Cross product

This article deals with the product of two vectors in space; for further meanings, see cross product (disambiguation) .

Cross product

The cross product , also vector product , vector product or outer product , is a link in the three-dimensional Euclidean vector space that assigns a vector to two vectors . In order to distinguish it from other products, in particular the scalar product , it is written in German and English-speaking countries with a cross as a multiplication symbol (see section Spellings ). The terms cross product and vector product go back to the physicist Josiah Willard Gibbs , the term outer product was coined by the mathematician Hermann Graßmann . ${\displaystyle \times }$

The cross product of the vectors and is a vector that is perpendicular to the plane spanned by the two vectors and forms a right system with them . The length of this vector corresponds to the area of the parallelogram that is spanned by vectors and . ${\displaystyle {\vec {a}}}$ ${\displaystyle {\vec {b}}}$ ${\displaystyle {\vec {a}}}$ ${\displaystyle {\vec {b}}}$

The cross product occurs in many places in physics, for example in electromagnetism when calculating the Lorentz force or the Poynting vector . In classical mechanics, it is used for rotational quantities such as torque and angular momentum or for apparent forces such as the Coriolis force .

Geometric definition

Right-hand rule

The cross product of two vectors and in the three-dimensional visual space is a vector that is orthogonal to and , and thus orthogonal to the plane spanned by and . ${\displaystyle {\vec {a}}\times {\vec {b}}}$ ${\displaystyle {\vec {a}}}$ ${\displaystyle {\vec {b}}}$ ${\displaystyle {\vec {a}}}$ ${\displaystyle {\vec {b}}}$ ${\displaystyle {\vec {a}}}$ ${\displaystyle {\vec {b}}}$

This vector is oriented in such a way that and in this order form a legal system . Mathematically this means that the three vectors and are oriented in the same way as the vectors , and the standard basis . In physical terms, it means that they behave like the thumb, index finger and the splayed middle finger of the right hand ( right-hand rule ). Rotating the first vector into the second vector results in the positive direction of the vector via the clockwise direction . ${\displaystyle {\vec {a}},{\vec {b}}}$ ${\displaystyle {\vec {a}}\times {\vec {b}}}$ ${\displaystyle {\vec {a}},{\vec {b}}}$ ${\displaystyle {\vec {a}}\times {\vec {b}}}$ ${\displaystyle {\vec {e}}_{1}}$ ${\displaystyle {\vec {e}}_{2}}$ ${\displaystyle {\vec {e}}_{3}}$ ${\displaystyle {\vec {a}}}$ ${\displaystyle {\vec {b}}}$ ${\displaystyle {\vec {a}}\times {\vec {b}}}$

The amount of is the surface area of from and spanned parallelogram on. Expressed by the angle enclosed by and applies ${\displaystyle {\vec {a}}\times {\vec {b}}}$ ${\displaystyle {\vec {a}}}$ ${\displaystyle {\vec {b}}}$ ${\displaystyle {\vec {a}}}$ ${\displaystyle {\vec {b}}}$ ${\displaystyle \theta }$

${\displaystyle |{\vec {a}}\times {\vec {b}}|=|{\vec {a}}|\,|{\vec {b}}|\,\sin \theta \,.}$

And denote the lengths of the vectors and , and is the sine of the angle they enclose . ${\displaystyle \vert {\vec {a}}\vert }$ ${\displaystyle \vert {\vec {b}}\vert }$ ${\displaystyle {\vec {a}}}$ ${\displaystyle {\vec {b}}}$ ${\displaystyle \sin \theta \,}$ ${\displaystyle \theta }$

In summary, then

${\displaystyle {\vec {a}}\times {\vec {b}}=(|{\vec {a}}|\,|{\vec {b}}|\,\sin \theta )\,{\vec {n}}\,,}$

where the vector is the unit vector perpendicular to and perpendicular to it, which complements it to form a legal system. ${\displaystyle {\vec {n}}}$ ${\displaystyle {\vec {a}}}$ ${\displaystyle {\vec {b}}}$

Spellings

Depending on the country, different spellings are sometimes used for the vector product. In English and German-speaking countries, the notation is usually used for the vector product of two vectors and , whereas in France and Italy, the notation is preferred. In Russia, the vector product is often written in the spelling or . ${\displaystyle {\vec {a}}}$ ${\displaystyle {\vec {b}}}$ ${\displaystyle {\vec {a}}\times {\vec {b}}}$ ${\displaystyle {\vec {a}}\wedge {\vec {b}}}$ ${\displaystyle [{\vec {a}}\ {\vec {b}}]}$ ${\displaystyle [{\vec {a}},{\vec {b}}]}$

The notation and the designation outer product are not only used for the vector product, but also for the link that assigns a so-called bivector to two vectors , see Graßmann algebra . ${\displaystyle {\vec {a}}\wedge {\vec {b}}}$

Calculation by component

In a right-handed Cartesian coordinate system or in real coordinate space with the standard scalar product and the standard orientation, the following applies to the cross product: ${\displaystyle \mathbb {R} ^{3}}$

${\displaystyle {\vec {a}}\times {\vec {b}}={\begin{pmatrix}a_{1}\\a_{2}\\a_{3}\end{pmatrix}}\times {\begin{pmatrix}b_{1}\\b_{2}\\b_{3}\end{pmatrix}}={\begin{pmatrix}a_{2}b_{3}-a_{3}b_{2}\\a_{3}b_{1}-a_{1}b_{3}\\a_{1}b_{2}-a_{2}b_{1}\end{pmatrix}}\,.}$

A numerical example:

${\displaystyle {\begin{pmatrix}1\\2\\3\end{pmatrix}}\times {\begin{pmatrix}-7\\8\\9\end{pmatrix}}={\begin{pmatrix}2\cdot 9-3\cdot 8\\3\cdot (-7)-1\cdot 9\\1\cdot 8-2\cdot (-7)\end{pmatrix}}={\begin{pmatrix}-6\\-30\\22\end{pmatrix}}\,.}$

A rule of thumb for this formula is based on a symbolic representation of the determinant . A matrix is noted in the first column of which the symbols , and stand for the standard basis . The second column is formed by the components of the vector and the third by those of the vector . This determinant is calculated according to the usual rules, for example by placing them by the first column developed ${\displaystyle (3\times 3)}$ ${\displaystyle {\vec {e}}_{1}}$ ${\displaystyle {\vec {e}}_{2}}$ ${\displaystyle {\vec {e}}_{3}}$ ${\displaystyle {\vec {a}}}$ ${\displaystyle {\vec {b}}}$

${\displaystyle {\begin{aligned}{\vec {a}}\times {\vec {b}}&=\det {\begin{pmatrix}{\vec {e}}_{1}&a_{1}&b_{1}\\{\vec {e}}_{2}&a_{2}&b_{2}\\{\vec {e}}_{3}&a_{3}&b_{3}\end{pmatrix}}\\&={\vec {e}}_{1}{\begin{vmatrix}a_{2}&b_{2}\\a_{3}&b_{3}\end{vmatrix}}-{\vec {e}}_{2}{\begin{vmatrix}a_{1}&b_{1}\\a_{3}&b_{3}\end{vmatrix}}+{\vec {e}}_{3}{\begin{vmatrix}a_{1}&b_{1}\\a_{2}&b_{2}\end{vmatrix}}\\&=(a_{2}\,b_{3}-a_{3}\,b_{2})\,{\vec {e}}_{1}+(a_{3}\,b_{1}-a_{1}\,b_{3})\,{\vec {e}}_{2}+(a_{1}\,b_{2}-\,a_{2}\,b_{1})\,{\vec {e}}_{3}\,,\end{aligned}}}$

or using the rule of Sarrus :

