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 .
![\ times](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703)
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 .
![{\ vec {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16)
![{\ vec {b}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216)
![{\ vec {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16)
![{\ vec {b}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216)
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
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 .
![\ vec {a} \ times \ vec {b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68affde71ee9058a22787425a052cf5a13f33f4d)
![{\ vec {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16)
![{\ vec {b}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216)
![{\ vec {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16)
![{\ vec {b}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216)
![{\ vec {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16)
![{\ vec {b}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216)
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 .
![{\ vec {a}}, {\ vec {b}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a58ee0ca2df6a9e4a7823885a6db21b214acf2b0)
![\ vec {a} \ times \ vec {b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68affde71ee9058a22787425a052cf5a13f33f4d)
![{\ vec {a}}, {\ vec {b}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a58ee0ca2df6a9e4a7823885a6db21b214acf2b0)
![\ vec {a} \ times \ vec {b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68affde71ee9058a22787425a052cf5a13f33f4d)
![\ vec e_1](https://wikimedia.org/api/rest_v1/media/math/render/svg/bad1143107d7fbfc245bd9fcdcc422d341ec852e)
![\ vec e_2](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c09c5782e951663cb81ed18fcfe715315d4e9c)
![\ vec e_3](https://wikimedia.org/api/rest_v1/media/math/render/svg/592d3c0a1e3620dbe7d533acca477356a903eaa8)
![{\ vec {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16)
![{\ vec {b}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216)
![\ vec {a} \ times \ vec {b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68affde71ee9058a22787425a052cf5a13f33f4d)
The amount of is the surface area of from and spanned parallelogram on. Expressed by the angle enclosed by and applies
![\ vec {a} \ times \ vec {b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68affde71ee9058a22787425a052cf5a13f33f4d)
![{\ vec {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16)
![{\ vec {b}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216)
![{\ vec {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16)
![\ theta](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af)
![| \ vec {a} \ times \ vec {b} | = | \ vec {a} | \, | \ vec {b} | \, \ sin \ theta \ ,.](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f6e1725bb0bf37d109e4fb7afcca6dbf8d6e30a)
And denote the lengths of the vectors and , and is the sine of the angle they enclose .
![\ vert \ vec {a} \ vert](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ccd9877d53ca7d224fd60dcac8b7eb274c8810c)
![\ vert \ vec {b} \ vert](https://wikimedia.org/api/rest_v1/media/math/render/svg/d59ea61a6ca66ba8fd7e046b8c06513dc7d01af7)
![{\ vec {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16)
![\ vec {b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216)
![\ sin \ theta \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d36af63acb13b05b295b62c463015069473f44f)
![\ theta](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af)
In summary, then
![\ vec {a} \ times \ vec {b} = (| \ vec {a} | \, | \ vec {b} | \, \ sin \ theta) \, \ vec {n} \ ,,](https://wikimedia.org/api/rest_v1/media/math/render/svg/cab0d537fb4e2a2b4a94126fefe2d6f825f47de6)
where the vector is the unit vector perpendicular to and perpendicular to it, which complements it to form a legal system.
![{\ vec {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49569db585c1b6306d5ffd91161775f67235fae0)
![{\ vec {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16)
![\ vec {b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216)
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 .
![{\ vec {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16)
![\ vec {b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216)
![\ vec {a} \ times \ vec {b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68affde71ee9058a22787425a052cf5a13f33f4d)
![\ vec {a} \ wedge \ vec {b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c6d20e3f1b04b8fb0dd76752065dc7bfedc9c33)
![{\ displaystyle [{\ vec {a}} \ {\ vec {b}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f525d35f8edddf7a3f90c4d8666e165cc0ac5a00)
![{\ displaystyle [{\ vec {a}}, {\ vec {b}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/008d8868e90625dc41200b2396382596923b8d9c)
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 .
