# Cross product

the cross product [itex] \ vec A \ times \ vec b< /math> (also cross product, vectorial product or vector product mentioned) two vectors [itex] \ vec A< /math> and [itex] \ 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 [itex] \ vec A< /math> and [itex] \ vec b< /math>.

The vectors [itex] \ vec A< /math> and [itex] \ 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.

## Mathematical representation

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

[itex]
``` \ 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:

[itex]
``` \ 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, [itex] \ vec {e}< /math> too the two vectors senkrechte unit vector, and [itex]|\ vec {A}|[/itex], [itex]|\ vec {b}|[/itex] the respective length (amount) of the vectors are.

[itex] \ 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:

[itex]
``` \ 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

being right lp rule

it gives math \ <vec> {e to two vectors}< /math>, those perpendicularlyon [itex] \ vec A< /math> and [itex] \ 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 [itex] \ vec A< /math>, [itex] \ vec b< /math> and [itex] \ 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:

[itex]
`\ 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 [itex] a_2< /math>. The indices are cyclically permutated. Then a juridical system always develops.

One can use numerical values simply:

[itex]
`\ 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:

The amount of [itex] \ vec A \ times \ vec b< /math> corresponds to the surface from [itex] \ vec A< /math> and [itex] \ vec b< /math> stretched parallelogram.

## important characteristics

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

To the cross product appliesthe commutative law not, but:

[itex] \ 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:

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

and

[itex] (\ 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:

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

[itex] (\ 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:

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

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

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

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

The vectors are [itex] \ vec {A}< /math> and [itex] \ 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:

[itex] \ 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 [itex] 3 \ times3< /math> - matrix results as the formal determinant:

[itex]
`\ 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 [itex] a_1< /math>,…, [itex] 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:

[itex]
`\ 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 [itex] \ vec {e_1}< /math>, [itex] \ vec {e_2}< /math> and [itex] \ vec {e_3}< /math>, applies that he itself as cross productthat in each case two remaining vectors to represent leaves.

Example:

[itex]
``` \ 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.