${\displaystyle {\begin{aligned}{\vec {a}}\times {\vec {b}}&=\det {\begin{pmatrix}{\vec {e}}_{1}&a_{1}&b_{1}\\{\vec {e}}_{2}&a_{2}&b_{2}\\{\vec {e}}_{3}&a_{3}&b_{3}\end{pmatrix}}\\&={\vec {e}}_{1}\,a_{2}\,b_{3}+a_{1}\,b_{2}\,{\vec {e}}_{3}+b_{1}\,{\vec {e}}_{2}\,a_{3}\\&\quad -{\vec {e}}_{3}\,a_{2}\,b_{1}-a_{3}\,b_{2}\,{\vec {e}}_{1}-b_{3}\,{\vec {e}}_{2}\,a_{1}\\&=(a_{2}\,b_{3}-a_{3}\,b_{2})\,{\vec {e}}_{1}+(a_{3}\,b_{1}-a_{1}\,b_{3})\,{\vec {e}}_{2}+(a_{1}\,b_{2}-\,a_{2}\,b_{1})\,{\vec {e}}_{3}\,.\end{aligned}}}$

With the Levi-Civita symbol , the cross product is written as ${\displaystyle \varepsilon _{ijk}}$

${\displaystyle {\vec {a}}\times {\vec {b}}=\sum _{i,j,k=1}^{3}\varepsilon _{ijk}a_{i}b_{j}{\vec {e}}_{k}\,.}$

properties

Bilinearity

The cross product is bilinear , that is, for all real numbers , and and all vectors , and holds ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle \gamma }$ ${\displaystyle {\vec {a}}}$ ${\displaystyle {\vec {b}}}$ ${\displaystyle {\vec {c}}}$

${\displaystyle {\begin{aligned}{\vec {a}}\times (\beta \,{\vec {b}}+\gamma \,{\vec {c}})=\beta \,({\vec {a}}\times {\vec {b}})+\gamma \,({\vec {a}}\times {\vec {c}})\,,\\(\alpha \,{\vec {a}}+\beta \,{\vec {b}})\times {\vec {c}}=\alpha \,({\vec {a}}\times {\vec {c}})+\beta \,({\vec {b}}\times {\vec {c}})\,.\end{aligned}}}$

The bilinearity also implies, in particular, the following behavior with regard to the scalar multiplication

${\displaystyle \ {\vec {a}}\times (\beta \,{\vec {b}})=\beta \,({\vec {a}}\times {\vec {b}})=(\beta \,{\vec {a}})\times {\vec {b}}\,.}$

Alternating figure

The cross product of a vector with itself or a collinear vector gives the zero vector

${\displaystyle {\vec {a}}\times r{\vec {a}}={\vec {0}}}$.

Bilinear maps for which this equation applies are named alternately.

Anti-commutativity

The cross product is anti-commutative . This means that if the arguments are swapped, it changes the sign:

${\displaystyle {\vec {a}}\times {\vec {b}}=-\,{\vec {b}}\times {\vec {a}}\,.}$

This follows from the property of being (1) alternating and (2) bilinear since

${\displaystyle {\vec {0}}{\mathrel {\stackrel {(1)}{=}}}({\vec {a}}+{\vec {b}})\times ({\vec {a}}+{\vec {b}}){\mathrel {\stackrel {(2)}{=}}}{\vec {a}}\times {\vec {a}}+{\vec {a}}\times {\vec {b}}+{\vec {b}}\times {\vec {a}}+{\vec {b}}\times {\vec {b}}{\mathrel {\stackrel {(1)}{=}}}{\vec {0}}+{\vec {a}}\times {\vec {b}}+{\vec {b}}\times {\vec {a}}+{\vec {0}}={\vec {a}}\times {\vec {b}}+{\vec {b}}\times {\vec {a}}}$

applies to all . ${\displaystyle {\vec {a}},{\vec {b}}\in \mathbb {R} ^{3}}$

Jacobi identity

The cross product is not associative . Instead, the Jacobi identity applies , i.e. the cyclic sum of repeated cross products vanishes:

${\displaystyle {\vec {a}}\times ({\vec {b}}\times {\vec {c}})+{\vec {b}}\times ({\vec {c}}\times {\vec {a}})+{\vec {c}}\times ({\vec {a}}\times {\vec {b}})={\vec {0}}}$

Because of this property and those mentioned above, the together with the cross product forms a Lie algebra . ${\displaystyle \mathbb {R} ^{3}}$

Relationship to the determinant

The following applies to each vector : ${\displaystyle {\vec {v}}}$

${\displaystyle {\vec {v}}\cdot ({\vec {a}}\times {\vec {b}})=\operatorname {det} ({\vec {v}},{\vec {a}},{\vec {b}})}$.

The painting point denotes the scalar product . The cross product is clearly determined by this condition:

The following applies to every vector : If two vectors and are given, then there is exactly one vector , so that applies to all vectors . This vector is . ${\displaystyle {\vec {v}}}$ ${\displaystyle {\vec {a}}}$ ${\displaystyle {\vec {b}}}$ ${\displaystyle {\vec {c}}}$ ${\displaystyle {\vec {v}}\cdot {\vec {c}}=\operatorname {det} ({\vec {v}},{\vec {a}},{\vec {b}})}$ ${\displaystyle {\vec {v}}}$ ${\displaystyle {\vec {c}}}$ ${\displaystyle {\vec {a}}\times {\vec {b}}}$

Graßmann identity

For the repeated cross product of three vectors (also called double vector product ) the Graßmann identity applies (also Graßmann evolution theorem , after Hermann Graßmann ). This is:

${\displaystyle {\vec {a}}\times ({\vec {b}}\times {\vec {c}})=({\vec {a}}\cdot {\vec {c}})\,{\vec {b}}-({\vec {a}}\cdot {\vec {b}})\,{\vec {c}}}$

or.

${\displaystyle ({\vec {a}}\times {\vec {b}})\times {\vec {c}}=({\vec {a}}\cdot {\vec {c}})\,{\vec {b}}\ -({\vec {b}}\cdot {\vec {c}})\,{\vec {a}}\,,}$

where the paint dots denote the scalar product . In physics, the spelling is often used

${\displaystyle {\vec {a}}\times ({\vec {b}}\times {\vec {c}})={\vec {b}}\,({\vec {a}}\cdot {\vec {c}})-{\vec {c}}\,({\vec {a}}\cdot {\vec {b}})\,,}$

used. According to this representation, the formula is also called the BAC-CAB formula . In index notation , the Graßmann identity is:

${\displaystyle \varepsilon _{ijk}\varepsilon _{klm}=\delta _{il}\delta _{jm}-\delta _{im}\delta _{jl}}$.

Here is the Levi-Civita symbol and the Kronecker delta . ${\displaystyle \varepsilon _{ijk}}$ ${\displaystyle \delta _{ij}}$

Lagrange identity

For the scalar product of two cross products applies

${\displaystyle {\begin{aligned}({\vec {a}}\times {\vec {b}})\cdot ({\vec {c}}\times {\vec {d}})&=({\vec {a}}\cdot {\vec {c}})({\vec {b}}\cdot {\vec {d}})-({\vec {b}}\cdot {\vec {c}})({\vec {a}}\cdot {\vec {d}})\\&=\det {\begin{pmatrix}({\vec {a}}\cdot {\vec {c}})&({\vec {a}}\cdot {\vec {d}})\\({\vec {b}}\cdot {\vec {c}})&({\vec {b}}\cdot {\vec {d}})\end{pmatrix}}\;.\end{aligned}}}$