![\ vec {a} \ wedge \ vec {b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c6d20e3f1b04b8fb0dd76752065dc7bfedc9c33)
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:
![\ mathbb {R} ^ {3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5)
![\ 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} \ ,.](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7bcae59e42f74dec355b2ee203e38ce480ead22)
A numerical example:
![\ 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} \ ,.](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c91e8af4ef2a688caa732a04e4a3da5e4c2b9f3)
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![(3 \ times 3)](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b10844ab463cf38a84943c2a4ef2459502973dd)
![\ vec e_1](https://wikimedia.org/api/rest_v1/media/math/render/svg/bad1143107d7fbfc245bd9fcdcc422d341ec852e)
![\ vec e_2](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c09c5782e951663cb81ed18fcfe715315d4e9c)
![\ vec e_3](https://wikimedia.org/api/rest_v1/media/math/render/svg/592d3c0a1e3620dbe7d533acca477356a903eaa8)
![{\ vec {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16)
![\ begin {align} \ 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 {align}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8687a595b32512c987099e6d3ab70de5d270c94c)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f606be4c331708769950f3bf0d6e15a4935f4a21)
With the Levi-Civita symbol , the cross product is written as
![\ varepsilon_ {ijk}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21525193117bdfc0f3ac71b8ec46e3b6d0637daf)
![\ vec {a} \ times \ vec {b} = \ sum_ {i, j, k = 1} ^ 3 \ varepsilon_ {ijk} a_i b_j \ vec e_k \ ,.](https://wikimedia.org/api/rest_v1/media/math/render/svg/630f7b5c4da0de16fd74603089c5f846666f30d0)
properties
Bilinearity
The cross product is bilinear , that is, for all real numbers , and and all vectors , and holds
![\alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3)
![\beta](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8)
![\gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a)
![{\ vec {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16)
![{\ vec {b}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216)
![\ vec c](https://wikimedia.org/api/rest_v1/media/math/render/svg/965bd8710781b710cbfdb79da0b4e3b097bef506)
![{\ 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63566b97d5096f4a24bca5b698936eb59cc79943)
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}} \ ,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4332fe3dfa52880275c038ad0d8dcf405553457)
Alternating figure
The cross product of a vector with itself or a collinear vector gives the zero vector
-
.
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:
![\ vec {a} \ times \ vec {b} = - \, \ vec {b} \ times \ vec {a} \ ,.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7280ad33cc678b48ef7aa443ee167b4596e36054)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff2af89edce466442a13451c60a8175dc5e06bf1)
applies to all .
![{\ displaystyle {\ vec {a}}, {\ vec {b}} \ in \ mathbb {R} ^ {3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1de62a30c014707e0bd5fbd33f04af950d587731)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2789ca9a8ce3ed92f3a20f00b9e51f43be333ce2)
Because of this property and those mentioned above, the together with the cross product forms a Lie algebra .
![\ mathbb {R} ^ {3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5)
Relationship to the determinant
The following applies to each vector :
![{\ vec {v}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753)
-
.
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 .
![{\ vec {v}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753)
![{\ vec {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16)
![{\ vec {b}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216)
![\ vec c](https://wikimedia.org/api/rest_v1/media/math/render/svg/965bd8710781b710cbfdb79da0b4e3b097bef506)
![{\ displaystyle {\ vec {v}} \ cdot {\ vec {c}} = \ operatorname {det} ({\ vec {v}}, {\ vec {a}}, {\ vec {b}}) }](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb1fe63c2f45f9b0c867fe39615e9e459ef6ecc1)
![{\ vec {v}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753)
![\ vec c](https://wikimedia.org/api/rest_v1/media/math/render/svg/965bd8710781b710cbfdb79da0b4e3b097bef506)
![{\ displaystyle {\ vec {a}} \ times {\ vec {b}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68affde71ee9058a22787425a052cf5a13f33f4d)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00a4098b81750ee3ba3c44d2adb9002de51d48db)
or.
![{\ displaystyle ({\ vec {a}} \ times {\ vec {b}}) \ times {\ vec {c}} = ({\ vec {a}} \ cdot {\ vec {c}}) \ , {\ vec {b}} \ - ({\ vec {b}} \ cdot {\ vec {c}}) \, {\ vec {a}} \ ,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38ff2c6532b8d6bf45c452fa2a3761cccf8d98b1)
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}}) \ ,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1ca97ddeda9322aaff32164cb383a2f99943535)
used. According to this representation, the formula is also called the BAC-CAB formula . In index notation , the Graßmann identity is:
-
.
Here is the Levi-Civita symbol and the Kronecker delta .
![\ varepsilon_ {ijk}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21525193117bdfc0f3ac71b8ec46e3b6d0637daf)
![\ delta _ {ij}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa75d04c11480d976e1396951e02cbb3c4f71568)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c6d061c595c3d2290606b8cb847c5bcce2faed5)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ff4ac4fa469f079d1bd4df28c1af58b895ba3d5)
so the following applies to the amount of the cross product:
![{\ displaystyle | {\ vec {a}} \ times {\ vec {b}} | = | {\ vec {a}} | \, | {\ vec {b}} | \, \ sin \ theta \; .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95be1713f977211721bb0d81c8ce3a2ace113a07)
Since , the angle between and , is always between 0 ° and 180 °, is![\ theta](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af)
![{\ vec {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16)
![{\ vec {b}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216)
Cross product made from two cross products
![{\ 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 }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c315967e84fd3c9bae4787b08e40f1a729abf62f)
Special cases:
![({\ vec {a}} \ times {\ vec {b}}) \ times ({\ vec {b}} \ times {\ vec {c}}) = {\ vec {b}} \ cdot \ det ({\ vec {a}}, {\ vec {b}}, {\ vec {c}})](https://wikimedia.org/api/rest_v1/media/math/render/svg/d77f589d0a4712a5131d7790c4e0c7d1d3fb5729)
![({\ vec {a}} \ times {\ vec {b}}) \ times ({\ vec {a}} \ times {\ vec {c}}) = {\ vec {a}} \ cdot \ det ({\ vec {a}}, {\ vec {b}}, {\ vec {c}})](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c35e7fbf4f815e87abe18d1d25a6680ae2ee871)
![({\ vec {a}} \ times {\ vec {b}}) \ times ({\ vec {a}} \ times {\ vec {b}}) = {\ vec {0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8d88eff15c4059054e6385c2874d13bef1d69e7)
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![\ vec {w}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b6c48cdaecf8d81481ea21b1d0c046bf34b68ec)
![{\ vec {v}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753)
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With
does the same as the cross product with , d. H. :
![\ vec {w}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b6c48cdaecf8d81481ea21b1d0c046bf34b68ec)
![{W} \ vec {v} = \ vec {w} \ times \ vec {v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da54e183189027e4373839f13b09894801a7273e)
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.
The matrix is called the cross product matrix . It is also referred to with .
![W.](https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7)
![[\ vec w] _ {\ times}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5df1ca3d4151f61cc9ccdd7a4b0e463f5ff5121f)
Given a skew-symmetric matrix, the following applies
![{W}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7023cea9f49a191d3bf1213ecb9d4fe8042e2881)
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,
where is the transpose of , and the associated vector is obtained from
![{W} ^ {{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18b785a353969bff1273d817be1c89de47d907bb)
![{W}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7023cea9f49a191d3bf1213ecb9d4fe8042e2881)
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.
Has the shape , then the following applies to the corresponding cross product matrix:
![{\ vec {w}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b6c48cdaecf8d81481ea21b1d0c046bf34b68ec)
![\ vec {w} = \ vec {b} \ times \ vec {a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9ac1603b8466e3c7b0a16687c03db9a264f9b0d)
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and for everyone .![W_ {ij} = a_i b_j - b_i a_j](https://wikimedia.org/api/rest_v1/media/math/render/svg/d23141fb34370662d4aa9afb469fb71ace30f4cf)
![i, j](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4cbf8bbc622154cda8208d6e339495fe16a1f9a)
Here “ ” denotes the dyadic product .
![\ otimes](https://wikimedia.org/api/rest_v1/media/math/render/svg/de29098f5a34ee296a505681a0d5e875070f2aea)
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
![(\ vec {a} \ times \ vec {b}) \ cdot \ vec {c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd51490c47dca0dd3f4d723dfbd0485c9a10f874)
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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5e10da7731a7a03ef25c8fa33b84d2325053812)
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
![\ nabla](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3d0e93b78c50237f9ea83d027e4ebbdaef354b2)
![{\ vec {V}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae195b5427677e7fd3302b9bc400b2c9cbbe3082)
![\ mathbb {R} ^ {3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5)
![\ 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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76fc4a22ce14a3ff684e6269ed0675da48fb7eac)
again a vector field, the rotation of .
