar is positive, the resulting vector points to the same direction, it is negative, into the opposite direction. The following diagram illustrates two examples (multiplication with -1 and 2):Whereby the vector also the direction to change can do x(-1)

To the vektoraddition and the multiplication with a scalar that applies Distributive law:

<math>r \cdot(\vec A + \vec b) = r \vec A + r \vec b</math>

The multiplication of a vector with a scalar is called frequently also s-multiplication.

Dot product

That Dot product (or Internal product) < two vectors;math> \vec a</math> and <math> \vec b</math>, so mentioned because the result is a scalar, noted as one <math> \vec A \cdot \vec b</math> and is defined as

<math>

 \vec{a} \cdot \vec{b} = \left| \vec{a} \right| \left| \vec{b} \right| \cos \varphi

</math>,

whereby <math> \varphi</math> between both vectors the included Angle is (see also Cosine).

In the cartesian coordinate system the dot product computes itself as

<math> \vec{a} \cdot \vec{b} = \begin{pmatrix}a_1 \ \ a_2 \ \ a_3 \end{pmatrix} \cdot \begin{pmatrix}b_1 \ \ b_2 \ \ b_3 \end{pmatrix} =

a_1b_1+a_2b_2+a_3b_3</math>

in particular applies to the square of a vector

<math> \vec{a} \cdot \vec{a} = \begin{pmatrix}a_1 \ \ a_2 \ \ a_3 \end{pmatrix} \cdot \begin{pmatrix}a_1 \ \ a_2 \ \ a_3 \end{pmatrix} = a_1^2+a_2^2+a_3^2</math>

The dot product leaves itself geometrical to that also as multiplication of the length of the first vector with the length Projection the second vector on the first vector understand. Therefore is the dot product of two orthogonal vectors standing one on the other always 0. This operation often becomes in that Physics used, for example around those Work to compute, if Strength and Away in the same direction do not run.

To the dot product that applies Commutative law and that Distributive law.

Angle between two vectors

That Angle <math> \varphi </math>between two vectors <math> \vec{a}</math> and <math> \vec{b}</math> can be computed with the following formula, from the dot product follows:

<math> \cos \varphi = \frac{\vec{a} \cdot \vec{b}}{\left| \vec{a} \right| \cdot \left| \vec{b} \right|}</math>

That that Dot product a direct entrance to the computation of Angles in vector geometry makes possible, its great importance for geometry constitutes.

Cross product

That Cross product (also vectorial product, vector product or Cross product) (noted < as;math> \vec A \times \vec b</math>) (spoken as "A cross b") two vectors in a three-dimensional Euclidean Vector space is a certain vector, that normally (perpendicularly in the sense of the dot product) on of <math> \vec a</math> and <math> \vec b</math> stands for stretched level.

In the usual three-dimensional area <math> \mathbb{R}^3</math> is the cross product of A and b defined as

<math>

 \vec{a} \times \vec{b} = \left| \vec{a} \right| \left| \vec{b} \right| \sin(\theta) \vec{n}

</math> whereby <math> \theta</math> of both vectors the included angle (see also Sine), and <math> \vec n</math> too the two vectors normal unit vector is.

This definition has however the problem, that there are two vectors, normally on <math> \vec a</math> and <math> \vec b</math> stand. Those determines the correct vector Orientation the vector space. The coordinate system used nowadays is "right-handed" (a "juridical system in such a way specified"), D.h. both those Axes of coordinates (x, y and z) and the vectors <math> \vec a</math>, <math> \vec b</math> and <math> \vec{a} \times \vec{b}</math> behave like thumbs, Index finger and middle finger of the right Hand, if one it in right angle to each other of the palm away-stretches (therefore it often Being right lp rule called).

The cross product can graphic be represented as:

In the cartesian coordinate system the cross product can be computed as:

<math>

 \vec{a} \times \vec{b} = \begin{pmatrix}a_1 \ \ a_2 \ \ a_3 \end{pmatrix} \times \begin{pmatrix}b_1 \ \ b_2 \ \ b_3 \end{pmatrix} = \begin{pmatrix} a_2b_3 - a_3b_2 \ \
   a_3b_1 - a_1b_3 \ \
   a_1b_2 - a_2b_1 \end{pmatrix} = \begin{pmatrix} 0 & -a_3 & a_2 \ \
   a_3 & 0 & -a_1 \ \
  -a_2 & a_1 & 0 \end{pmatrix} \begin{pmatrix} b_1 \ \ b_2 \ \ b_3 \end{pmatrix}

</math>

The amount of <math> \vec A \times \vec b</math> corresponds to the surface from <math> \vec a</math> and <math> \vec b</math> stretched Parallelogram.

To the cross product that does not apply Commutative law separate the so-called anti-commutative law:

<math> \vec{a} \times \vec{b} = - \vec{b} \times \vec{a}</math>

It gives a verallgemeinerung up of the cross product ndimensional areas, the however no longer only two vectors link, separates n-1 vectors. The cross product of these vectors is a vector, on all perpendicularly and its length and sense of direction of the lengths and the order of the arguments stands depends. See in addition: Cross product.

See also

• Analytic geometry
• Spatprodukt
• Vector diagram
• Pseudo vector
• Homogeneous coordinates

History

• the vector analysis of Hermann Guenter was justified Hermann_Grassmann(15.04.1809 - 26.09.1877)

Literature

• Kurt Bohner, Peter Ihlenburg, Roland Ott: Mathematics for vocational High Schools - linear algebra vectorial geometry Merkur publishing house Rinteln, 1. Edition 2004, ISBN 3-8120-0552-2

Web on the left of

- word origin, Synonymous one and translations

• Javaapplet for the illustration of the vektoraddition
• Javaapplet for the illustration of the cross product