Editing Cross product (section)

== Definition ==
[[File:Right hand rule cross product.svg|thumb|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 {{nowrap|1='''a''' × '''b'''}}. In [[physics]] and [[applied mathematics]], the wedge notation {{nowrap|1='''a''' ∧ '''b'''}} is often used (in conjunction with the name ''vector product''),<ref>{{cite book|author1=Jeffreys, H. |author2=Jeffreys, B. S. |title=Methods of mathematical physics |year=1999 |publisher=Cambridge University Press |oclc=41158050}}</ref><ref>{{cite book |author1=Acheson, D. J. |author-link=David Acheson (mathematician) |title=Elementary Fluid Dynamics |year=1990 |publisher=Oxford University Press |isbn=0198596790}}</ref><ref>{{cite book |author1=Howison, Sam |title=Practical Applied Mathematics |year=2005 |publisher=Cambridge University Press |isbn=0521842743}}</ref> although in pure mathematics such notation is usually reserved for just the exterior product, an abstraction of the vector product to {{mvar|n}} dimensions.

The cross product {{nowrap|'''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]]<ref name=":1" /><!-- this is how first time students, who also use right-hand coordinates, learn --> and a magnitude equal to the area of the [[parallelogram]] that the vectors span.<ref name=":2" />

The cross product is defined by the formula<ref>{{harvnb|Wilson|1901|page=60–61}}.</ref><ref name=Cullen>{{cite book |title=Advanced engineering mathematics |author1=Dennis G. Zill |author2=Michael R. Cullen |edition=3rd |year=2006 |publisher=Jones & Bartlett Learning |chapter-url=https://books.google.com/books?id=x7uWk8lxVNYC&pg=PA324 |page=324 |chapter=Definition 7.4: Cross product of two vectors |isbn=0-7637-4591-X}}</ref>

: <math>\mathbf{a} \times \mathbf{b} = \| \mathbf{a} \| \| \mathbf{b} \| \sin(\theta) \, \mathbf{n},</math>

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 [[Magnitude (vector)|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 [[Orientation (vector space)|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 ===
[[File:Cross product.gif|left|thumb|The cross product {{nowrap|'''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 [[Anticommutativity|anti-commutative]]; that is, {{nowrap|1='''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''. See {{Section link||Handedness}} for more detail.