![{\ vec {V}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae195b5427677e7fd3302b9bc400b2c9cbbe3082)
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 .
![{\ vec {V}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae195b5427677e7fd3302b9bc400b2c9cbbe3082)
![\ tfrac \ partial {\ partial x_i} V_j](https://wikimedia.org/api/rest_v1/media/math/render/svg/0076ac0d76cf7c6b54db5ac01fd5e90ede6ab250)
![\ tfrac \ partial {\ partial x_i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7626b109c540e4f4e1cf8963d2848302b73e820d)
![V_ {j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2c9bb9af363d8550bde7bfaee674d3fb2bba343)
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.
![n \ ge 2](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bf67f9d06ca3af619657f8d20ee1322da77174)
![\ mathbb {R} ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d)
![\ mathbb {R} ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d)
![n-1](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521)
The cross product of the vectors is characterized in that for each vector is considered
![\ vec a_1 \ times \ vec a_2 \ times \ cdots \ times \ vec a_ {n-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eba054b04af8d0fb0a7226fe9ce1b9cca62a95cb)
![\ vec a_1, \ dots, \ vec a_ {n-1} \ in \ R ^ n](https://wikimedia.org/api/rest_v1/media/math/render/svg/81e55ed5726835fab6809cf39896818a9b5d468e)
![{\ vec {v}} \ in \ mathbb {R} ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2af402e09ae98eea018b8e53be88ef128e4f8916)
![\ 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}).](https://wikimedia.org/api/rest_v1/media/math/render/svg/d011c0a45cc5d5a027c95ec134eb13b833d6105d)
The cross product in coordinates can be calculated as follows. Let it be the associated -th canonical unit vector . For vectors
![\ mathbb {R} ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d)
![\ vec e_i](https://wikimedia.org/api/rest_v1/media/math/render/svg/a86dbaf7593cb1a89a4d90d740b2c8b010551cbd)
![i](https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20)
![n-1](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521)
![\ 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 \ R ^ n](https://wikimedia.org/api/rest_v1/media/math/render/svg/3280bebee3674ae3020be8daa0be52fb61d3f712)
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}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88f43d9ad6d934a25bda10846b780f29b2c1cf02)
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 .
![\ vec a_1 \ times \ vec a_2 \ times \ cdots \ times \ vec a_ {n-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eba054b04af8d0fb0a7226fe9ce1b9cca62a95cb)
![{\ displaystyle {\ vec {a}} _ {1}, {\ vec {a}} _ {2}, \ dotsc, {\ vec {a}} _ {n-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3bad0286f3c7f4ab93450384ee0b44f7b675d20)
![{\ 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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85cab41d54e9ef761f9eaed1d5f93345a73a30c2)
![\ vec a_1 \ times \ vec a_2 \ times \ cdots \ times \ vec a_ {n-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eba054b04af8d0fb0a7226fe9ce1b9cca62a95cb)
![(n-1)](https://wikimedia.org/api/rest_v1/media/math/render/svg/df88c6333caaf6471cf277f24b802ff9931b133e)
![{\ displaystyle {\ vec {a}} _ {1}, {\ vec {a}} _ {2}, \ dotsc, {\ vec {a}} _ {n-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3bad0286f3c7f4ab93450384ee0b44f7b675d20)
For you don't get a product, just a linear mapping
![n = 2](https://wikimedia.org/api/rest_v1/media/math/render/svg/a02c8bd752d2cc859747ca1f3a508281bdbc3b34)
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,
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.
![\ mathbb {R} ^ {3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![(\ 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})](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce3553ccd6f758fda1f87d9b780f87309bf25b61)
![(\ 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})](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8f1d66e5ad5566973ecb2f03a838a0090ab1f9d)
![\ mathbb {R} ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d)
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:
Web links
swell
Individual evidence
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↑ Max Päsler: Basics of vector and tensor calculus . Walter de Gruyter, 1977, ISBN 3-11-082794-8 , pp. 33 .
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^ 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
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↑ 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)@1@ 2Template: Toter Link / elearning.physik.uni-frankfurt.de