This gives us the square of the norm

${\displaystyle {\begin{aligned}|{\vec {a}}\times {\vec {b}}|^{2}&=|{\vec {a}}|^{2}\,|{\vec {b}}|^{2}-({\vec {a}}\cdot {\vec {b}})^{2}\\&=|{\vec {a}}|^{2}|{\vec {b}}|^{2}(1-\cos ^{2}\theta )\\&=|{\vec {a}}|^{2}|{\vec {b}}|^{2}\sin ^{2}\theta \;,\end{aligned}}}$

so the following applies to the amount of the cross product:

${\displaystyle |{\vec {a}}\times {\vec {b}}|=|{\vec {a}}|\,|{\vec {b}}|\,\sin \theta \;.}$

Since , the angle between and , is always between 0 ° and 180 °, is ${\displaystyle \theta }$ ${\displaystyle {\vec {a}}}$ ${\displaystyle {\vec {b}}}$ ${\displaystyle \sin \theta \geq 0.}$

Cross product made from two cross products

${\displaystyle {\begin{aligned}({\vec {a}}\times {\vec {b}})\times ({\vec {c}}\times {\vec {d}})&={\vec {b}}\cdot \det({\vec {a}},{\vec {c}},{\vec {d}})-{\vec {a}}\cdot \det({\vec {b}},{\vec {c}},{\vec {d}})\\&={\vec {c}}\cdot \det({\vec {a}},{\vec {b}},{\vec {d}})-{\vec {d}}\cdot \det({\vec {a}},{\vec {b}},{\vec {c}})\end{aligned}}}$

Special cases:

${\displaystyle ({\vec {a}}\times {\vec {b}})\times ({\vec {b}}\times {\vec {c}})={\vec {b}}\cdot \det({\vec {a}},{\vec {b}},{\vec {c}})}$

${\displaystyle ({\vec {a}}\times {\vec {b}})\times ({\vec {a}}\times {\vec {c}})={\vec {a}}\cdot \det({\vec {a}},{\vec {b}},{\vec {c}})}$

${\displaystyle ({\vec {a}}\times {\vec {b}})\times ({\vec {a}}\times {\vec {b}})={\vec {0}}}$

Cross product matrix

For a fixed vector , the cross product defines a linear mapping that maps a vector onto the vector . This can with a skew-symmetric tensor second stage be identified . When using the standard basis , the linear mapping corresponds to a matrix operation . The skew-symmetric matrix ${\displaystyle {\vec {w}}}$ ${\displaystyle {\vec {v}}}$ ${\displaystyle {\vec {w}}\times {\vec {v}}}$ ${\displaystyle \lbrace {\vec {e}}_{1},{\vec {e}}_{2},{\vec {e}}_{3}\rbrace }$

${\displaystyle {W}=\sum _{i=1}^{3}({\vec {w}}\times {\vec {e}}_{i})\otimes {\vec {e}}_{i}=\left({\begin{array}{ccc}0&-w_{3}&w_{2}\\w_{3}&0&-w_{1}\\-w_{2}&w_{1}&0\end{array}}\right)}$    With    ${\displaystyle \displaystyle {\vec {w}}=\sum _{i=1}^{3}w_{i}{\vec {e}}_{i}=\left({\begin{array}{c}w_{1}\\w_{2}\\w_{3}\end{array}}\right)}$

does the same as the cross product with , d. H. : ${\displaystyle {\vec {w}}}$ ${\displaystyle {W}{\vec {v}}={\vec {w}}\times {\vec {v}}}$

${\displaystyle \left({\begin{array}{ccc}0&-w_{3}&w_{2}\\w_{3}&0&-w_{1}\\-w_{2}&w_{1}&0\end{array}}\right)\left({\begin{array}{c}v_{1}\\v_{2}\\v_{3}\end{array}}\right)=\left({\begin{array}{c}-w_{3}v_{2}+w_{2}v_{3}\\w_{3}v_{1}-w_{1}v_{3}\\-w_{2}v_{1}+w_{1}v_{2}\end{array}}\right)=\left({\begin{array}{c}w_{1}\\w_{2}\\w_{3}\end{array}}\right)\times \left({\begin{array}{c}v_{1}\\v_{2}\\v_{3}\end{array}}\right)}$.

