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sional space time 4 see generalizations below for other dimensions definition edit finding the direction of the cross product by the right hand rule the cross product of two vectors a and b is defined only in three dimensional space and is denoted by a b in physics and applied mathematics the wedge notation a b is often used in conjunction with the name vector product 5 6 7 although in pure mathematics such notation is usually reserved for just the exterior product an abstraction of the vector product to n dimensions the cross product a b is defined as a vector c that is perpendicular orthogonal to both a and b with a direction given by the right hand rule 1 and a magnitude equal to the area of the parallelogram that the vectors span 2 the cross product is defined by the formula 8 9 a b a b sin θ n displaystyle mathbf a times mathbf b left mathbf a right left mathbf b right sin theta mathbf n where θ is the angle between a and b in the plane containing them hence it is between 0 and 180 a and b are the magnitudes of vectors a and b n is a unit vector perpendicular to the plane containing a and b with direction such that the ordered set a b n is positively oriented if the vectors a and b are parallel that is the angle θ between them is either 0 or 180 by the above formula the cross product of a and b is the zero vector 0 direction edit the cross product a b vertical in purple changes as the angle between the vectors a blue and b red changes the cross product is always orthogonal to both vectors and has magnitude zero when the vectors are parallel and maximum magnitude a b when they are orthogonal the direction of the vector n depends on the chosen orientation of the space conventionally it is given by the right hand rule where one simply points the forefinger of the right hand in the direction of a and the middle finger in the direction of b then the vector n is coming out of the thumb see the adjacent picture using this rule implies that the cross product is anti commutative that is b a a b by pointing the forefinger toward b first and then pointing the middle finger toward a the thumb will be forced in the opposite direction reversing the sign of the product vector as the cross product operator depends on the orientation of the space in general the cross product of two vectors is not a true vector but a pseudovector names and origin edit according to sarrus s rule the determinant of a 3 3 matrix involves multiplications between matrix elements identified by crossed diagonals in 1842 william rowan hamilton first described the algebra of quaternions and the non commutative hamilton product in particular when the hamilton product of two vectors that is pure quaternions with zero scalar part is performed it results in a quaternion with a scalar and vector part the scalar and vector part of this hamilton product corresponds to the negative of dot product and cross product of the two vectors in 1881 josiah willard gibbs 10 and independently oliver heaviside introduced the notation for both the dot product and the cross product using a period a b and an a b respectively to denote them 11 in 1877 to emphasize the fact that the result of a dot product is a scalar while the result of a cross product is a vector william kingdon clifford coined the alternative names scalar product and vector product for the two operations 11 these alternative names are still widely used in the literature both the cross notation a b and the name cross product were possibly inspired by the fact that each scalar component of a b is computed by multiplying non corresponding components of a and b conversely a dot product a b involves multiplications between corresponding components of a and b as explained below the cross product can be expressed in the form of a determinant of a special 3 3 matrix according to sarrus s rule this involves multiplications between matrix elements identified by crossed diagonals computing edit coordinate notation edit standard basis vectors i j k and vector components of a denoted here a x a y a z if i j k displaystyle mathbf color red i mathbf color green j mathbf color blue k is a positively oriented orthonormal basis the basis vectors satisfy the following equalities 1 i j k j k i k i j displaystyle begin alignedat 2 mathbf color red i times mathbf color green j mathbf color blue k mathbf color green j times mathbf color blue k mathbf color red i mathbf color blue k times mathbf color red i mathbf color green j end alignedat a mnemonic for these formulas is that they can be deduced from any other of them by a cyclic permutation of the basis vectors this mnemonic applies also to many formulas given in this article the anticommutativity of the cross product implies that j i k k j i i k j displaystyle begin alignedat 2 mathbf color green j times mathbf color red i mathbf color blue k mathbf color blue k times mathbf color green j mathbf color red i mathbf color red i times mathbf color blue k mathbf color green j end alignedat the anticommutativity of the cross product and the obvious lack of linear independence also implies that i i j j k k 0 displaystyle mathbf color red i times mathbf color red i mathbf color green j times mathbf color green j mathbf color blue k times mathbf color blue k mathbf 0 the zero vector these equalities together with the distributivity and linearity of the cross product though neither follows easily from the definition given above are sufficient to determine the cross product of any two vectors a and b each vector can be defined as the sum of three orthogonal components parallel to the standard basis vectors a a 1 i a 2 j a 3 k b b 1 i b 2 j b 3 k displaystyle begin alignedat 3 mathbf a a_ 1 mathbf color red i a_ 2 mathbf color green j a_ 3 mathbf color blue k mathbf b b_ 1 mathbf color red i b_ 2 mathbf color green j b_ 3 mathbf color blue k end alignedat their cross product a b can be expanded using distributivity a b a 1 i a 2 j a 3 k b 1 i b 2 j b 3 k a 1 b 1 i i a 1 b 2 i j a 1 b 3 i k a 2 b 1 j i a 2 b 2 j j a 2 b 3 j k a 3 b 1 k i a 3 b 2 k j a 3 b 3 k k displaystyle begin aligned mathbf a times mathbf b a_ 1 mathbf color red i a_ 2 mathbf color green j a_ 3 mathbf color blue k times b_ 1 mathbf color red i b_ 2 mathbf color green j b_ 3 mathbf color blue k 1ex a_ 1 b_ 1 mathbf color red i times mathbf color red i a_ 1 b_ 2 mathbf color red i times mathbf color green j a_ 1 b_ 3 mathbf color red i times mathbf color blue k a_ 2 b_ 1 mathbf color green j times mathbf color red i a_ 2 b_ 2 mathbf color green j times mathbf color green j a_ 2 b_ 3 mathbf color green j times mathbf color blue k a_ 3 b_ 1 mathbf color blue k times mathbf color red i a_ 3 b_ 2 mathbf color blue k times mathbf color green j a_ 3 b_ 3 mathbf color blue k times mathbf color blue k end aligned this can be interpreted as the decomposition of a b into the sum of nine simpler cross products involving vectors aligned with i j or k each one of these nine cross products operates on two vectors that are easy to handle as they are either parallel or orthogonal to each other from this decomposition by using the above mentioned equalities and collecting similar terms we obtain a b a 1 b 1 0 a 1 b 2 k a 1 b 3 j a 2 b 1 k a 2 b 2 0 a 2 b 3 i a 3 b 1 j a 3 b 2 i a 3 b 3 0 a 2 b 3 a 3 b 2 i a 3 b 1 a 1 b 3 j a 1 b 2 a 2 b 1 k displaystyle begin aligned mathbf a times mathbf b quad a_ 1 b_ 1 mathbf 0 a_ 1 b_ 2 mathbf color blue k a_ 1 b_ 3 mathbf color green j a_ 2 b_ 1 mathbf color blue k a_ 2 b_ 2 mathbf 0 a_ 2 b_ 3 mathbf color red i a_ 3 b_ 1 mathbf color green j a_ 3 b_ 2 mathbf color red i a_ 3 b_ 3 mathbf 0 1ex a_ 2 b_ 3 a_ 3 b_ 2 mathbf color red i a_ 3 b_ 1 a_ 1 b_ 3 mathbf color green j a_ 1 b_ 2 a_ 2 b_ 1 mathbf color blue k end aligned meaning that the three scalar components of the resulting vector s s 1 i s 2 j s 3 k a b are s 1 a 2 b 3 a 3 b 2 s 2 a 3 b 1 a 1 b 3 s 3 a 1 b 2 a 2 b 1 displaystyle begin aligned s_ 1 a_ 2 b_ 3 a_ 3 b_ 2 s_ 2 a_ 3 b_ 1 a_ 1 b_ 3 s_ 3 a_ 1 b_ 2 a_ 2 b_ 1 end aligned using column vectors we can represent the same result as follows s 1 s 2 s 3 a 2 b 3 a 3 b 2 a 3 b 1 a 1 b 3 a 1 b 2 a 2 b 1 displaystyle begin bmatrix s_ 1 s_ 2 s_ 3 end bmatrix begin bmatrix 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 bmatrix matrix notation edit use of sarrus s rule to find the cross product of a and b the cross product can also be expressed as the formal determinant note 1 1 a b i j k a 1 a 2 a 3 b 1 b 2 b 3 displaystyle mathbf a times b begin vmatrix mathbf i mathbf j mathbf k a_ 1 a_ 2 a_ 3 b_ 1 b_ 2 b_ 3 end vmatrix this determinant can be computed using sarrus s rule or cofactor expansion using sarrus s rule it expands to a b a 2 b 3 i a 3 b 1 j a 1 b 2 k a 3 b 2 i a 1 b 3 j a 2 b 1 k a 2 b 3 a 3 b 2 i a 1 b 3 a 3 b 1 j a 1 b 2 a 2 b 1 k displaystyle begin aligned mathbf a times mathbf b a_ 2 b_ 3 mathbf i a_ 3 b_ 1 mathbf j a_ 1 b_ 2 mathbf k a_ 3 b_ 2 mathbf i a_ 1 b_ 3 mathbf j a_ 2 b_ 1 mathbf k a_ 2 b_ 3 a_ 3 b_ 2 mathbf i a_ 1 b_ 3 a_ 3 b_ 1 mathbf j a_ 1 b_ 2 a_ 2 b_ 1 mathbf k end aligned 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 p...
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