# Cross product

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:

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

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}} | ^ {2nd } | {\ 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

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

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: