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which gives the components of the resulting vector directly using levi civita tensors edit in any basis the cross product a b displaystyle a times b is given by the tensorial formula e i j k a i b j displaystyle e_ ijk a i b j where e i j k displaystyle e_ ijk is the covariant levi civita tensor we note the position of the indices that corresponds to the intrinsic formula given here in an orthonormal basis having the same orientation as the space a b displaystyle a times b is given by the pseudo tensorial formula ε i j k a i b j displaystyle varepsilon _ ijk a i b j where ε i j k displaystyle varepsilon _ ijk is the levi civita symbol which is a pseudo tensor that is the formula used for everyday physics but it works only for this special choice of basis in any orthonormal basis a b displaystyle a times b is given by the pseudo tensorial formula 1 b ε i j k a i b j displaystyle 1 b varepsilon _ ijk a i b j where 1 b 1 displaystyle 1 b pm 1 indicates whether the basis has the same orientation as the space or not the latter formula avoids having to change the orientation of the space when we inverse an orthonormal basis properties edit geometric meaning edit see also triple product figure 1 the area of a parallelogram as the magnitude of a cross product figure 2 three vectors defining a parallelepiped the magnitude of the cross product can be interpreted as the positive area of the parallelogram having a and b as sides see figure 1 1 a b a b sin θ displaystyle left mathbf a times mathbf b right left mathbf a right left mathbf b right left sin theta right indeed one can also compute the volume v of a parallelepiped having a b and c as edges by using a combination of a cross product and a dot product called scalar triple product see figure 2 a b c b c a c a b displaystyle mathbf a cdot mathbf b times mathbf c mathbf b cdot mathbf c times mathbf a mathbf c cdot mathbf a times mathbf b since the result of the scalar triple product may be negative the volume of the parallelepiped is given by its absolute value v a b c displaystyle v mathbf a cdot mathbf b times mathbf c because the magnitude of the cross product goes by the sine of the angle between its arguments the cross product can be thought of as a measure of perpendicularity in the same way that the dot product is a measure of parallelism given two unit vectors their cross product has a magnitude of 1 if the two are perpendicular and a magnitude of zero if the two are parallel the dot product of two unit vectors behaves just oppositely it is zero when the unit vectors are perpendicular and 1 if the unit vectors are parallel unit vectors enable two convenient identities the dot product of two unit vectors yields the cosine which may be positive or negative of the angle between the two unit vectors the magnitude of the cross product of the two unit vectors yields the sine which will always be positive algebraic properties edit cross product scalar multiplication left decomposition of b into components parallel and perpendicular to a right scaling of the perpendicular components by a positive real number r if negative b and the cross product are reversed cross product distributivity over vector addition left the vectors b and c are resolved into parallel and perpendicular components to a right the parallel components vanish in the cross product only the perpendicular components shown in the plane perpendicular to a remain 12 the two nonequivalent triple cross products of three vectors a b c in each case two vectors define a plane the other is out of the plane and can be split into parallel and perpendicular components to the cross product of the vectors defining the plane these components can be found by vector projection and rejection the triple product is in the plane and is rotated as shown if the cross product of two vectors is the zero vector that is a b 0 then either one or both of the inputs is the zero vector a 0 or b 0 or else they are parallel or antiparallel a b so that the sine of the angle between them is zero θ 0 or θ 180 and sin θ 0 the self cross product of a vector is the zero vector a a 0 displaystyle mathbf a times mathbf a mathbf 0 the cross product is anticommutative a b b a displaystyle mathbf a times mathbf b mathbf b times mathbf a distributive over addition a b c a b a c displaystyle mathbf a times mathbf b mathbf c mathbf a times mathbf b mathbf a times mathbf c and compatible with scalar multiplication so that r a b a r b r a b displaystyle r mathbf a times mathbf b mathbf a times r mathbf b r mathbf a times mathbf b it is not associative but satisfies the jacobi identity a b c b c a c a b 0 displaystyle mathbf a times mathbf b times mathbf c mathbf b times mathbf c times mathbf a mathbf c times mathbf a times mathbf b mathbf 0 distributivity linearity and jacobi identity show that the r 3 vector space together with vector addition and the cross product forms a lie algebra the lie algebra of the real orthogonal group