Editing Cross product (section)

== Properties ==

=== Geometric meaning ===
{{See also|Triple product}}
[[File:Cross product parallelogram.svg|right|thumb|Figure 1. The area of a parallelogram as the magnitude of a cross product]]
[[File:Parallelepiped volume.svg|right|thumb|240px|Figure 2. Three vectors defining a parallelepiped]]

The [[Euclidean norm|magnitude]] of the cross product can be interpreted as the positive [[area]] of the [[parallelogram]] having '''a''' and '''b''' as sides (see Figure 1):<ref name=":1" />
<math display="block"> \left\| \mathbf{a} \times \mathbf{b} \right\| = \left\| \mathbf{a} \right\| \left\| \mathbf{b} \right\| \left| \sin \theta \right| .</math>

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):

:<math>
  \mathbf{a}\cdot(\mathbf{b}\times \mathbf{c})=
  \mathbf{b}\cdot(\mathbf{c}\times \mathbf{a})=
  \mathbf{c}\cdot(\mathbf{a}\times \mathbf{b}).
</math>

Since the result of the scalar triple product may be negative, the volume of the parallelepiped is given by its absolute value:

:<math>V = |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|.</math>

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 ===
[[File:Cross product scalar multiplication.svg|350px|thumb|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).]]

[[File:Cross product distributivity.svg|350px|thumb|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.<ref>{{cite book|title=Vector Analysis|author1=M. R. Spiegel |author2=S. Lipschutz |author3=D. Spellman |series=Schaum's outlines|year=2009|page=29|publisher=McGraw Hill|isbn=978-0-07-161545-7}}</ref>]]

[[File:Cross product triple.svg|thumb|350px|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 [[vector rejection|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, {{nowrap|1='''a''' × '''b''' = '''0'''}}), then either one or both of the inputs is the zero vector, ({{nowrap|1='''a''' = '''0'''}} or {{nowrap|1='''b''' = '''0'''}}) or else they are parallel or antiparallel ({{nowrap|'''a''' ∥ '''b'''}}) so that the sine of the angle between them is zero ({{nowrap|1=''θ'' = 0°}} or {{nowrap|1=''θ'' = 180°}} and {{nowrap|1=sin&nbsp;''θ'' = 0}}).

The self cross product of a vector is the zero vector:
:<math>\mathbf{a} \times \mathbf{a} = \mathbf{0}.</math>
The cross product is [[anticommutativity|anticommutative]],
:<math>\mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a}),</math>
[[distributive property|distributive]] over addition,
: <math>\mathbf{a} \times (\mathbf{b} + \mathbf{c}) = (\mathbf{a} \times \mathbf{b}) + (\mathbf{a} \times \mathbf{c}),</math>
and compatible with scalar multiplication so that
:<math>(r\,\mathbf{a}) \times \mathbf{b} = \mathbf{a} \times (r\,\mathbf{b}) = r\,(\mathbf{a} \times \mathbf{b}).</math>

It is not [[associative]], but satisfies the [[Jacobi identity]]:
:<math>\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}.</math>
Distributivity, linearity and Jacobi identity show that the '''R'''<sup>3</sup> [[Real coordinate space|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, {{nowrap|1='''a''' × '''b''' = '''a''' × '''c'''}} with {{nowrap|'''a''' ≠ '''0'''}} does not imply {{nowrap|1='''b''' = '''c'''}}, but only that:
:<math> \begin{align}
  \mathbf{0} &= (\mathbf{a} \times \mathbf{b}) - (\mathbf{a} \times \mathbf{c})\\
             &= \mathbf{a} \times (\mathbf{b} - \mathbf{c}).\\
\end{align}</math>

This can be the case where '''b''' and '''c''' cancel, but additionally where '''a''' and {{nowrap|'''b''' − '''c'''}} are parallel; that is, they are related by a scale factor ''t'', leading to:
:<math>\mathbf{c} = \mathbf{b} + t\,\mathbf{a},</math>
for some scalar ''t''.

If, in addition to {{nowrap|1='''a''' × '''b''' = '''a''' × '''c'''}} and {{nowrap|'''a''' ≠ '''0'''}} as above, it is the case that {{nowrap|1='''a''' ⋅ '''b''' = '''a''' ⋅ '''c'''}} then
:<math>\begin{align}
  \mathbf{a} \times (\mathbf{b} - \mathbf{c}) &= \mathbf{0} \\
   \mathbf{a} \cdot (\mathbf{b} - \mathbf{c}) &= 0,
\end{align}</math>
As {{nowrap|1='''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: {{nowrap|1='''b''' = '''c'''}}.

From the geometrical definition, the cross product is invariant under proper [[rotation (mathematics)|rotations]] about the axis defined by {{nowrap|'''a''' × '''b'''}}. In formulae:
:<math>(R\mathbf{a}) \times (R\mathbf{b}) = R(\mathbf{a} \times \mathbf{b})</math>, where <math>R</math> is a [[rotation matrix]] with <math>\det(R)=1</math>.

More generally, the cross product obeys the following identity under [[matrix (mathematics)|matrix]] transformations:
:<math>(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}) </math>
where <math>M</math> is a 3-by-3 [[matrix (mathematics)|matrix]] and <math>\left(M^{-1}\right)^\mathrm{T}</math> is the [[transpose]] of the [[inverse matrix|inverse]] and <math>\operatorname{cof}</math> is the cofactor matrix. It can be readily seen how this formula reduces to the former one if <math>M</math> is a rotation matrix. If <math>M</math> is a 3-by-3 symmetric matrix applied to a generic cross product <math>\mathbf{a} \times \mathbf{b}</math>, the following relation holds true: 
:<math>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}</math>
The cross product of two vectors lies in the [[null space]] of the {{nowrap|2 × 3}} matrix with the vectors as rows:
:<math>\mathbf{a} \times \mathbf{b} \in NS\left(\begin{bmatrix}\mathbf{a} \\ \mathbf{b}\end{bmatrix}\right).</math>
For the sum of two cross products, the following identity holds:
:<math>\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}.</math>

=== Differentiation ===
{{Main|Vector-valued_function#Derivative_and_vector_multiplication|l1=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:
:<math>\frac{d}{dt}(\mathbf{a} \times \mathbf{b}) = \frac{d\mathbf{a}}{dt} \times \mathbf{b} + \mathbf{a} \times \frac{d\mathbf{b}}{dt} ,</math>

where '''a''' and '''b''' are vectors that depend on the real variable ''t''.

=== Triple product expansion ===
{{Main|Triple product}}

The cross product is used in both forms of the triple product. The [[scalar triple product]] of three vectors is defined as

:<math>\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}), </math>

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:

:<math>\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) = \mathbf{c} \cdot (\mathbf{a} \times \mathbf{b}), </math>

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

:<math>\begin{align}
\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{align}</math>

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 [[gradient]]s and useful in [[vector calculus]], is
:<math>\begin{align}
  \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{align}</math>

where ∇<sup>2</sup> is the [[vector Laplacian]] operator.

Other identities relate the cross product to the scalar triple product:
:<math>\begin{align}
(\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{align}</math>

where ''I'' is the identity matrix.

=== Alternative formulation ===
The cross product and the dot product are related by:
:<math> \left\| \mathbf{a} \times \mathbf{b} \right\| ^2  = \left\| \mathbf{a}\right\|^2  \left\|\mathbf{b}\right\|^2 - (\mathbf{a} \cdot \mathbf{b})^2 .</math>

The right-hand side is the [[Gramian matrix|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:

:<math> \mathbf{a \cdot b} = \left\| \mathbf a \right\| \left\| \mathbf b \right\| \cos \theta , </math>

the above given relationship can be rewritten as follows:

:<math>  \left\| \mathbf{a \times b} \right\|^2 = \left\| \mathbf{a} \right\| ^2 \left\| \mathbf{b}\right \| ^2 \left(1-\cos^2 \theta \right) .</math>

Invoking the [[Pythagorean trigonometric identity]] one obtains:
:<math> \left\| \mathbf{a} \times \mathbf{b} \right\| = \left\| \mathbf{a} \right\| \left\| \mathbf{b} \right\| \left| \sin \theta \right| ,</math>

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|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.<ref name=Massey>{{cite journal |title=Cross products of vectors in higher dimensional Euclidean spaces |author=WS Massey |journal=The American Mathematical Monthly |volume=90 |date=Dec 1983 |pages=697–701 |issue=10 |doi=10.2307/2323537 |publisher=The American Mathematical Monthly, Vol. 90, No. 10 |jstor=2323537}}</ref>

=== Cross product inverse ===

Given two vectors {{math|'''a'''}} and {{math|'''c'''}} with {{nowrap|1='''a'''≠'''0'''}}, the equation {{nowrap|1='''a''' × '''b''' = '''c'''}} admits solutions for {{math|'''b'''}} if and only if {{math|'''a'''}} is orthogonal to {{math|'''c'''}} (that is, if {{nowrap|1='''a''' ⋅ '''c''' = 0}}). In that case, there exists an infinite family of solutions for {{math|'''b'''}}, which are
:<math> \mathbf{b} = \frac{\mathbf{c} \times \mathbf{a}}{\left\| \mathbf{a} \right\|^2} + t \mathbf{a} ,</math>
where {{nowrap|1=''t''}} is an arbitrary constant.

This can be derived using the triple product expansion:
:<math>
\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} </math>
Rearrange to solve for {{nowrap|1='''b'''}} to give
:<math>
\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}
</math>
The coefficient of the last term can be simplified to just the arbitrary constant {{nowrap|1=''t''}} to yield the result shown above.

=== Lagrange's identity ===
The relation
:<math>
  \left\| \mathbf{a} \times \mathbf{b} \right\|^2 \equiv
  \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} \equiv
  \left\| \mathbf{a} \right\| ^2  \left\| \mathbf{b} \right\| ^2 - (\mathbf{a} \cdot \mathbf{b})^2
</math>

can be compared with another relation involving the right-hand side, namely [[Lagrange's identity]] expressed as<ref name=Boichenko>{{cite book |title=Dimension theory for ordinary differential equations |author1=Vladimir A. Boichenko |author2=Gennadiĭ Alekseevich Leonov |author3=Volker Reitmann |url=https://books.google.com/books?id=9bN1-b_dSYsC&pg=PA26 |page=26 |isbn=3-519-00437-2 |year=2005 |publisher=Vieweg+Teubner Verlag}}</ref>

:<math>
  \sum_{1 \le i < j \le n} \left( a_ib_j - a_jb_i \right)^2 \equiv
  \left\| \mathbf a \right\|^2 \left\| \mathbf b \right\|^2 - ( \mathbf{a \cdot b } )^2,
</math>

where '''a''' and '''b''' may be ''n''-dimensional vectors. This also shows that the [[Riemannian volume form]] for surfaces is exactly the [[Volume form|surface element]] from vector calculus. In the case where {{nowrap|1=''n'' = 3}}, combining these two equations results in the expression for the magnitude of the cross product in terms of its components:<ref name=Lounesto1>{{cite book |url=https://books.google.com/books?id=kOsybQWDK4oC&q=%22which+in+coordinate+form+means+Lagrange%27s+identity%22&pg=PA94 |author=Pertti Lounesto |page=94 |title=Clifford algebras and spinors |isbn=0-521-00551-5 |edition=2nd |publisher=Cambridge University Press |year=2001}}</ref>
 
:<math>\begin{align}
  \|\mathbf{a} \times \mathbf{b}\|^2
   &\equiv \sum_{1 \le i < j \le 3} (a_ib_j - a_jb_i)^2 \\
   &\equiv (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{align}</math>

The same result is found directly using the components of the cross product found from
:<math>\mathbf{a} \times \mathbf{b} \equiv \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}.</math>

In '''R'''<sup>3</sup>, Lagrange's equation is a special case of the multiplicativity {{nowrap|1={{abs|'''vw'''}} = {{abs|'''v'''}}{{abs|'''w'''}}}} of the norm in the [[Quaternion#Algebraic properties|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]]:<ref name=Liu/><ref name=Weisstein>by {{cite book |author=Eric W. Weisstein |chapter=Binet-Cauchy identity |title=CRC concise encyclopedia of mathematics |chapter-url=https://books.google.com/books?id=8LmCzWQYh_UC&pg=PA228 |page=228 |isbn=1-58488-347-2 |edition=2nd |year=2003 |publisher=CRC Press}}</ref>

:<math>
  (\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{c} \times \mathbf{d}) \equiv
  (\mathbf{a} \cdot \mathbf{c})(\mathbf{b} \cdot \mathbf{d}) - (\mathbf{a} \cdot \mathbf{d})(\mathbf{b} \cdot \mathbf{c}).
</math>

If {{nowrap|1='''a''' = '''c'''}} and {{nowrap|1='''b''' = '''d'''}}, this simplifies to the formula above.

=== Infinitesimal generators of rotations ===
{{further|Infinitesimal rotation matrix#Generators of rotations}}

The cross product conveniently describes the infinitesimal generators of [[rotation (mathematics)|rotation]]s in '''R'''<sup>3</sup>. Specifically, if '''n''' is a unit vector in '''R'''<sup>3</sup> and ''R''(''φ'', '''n''') denotes a rotation about the axis through the origin specified by '''n''', with angle φ (measured in radians, counterclockwise when viewed from the tip of '''n'''), then
:<math>\left.{d\over d\phi} \right|_{\phi=0} R(\phi,\boldsymbol{n}) \boldsymbol{x} = \boldsymbol{n} \times \boldsymbol{x}</math>
for every vector '''x''' in '''R'''<sup>3</sup>. The cross product with '''n''' therefore describes the infinitesimal generator of the rotations about '''n'''. These infinitesimal generators form the [[Lie algebra]] '''so'''(3) of the [[rotation group SO(3)]], and we obtain the result that the Lie algebra '''R'''<sup>3</sup> with cross product is isomorphic to the Lie algebra '''so'''(3).