Cross product

the cross product <math> \ vec A \ times \ vec b< /math> (also cross product, vectorial product or vector product mentioned) two vectors <math> \ vec A< /math> and <math> \ vec b< /math> in a three-dimensional vector space a pseudo vector is perpendicular, on of both vectors the stretched levelstands. The length of this vector corresponds to the surface of the parallelogram with the sides <math> \ vec A< /math> and <math> \ vec b< /math>.

The vectors <math> \ vec A< /math> and <math> \ vec b< /math> form with the vector of their cross product a juridical system. Cross and dot product are over that Spatprodukt links with one another.

Table of contents

Mathematical representation

the cross product is written with a cross as multiplication symbol:

<math>
 \ vec {C} = \ vec {A} \ times \ vec {b}

< /math>

In the usual three-dimensional area R 3 one can in such a way define the cross product of A and b:

<math>
 \ vec {A} \ times \ vec {b}= \ left|\ vec {A} \ right| \ left|\ vec {b} \ right|\ sin (\ theta) \ \ vec {e cdot}

< /math> how sin (θ) the sine of both vectors of included angle θ, <math> \ vec {e}< /math> too the two vectors senkrechte unit vector, and <math>|\ vec {A}|</math>, <math>|\ vec {b}|</math> the respective length (amount) of the vectors are.

<math> \ vec {C}< /math> is a pseudo vector. It is a simplified way of writing forthe elements of an antisymmetric tensor different of zero. Only by accident it has 3 exactly 3 components, which can be represented as vector in the same area in R. In R 4 for example there is 4, but 6 not independentElements.

The components in R 3 read:

<math>
 \ begin {pmatrix}  0             & - a_2b_1+a_1b_2 & - a_3b_1+a_1b_3 \ \  a_2b_1-a_1b_2 & 0             &   - a_3b_2+a_2b_3 \ \  a_3b_1-a_1b_3 & a_3b_2-a_2b_3 & 0\ end {pmatrix}

< to /math>

orientation

Rechte-Hand-Regel
being right lp rule

it gives math \ <vec> {e to two vectors}< /math>, those perpendicularlyon <math> \ vec A< /math> and <math> \ vec b< /math> stand and the appropriate length have; these wise in opposite directions. The orientation of the vector space determines the correct vector. The coordinate system used nowadays is “right-handed” (a juridical system in such a way specified), i.e. both the 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 fingers and middle fingers of the right hand, if one away-stretches it to each other in the right angle from the palm (called therefore often being right lp rule).

component way computation

in normal R 3 one can compute the cross product simply komponentenweise:

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

< to /math>

Each line contains thereby in the cross product thoseDifference of the products crosswise the other two lines, beginning with <math> a_2< /math>. The indices are cyclically permutated. Then a juridical system always develops.

One can use numerical values simply:

<math>
\ begin {pmatrix} to 1 \ \ 2 \ \ 3 \ end {pmatrix} \ times \ begin {pmatrix} - to 7 \ \ 8 \ \ 9 \ end {pmatrix}= \ begin {pmatrix} 2 \ cdot 9 - 3 \ 8 \ \ 3 \ cdot (- 7) - 1 cdot \ cdot 9 \ \ 1 \ cdot 8 - 2 \ cdot (- 7) \ end {pmatrix} = \ begin {pmatrix} to -6 \ \ -30 \ \ 22 \ end {pmatrix}

< to /math>

diagram

The cross product can graphic be represented as:

Bild:crossproduct.png

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.

important characteristics

<math> \ vec {A} \ times \ vec {A} = \ vec {0}< /math>
<math> \ vec {A} \ times \ vec {0} = \ vec {0}< /math>

To the cross product appliesthe commutative law not, but:

<math> \ vec {A} \ times \ vec {b} = - (\ vec {b} \ times \ vec {A})< /math>

With permutation of the vectors thus the sign changes. One says: The cross product is anti-commutative or also inclined symmetrical.

Two distributive laws apply:

<math> \ vec {A} \ times (\ vec {b} + \ vec {C}) = \ vec {A} \ times \ vec {b} + \ vec {A} \ times \ vec {C}< /math>

and

<math> (\ vec {A} + \ vec {b}) \ times \ vec {C} = \ vec {A} \ times \ vec {C} + \ vec {b} \ times \ vec {C}< /math>.

Furthermore the associative law of the scalar multiplication applies:

<math> \ alpha< /math> out math <\> mathbb {R is}< /math>.

<math> (\ alpha \ vec {A}) \ times \ vec {b} = \ alpha (\ vec {A} \ times \ vec {b}) = \ vec {A} \ times (\ alpha \ vec {b})< /math>.

