page_title: Cross product
section_title: Cross product as an external product
hierarchical_section_title: Cross product / Cross product as an external product
context_page_description: In mathematics, the cross product or vector product is a binary operation on two vectors in three-dimensional space and is denoted by the symbol. Given two linearly independent vectors a and b, the cross product, a × b, is a vector that is perpendicular to both a and b and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product.
  If two vectors have the same direction or if either one has zero length, then their cross product is zero. More generally, the magnitude of the product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two perpendicular vectors is the product of their lengths. The cross product is anticommutative and is distributive over addition. The space together with the cross product is an algebra over the real numbers, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket.
  Like the dot product, it depends on the metric of Euclidean space, but unlike the dot product, it also depends on a choice of orientation or "handedness".
context_section_description: The cross product can be defined in terms of the exterior product. It can be generalized to an external product in other than three dimensions. This view allows for a natural geometric interpretation of the cross product. In exterior algebra the exterior product of two vectors is a bivector. A bivector is an oriented plane element, in much the same way that a vector is an oriented line element. Given two vectors a and b, one can view the bivector a ∧ b as the oriented parallelogram spanned by a and b. The cross product is then obtained by taking the Hodge star of the bivector a ∧ b, mapping 2-vectors to vectors:
  This can be thought of as the oriented multi-dimensional element "perpendicular" to the bivector. Only in three dimensions is the result an oriented one-dimensional element – a vector – whereas, for example, in four dimensions the Hodge dual of a bivector is two-dimensional – a bivector. So, only in three dimensions can a vector cross product of a and b be defined as the vector dual to the bivector a ∧ b: it is perpendicular to the bivector, with orientation dependent on the coordinate system's handedness, and has the same magnitude relative to the unit normal vector as a ∧ b has relative to the unit bivector; precisely the properties described above.
caption: The cross product in relation to the exterior product. In red are the orthogonal unit vector, and the "parallel" unit bivector.