The matrix is called the cross product matrix . It is also referred to with . ${\displaystyle W}$ ${\displaystyle [{\vec {w}}]_{\times }}$

Given a skew-symmetric matrix, the following applies ${\displaystyle {W}}$

${\displaystyle {W}=\sum _{i=1}^{3}\sum _{j=1}^{3}W_{ij}{\vec {e}}_{i}\otimes {\vec {e}}_{j}=-W^{T}}$,

where is the transpose of , and the associated vector is obtained from ${\displaystyle {W}^{T}}$ ${\displaystyle {W}}$

${\displaystyle {\vec {w}}=-{\frac {1}{2}}\sum _{i=1}^{3}\sum _{j=1}^{3}W_{ij}{\vec {e}}_{i}\times {\vec {e}}_{j}}$.

Has the shape , then the following applies to the corresponding cross product matrix: ${\displaystyle {\vec {w}}}$ ${\displaystyle {\vec {w}}={\vec {b}}\times {\vec {a}}}$

${\displaystyle {W}=[{\vec {w}}]_{\times }={\vec {a}}\otimes {\vec {b}}-{\vec {b}}\otimes {\vec {a}}}$and for everyone . ${\displaystyle W_{ij}=a_{i}b_{j}-b_{i}a_{j}}$ ${\displaystyle i,j}$

Here “ ” denotes the dyadic product . ${\displaystyle \otimes }$

Polar and axial vectors

When applying the cross product to vector physical quantities , the distinction between polar or shear vectors (these are those that behave like differences between two position vectors, for example speed , acceleration , force , electric field strength ) on the one hand and axial or rotation vectors , also called pseudo vectors , on the other hand (these are those that behave like axes of rotation, for example angular velocity , torque , angular momentum , magnetic flux density ) an important role.

Polar or shear vectors are assigned the signature (or parity ) +1, axial or rotation vectors the signature −1. When two vectors are multiplied by vector, these signatures are multiplied: two vectors with the same signature produce an axial product, two with a different signature produce a polar vector product. In operational terms: a vector transfers its signature to the cross product with another vector if this is axial; if the other vector is polar, the cross product gets the opposite signature.

Operations derived from the cross product

Late product

→ Main article : late product

The combination of cross and scalar product in the form

${\displaystyle ({\vec {a}}\times {\vec {b}})\cdot {\vec {c}}}$

is called a late product. The result is a number which corresponds to the oriented volume of the parallelepiped spanned by the three vectors . The late product can also be represented as a determinant of the named three vectors

${\displaystyle V=({\vec {a}}\times {\vec {b}})\cdot {\vec {c}}=\det \left({\vec {a}},{\vec {b}},{\vec {c}}\right).}$

rotation

→ Main article : Rotation of a vector field

In vector analysis , the cross product is used together with the Nabla operator to denote the differential operator "rotation". If a vector field is im , then is ${\displaystyle \nabla }$ ${\displaystyle {\vec {V}}}$ ${\displaystyle \mathbb {R} ^{3}}$

${\displaystyle \operatorname {rot} {\vec {V}}=\nabla \times {\vec {V}}={\begin{pmatrix}{\frac {\partial }{\partial x_{1}}}\\[.5em]{\frac {\partial }{\partial x_{2}}}\\[.5em]{\frac {\partial }{\partial x_{3}}}\end{pmatrix}}\times {\begin{pmatrix}V_{1}\\[.5em]V_{2}\\[.5em]V_{3}\end{pmatrix}}={\begin{pmatrix}{\frac {\partial }{\partial x_{2}}}V_{3}-{\frac {\partial }{\partial x_{3}}}V_{2}\\[.5em]{\frac {\partial }{\partial x_{3}}}V_{1}-{\frac {\partial }{\partial x_{1}}}V_{3}\\[.5em]{\frac {\partial }{\partial x_{1}}}V_{2}-{\frac {\partial }{\partial x_{2}}}V_{1}\end{pmatrix}}={\begin{pmatrix}{\frac {\partial V_{3}}{\partial x_{2}}}-{\frac {\partial V_{2}}{\partial x_{3}}}\\[.5em]{\frac {\partial V_{1}}{\partial x_{3}}}-{\frac {\partial V_{3}}{\partial x_{1}}}\\[.5em]{\frac {\partial V_{2}}{\partial x_{1}}}-{\frac {\partial V_{1}}{\partial x_{2}}}\end{pmatrix}}}$