in 3 dimensions so 3 the cross product does not obey the cancellation law that is a b a c with a 0 does not imply b c but only that 0 a b a c a b c displaystyle begin aligned mathbf 0 mathbf a times mathbf b mathbf a times mathbf c mathbf a times mathbf b mathbf c end aligned this can be the case where b and c cancel but additionally where a and b c are parallel that is they are related by a scale factor t leading to c b t a displaystyle mathbf c mathbf b t mathbf a for some scalar t if in addition to a b a c and a 0 as above it is the case that a b a c then a b c 0 a b c 0 displaystyle begin aligned mathbf a times mathbf b mathbf c mathbf 0 mathbf a cdot mathbf b mathbf c 0 end aligned as b c cannot be simultaneously parallel for the cross product to be 0 and perpendicular for the dot product to be 0 to a it must be the case that b and c cancel b c from the geometrical definition the cross product is invariant under proper rotations about the axis defined by a b in formulae r a r b r a b displaystyle r mathbf a times r mathbf b r mathbf a times mathbf b where r displaystyle r is a rotation matrix with det r 1 displaystyle det r 1 more generally the cross product obeys the following identity under matrix transformations m a m b det m m 1 t a b cof m a b displaystyle m mathbf a times m mathbf b det m left m 1 right mathrm t mathbf a times mathbf b operatorname cof m mathbf a times mathbf b where m displaystyle m is a 3 by 3 matrix and m 1 t displaystyle left m 1 right mathrm t is the transpose of the inverse and cof displaystyle operatorname cof is the cofactor matrix it can be readily seen how this formula reduces to the former one if m displaystyle m is a rotation matrix if m displaystyle m is a 3 by 3 symmetric matrix applied to a generic cross product a b displaystyle mathbf a times mathbf b the following relation holds true m a b tr m a b a m b b m a displaystyle m mathbf a times mathbf b operatorname tr m mathbf a times mathbf b mathbf a times m mathbf b mathbf b times m mathbf a the cross product of two vectors lies in the null space of the 2 3 matrix with the vectors as rows a b n s a b displaystyle mathbf a times mathbf b in ns left begin bmatrix mathbf a mathbf b end bmatrix right for the sum of two cross products the following identity holds a b c d a c b d a d c b displaystyle mathbf a times mathbf b mathbf c times mathbf d mathbf a mathbf c times mathbf b mathbf d mathbf a times mathbf d mathbf c times mathbf b differentiation edit main article vector valued function derivative and vector multiplication the product rule of differential calculus applies to any bilinear operation and therefore also to the cross product d d t a b d a d t b a d b d t displaystyle frac d dt mathbf a times mathbf b frac d mathbf a dt times mathbf b mathbf a times frac d mathbf b dt where a and b are vectors that depend on the real variable t triple product expansion edit main article triple product the cross product is used in both forms of the triple product the scalar triple product of three vectors is defined as a b c displaystyle mathbf a cdot mathbf b times mathbf c it is the signed volume of the parallelepiped with edges a b and c and as such the vectors can be used in any order that s an even permutation of the above ordering the following therefore are equal a b c b c a c a b displaystyle mathbf a cdot mathbf b times mathbf c mathbf b cdot mathbf c times mathbf a mathbf c cdot mathbf a times mathbf b the vector triple product is the cross product of a vector with the result of another cross product and is related to the dot product by the following formula a b c b a c c a b a b c b c a a b c displaystyle begin aligned mathbf a times mathbf b times mathbf c mathbf b mathbf a cdot mathbf c mathbf c mathbf a cdot mathbf b mathbf a times mathbf b times mathbf c mathbf b mathbf c cdot mathbf a mathbf a mathbf b cdot mathbf c end aligned the mnemonic bac minus cab is used to remember the order of the vectors in the right hand member this formula is used in physics to simplify vector calculations a special case regarding gradients and useful in vector calculus is f f f f 2 f displaystyle begin aligned nabla times nabla times mathbf f nabla nabla cdot mathbf f nabla cdot nabla mathbf f nabla nabla cdot mathbf f nabla 2 mathbf f end aligned where 2 is the vector laplacian operator other identities relate the cross product to the scalar triple product a b a c a b c a a b c d b t c t a i c a t d a c b d a d b c displaystyle begin aligned mathbf a times mathbf b times mathbf a times mathbf c mathbf a cdot mathbf b times mathbf c mathbf a mathbf a times mathbf b cdot mathbf c times mathbf d mathbf b mathrm t left left mathbf c mathrm t mathbf a right i mathbf c mathbf a mathrm t right mathbf d mathbf a cdot mathbf c mathbf b cdot mathbf d mathbf a cdot mathbf d mathbf b cdot mathbf c end aligned where i is the identity matrix alternative formulation edit the cross product and the dot product are related by a b 2 a 2 b 2 a b 2 displaystyle left mathbf a times mathbf b right 2 left mathbf a right 2 left mathbf b right 2 mathbf a cdot mathbf b 2 the right hand side is the gram determinant of a and b the square of the area of the parallelogram defined by the vectors this condition determines the magnitude of the cross product namely since the dot product is defined in terms of the angle θ between the two vectors as a b a b cos θ displaystyle mathbf a cdot b left mathbf a right left mathbf b right cos theta the above given relationship can be rewritten as follows a b 2 a 2 b 2 1 cos 2 θ displaystyle left mathbf a times b right 2 left mathbf a right 2 left mathbf b right 2 left 1 cos 2 theta right invoking the pythagorean trigonometric identity one obtains a b a b sin θ displaystyle left mathbf a times mathbf b right left mathbf a right left mathbf b right left sin theta right which is the magnitude of the cross product expressed in terms of θ equal to the area of the parallelogram defined by a and b see definition above the combination of this requirement and the property that the cross product be orthogonal to its constituents a and b provides an alternative definition of the cross product 13 cross product inverse edit given two vectors a and c with a 0 the equation a b c admits solutions for b if and only if a is orthogonal to c that is if a c 0 in that case there exists an infinite family of solutions for b which are b c a a 2 t a displaystyle mathbf b frac mathbf c times mathbf a left mathbf a right 2 t mathbf a where t is an arbitrary constant this can be derived using the triple product expansion c a a b a a 2 b a b a displaystyle mathbf c times mathbf a mathbf a times mathbf b times mathbf a left mathbf a right 2 mathbf b mathbf a cdot mathbf b mathbf a rearrange to solve for b to give b c a a 2 a b a 2 a displaystyle mathbf b frac mathbf c times mathbf a left mathbf a right 2 frac mathbf a cdot mathbf b left mathbf a right 2 mathbf a the coefficient of the last term can be simplified to just the arbitrary constant t to yield the result shown above lagrange s identity edit the relation a b 2 det a a a b a b b b a 2 b 2 a b 2 displaystyle left mathbf a times mathbf b right 2 det begin bmatrix mathbf a cdot mathbf a mathbf a cdot mathbf b mathbf a cdot mathbf b mathbf b cdot mathbf b end bmatrix left mathbf a right 2 left mathbf b right 2 mathbf a cdot mathbf b 2 can be compared with another relation involving the right hand side namely lagrange s identity expressed as 14 1 i j n a i b j a j b i 2 a 2 b 2 a b 2 displaystyle sum _ 1 leq i j leq n left a_ i b_ j a_ j b_ i right 2 left mathbf a right 2 left mathbf b right 2 mathbf a cdot b 2 where a and b may be n dimensional vectors this also shows that the riemannian volume form for surfaces is exactly the surface element from vector calculus in the case where n 3 combining these two equations results in the expression for the magnitude of the cross product in terms of its components 15 a b 2 1 i j 3 a i b j a j b i 2 a 1 b 2 b 1 a 2 2 a 2 b 3 a 3 b 2 2 a 3 b 1 a 1 b 3 2 displaystyle begin aligned mathbf a times mathbf b 2 sum _ 1 leq i j leq 3 a_ i b_ j a_ j b_ i 2 a_ 1 b_ 2 b_ 1 a_ 2 2 a_ 2 b_ 3 a_ 3 b_ 2 2 a_ 3 b_ 1 a_ 1 b_ 3 2 end aligned the same result is found directly using the components of the cross product found from a b det i j k a 1 a 2 a 3 b 1 b 2 b 3 displaystyle mathbf a times mathbf b det begin bmatrix hat mathbf i hat mathbf j hat mathbf k a_ 1 a_ 2 a_ 3 b_ 1 b_ 2 b_ 3 end bmatrix in r 3 lagrange s equation is a special case of the multiplicativity vw v w of the norm in the quaternion algebra it is a special case of another formula also sometimes called lagrange s identity which is the three dimensional case of the binet cauchy identity 16 17 a b c d a c b d a d b c displaystyle mathbf a times mathbf b cdot mathbf c times mathbf d mathbf a cdot mathbf c mathbf b cdot mathbf d mathbf a cdot mathbf d mathbf b cdot mathbf c if a c and b d this simplifies to the formula above alternative ways to compute edit conversion to matrix multiplication edit the vector cross product also can be expressed as the product of a skew symmetric matrix and a vector 16 a b a b 0 a 3 a 2 a 3 0 a 1 a 2 a 1 0 b 1 b 2 b 3 a b b t a 0 b 3 b 2 b 3 0 b 1 b 2 b 1 0 a 1 a 2 a 3 displaystyle begin aligned mathbf a times mathbf b mathbf a _ times mathbf b begin bmatrix 0 a_ 3 a_ 2 a_ 3 0 a_ 1 a_ 2 a_ 1 0 end bmatrix begin bmatrix b_ 1 b_ 2 b_ 3 end bmatrix mathbf a times mathbf b mathbf b _ times mathrm t mathbf a begin bmatrix 0 b_ 3 b_ 2 b_ 3 0 b_ 1 b_ 2 b_ 1 0 end bmatrix begin bmatrix a_ 1 a_ 2 a_ 3 end bmatrix end aligned where the superscript t refers to the transpose operation and a is defined by a d e f 0 a 3 a 2 a 3 0 a 1 a 2 a 1 0 displaystyle mathbf a _ times stackrel rm def begin bmatrix 0 a_ 3 a_ 2 a_ 3 0 a_ 1 a_ 2 a_ 1 0 end bmatrix the columns a i of the skew symmetric matrix for a vector a can be also obtained by calculating the cross product with unit vectors that is a i a e i i 1 2 3 disp...
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