For the square of the standard one receives:

<math>|\ vec {A} \ times \ vec {b}|^2 = |\ vec {A}|^2 \ cdot|\ vec {b}|^2 - <\ vec {A}; \ vec {b}> ^2< /math>

or more simply: <math>|\ vec {A} \ times \ vec {b}|^2 = |\ vec {A}|^2 \ cdot|\ vec {b}|^2- (\ vec {A} \ cdot \ vec {b}) ^2< /math>.

For between the vectors <math> \ vec {the A}< /math> and <math> \ vec {b}< /math> stretched not over-blunt angle <math> \ phi< /math> applies:

<math>|\ vec {A} \ times \ vec {b}| = |\ vec {A}|\ cdot|\ vec {b}|\ \ sin (\ phi) /math< cdot>.

The vectors are <math> \ vec {A}< /math> and <math> \ vec {b}< /math> parallel, then their cross product is the zero-vector.

grass man identity

the grass man identity(also Grassmann expansion theorem after Hermann grass man, also BAC-CAB rule mentioned) is suitable, in order to simplify physical vector computations. It reads:

<math> \ vec {A} \ times (\ vec {b} \ times \ vec {C}) = \ vec {b} (\ vec {A} \ cdot \ vec {C}) - \ vec {C} (\ vec {A} \ cdot \ vec {b})< /math>,

a noticing set for this formula is “ABC = BAC minus CAB”.

Note:The grass man identity applies only togenuine vectors and not for vektorwertige operators (like e.g. for the Del).

Verallgemeinerung

it gives a Verallgemeinerung of the cross product on n - dimensional Euclidean areas, which links however no longer only two vectors, but n to -1 vectors.The cross product of these vectors is a vector, which on all normally (perpendicularly in the sense of the dot product) and whose length and sense of direction stands depends on the lengths and the order of the arguments.

motivation

the cross product of one <math> 3 \ times3< /math> - matrix results as the formal determinant:

<math>
\ vec {A} \ times \ vec {b} = \ begin {pmatrix} a_1 \ \ a_2 \ \ a_3 \ end {pmatrix} \ times \ begin {pmatrix} b_1 \ \ to b_2 \ \ b_3 \ end {pmatrix} = \ December \ begin {pmatrix} \ vec {e_x} & \ vec {e_y} & \ vec {e_z} \ \ a_1 & a_2 & a_3 \ \ b_1 &b_2 & b_3 \ end {pmatrix} = \ vec {e_x} \ cdot (a_2b_3 - b_2a_3) + \ vec {e_y} (b_1a_3-a_1b_3) + \ vec {e_z} (a_1b_2-b_1a_2)

< to /math>.

Now V is a n - dimensional Euclidean K - vector space and <math> a_1< /math>,…, <math> a_ {n-1}< /math> Vectors of V. Then one defines the cross product as formal determinant

< math> a_1 \ times \ ldots \ times a_ {n-1}: = \ December (E, a_1, \ ldots, a_ {n-1})< /math>,

whereby the A i are understood as coordinate vectors concerning an orthonormal basis and E is the column vector, whose components are the basis vectors. There that not elements of K (but of V) are, e.g. is a formal determinant. by development after the first column to be computed knows and a vector to V supplies.

A still large Verallgemeinerung of the cross product places the vector product of linear forms (and/orstill more generally of (alternating) multi-linear forms). Then any number of vectors can be linked, the result is however generally no more vector.

derivative of the calculation formula in R 3

for R 3 alsothe canonical dot product and the orthonormal basis {e 1 = (1,0,0), e 2 = (0,1,0), e 3 = (0,0,1)} also the formula for the components follows from the general definition:

<math>
\ vec {A} \ times \ vec {b} = \ begin {pmatrix} a_1 \ \ a_2 \ \ a_3 \ end {pmatrix} \ times \ begin {pmatrix} b_1 \ \ to b_2 \ \b_3 \ end {pmatrix} = \ begin {pmatrix} to a_2b_3 - a_3b_2 \ \ a_3b_1 - a_1b_3 \ \ a_1b_2 - a_2b_1 \ end {pmatrix}

< to /math>

further characteristics

for each unit vector in R 3, speak <math> \ vec {e_1}< /math>, <math> \ vec {e_2}< /math> and <math> \ vec {e_3}< /math>, applies that he itself as cross productthat in each case two remaining vectors to represent leaves.

Example:

<math>
 \ vec {e_1} \ times \ vec {e_2} =\ begin {pmatrix}  1 \ \  0 \ \  0\ end {pmatrix}\ times\ begin {pmatrix}  0 \ \  1 \ \  0\ end {pmatrix}=\ begin {pmatrix}  0 \ \  0 \ \  1\ end {pmatrix}=\ vec {e_3}

< /math>

applications

some physical dimension can with the help of the cross product be computed. Examples of it are the torque or the Lorentzkraft.

Web on the left of

 

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