again a vector field, the rotation of . ${\displaystyle {\vec {V}}}$

Formally, this vector field is calculated as the cross product of the Nabla operator and the vector field . The expressions occurring here are not products, but applications of the differential operator to the function . Therefore, the calculation rules listed above, such as B. the Graßmann identity is not valid in this case. Instead, special calculation rules apply to double cross products with the Nabla operator . ${\displaystyle {\vec {V}}}$ ${\displaystyle {\tfrac {\partial }{\partial x_{i}}}V_{j}}$ ${\displaystyle {\tfrac {\partial }{\partial x_{i}}}}$ ${\displaystyle V_{j}}$

Cross product in n-dimensional space

The cross product can be generalized to n-dimensional space for any dimension . The cross product im is not a product of two factors, but of factors. ${\displaystyle n\geq 2}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle n-1}$

The cross product of the vectors is characterized in that for each vector is considered ${\displaystyle {\vec {a}}_{1}\times {\vec {a}}_{2}\times \cdots \times {\vec {a}}_{n-1}}$ ${\displaystyle {\vec {a}}_{1},\dots ,{\vec {a}}_{n-1}\in \mathbb {R} ^{n}}$ ${\displaystyle {\vec {v}}\in \mathbb {R} ^{n}}$

${\displaystyle {\vec {v}}\cdot ({\vec {a}}_{1}\times {\vec {a}}_{2}\times \cdots \times {\vec {a}}_{n-1})=\operatorname {det} ({\vec {v}},{\vec {a}}_{1},\dots ,{\vec {a}}_{n-1}).}$

The cross product in coordinates can be calculated as follows. Let it be the associated -th canonical unit vector . For vectors ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle {\vec {e}}_{i}}$ ${\displaystyle i}$ ${\displaystyle n-1}$

${\displaystyle {\vec {a}}_{1}={\begin{pmatrix}a_{11}\\a_{21}\\\vdots \\a_{n1}\end{pmatrix}},\ {\vec {a}}_{2}={\begin{pmatrix}a_{12}\\a_{22}\\\vdots \\a_{n2}\end{pmatrix}},\ \dots ,\ {\vec {a}}_{n-1}={\begin{pmatrix}a_{1\,(n-1)}\\a_{2\,(n-1)}\\\vdots \\a_{n\,(n-1)}\end{pmatrix}}\in \mathbb {R} ^{n}}$

applies

${\displaystyle {\vec {a}}_{1}\times {\vec {a}}_{2}\times \cdots \times {\vec {a}}_{n-1}=\det {\begin{pmatrix}{\vec {e}}_{1}&a_{11}&\cdots &a_{1(n-1)}\\{\vec {e}}_{2}&a_{21}&\cdots &a_{2(n-1)}\\\vdots &\vdots &\ddots &\vdots \\{\vec {e}}_{n}&a_{n1}&\dots &a_{n(n-1)}\end{pmatrix}},}$

analogous to the above-mentioned calculation with the help of a determinant.

The vector is orthogonal to . The orientation is such that the vectors in this order form a legal system. The amount of equal to the dimensional volume of from spanned Parallelotops . ${\displaystyle {\vec {a}}_{1}\times {\vec {a}}_{2}\times \cdots \times {\vec {a}}_{n-1}}$ ${\displaystyle {\vec {a}}_{1},{\vec {a}}_{2},\dotsc ,{\vec {a}}_{n-1}}$ ${\displaystyle {\vec {a}}_{1}\times {\vec {a}}_{2}\times \cdots \times {\vec {a}}_{n-1},{\vec {a}}_{1},{\vec {a}}_{2},\dotsc ,{\vec {a}}_{n-1}}$ ${\displaystyle {\vec {a}}_{1}\times {\vec {a}}_{2}\times \cdots \times {\vec {a}}_{n-1}}$ ${\displaystyle (n-1)}$ ${\displaystyle {\vec {a}}_{1},{\vec {a}}_{2},\dotsc ,{\vec {a}}_{n-1}}$

For you don't get a product, just a linear mapping ${\displaystyle n=2}$

${\displaystyle \mathbb {R} ^{2}\to \mathbb {R} ^{2};\ {\begin{pmatrix}a_{1}\\a_{2}\end{pmatrix}}\mapsto {\begin{pmatrix}a_{2}\\-a_{1}\end{pmatrix}}}$,

the rotation by 90 ° clockwise.

This also shows that the component vectors of the cross product including the result vector in this order - unlike the usual - generally do not form a legal system; these arise only in real vector spaces with odd , with even the result vector forms a link system with the component vectors. This is in turn due to the fact that the basis in spaces of even dimensions is not the same as the basis , which by definition (see above) is a legal system . A small change in the definition would mean that the vectors in the first-mentioned order always form a legal system, namely if the column of the unit vectors in the symbolic determinant were set to the far right, this definition has not been accepted. ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle ({\vec {a}}_{1},{\vec {a}}_{2},\dotsc ,{\vec {a}}_{n-1},{\vec {a}}_{1}\times {\vec {a}}_{2}\times \dotsb \times {\vec {a}}_{n-1})}$ ${\displaystyle ({\vec {a}}_{1}\times {\vec {a}}_{2}\times \dotsb \times {\vec {a}}_{n-1},{\vec {a}}_{1},{\vec {a}}_{2},\dotsc ,{\vec {a}}_{n-1})}$ ${\displaystyle \mathbb {R} ^{n}}$

An even further generalization leads to the Graßmann algebras . These algebras are used, for example, in formulations of differential geometry , which allow the rigorous description of classical mechanics ( symplectic manifolds ), quantum geometry and, first and foremost, general relativity . In the literature, the cross product in the higher-dimensional and possibly curved space is usually written out in index with the Levi-Civita symbol .

Applications

The cross product is used in many areas of mathematics and physics, including the following topics:

• Mathematical calculation of the torque
• Distance formula of skewed straight lines.

Web links

Commons : cross product  - collection of images, videos and audio files

Wiktionary: cross product  - explanations of meanings, word origins, synonyms, translations

• Video: Vector Product 1 . Jörn Loviscach 2010, made available by the Technical Information Library (TIB), doi : 10.5446 / 9743 .
• Video: Vector Product 2 . Jörn Loviscach 2010, made available by the Technical Information Library (TIB), doi : 10.5446 / 9744 .

swell

• Gerd Fischer: Lineare Algebra , Vieweg-Verlag, ISBN 3-528-97217-3 .

Individual evidence

1. ↑ Max Päsler: Basics of vector and tensor calculus . Walter de Gruyter, 1977, ISBN 3-11-082794-8 , pp. 33 .
2. ^ ^{A b c d e} Herbert Amann, Joachim Escher : Analysis. 2nd volume 2nd corrected edition. Birkhäuser-Verlag, Basel et al. 2006, ISBN 3-7643-7105-6 ( basic studies in mathematics ), pp. 312-313
3. ↑ Duplicate vector product  ( page no longer available , search in web archives )  Info: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice. (Website from elearning.physik.uni-frankfurt.de, accessed on June 5, 2015, password protected)