**Cross Product As An Exterior Product**

The cross product can be viewed in terms of the exterior product. This view allows for a natural geometric interpretation of the cross product. In exterior algebra the exterior product (or wedge 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 dual 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 line element – a vector – whereas, for example, in 4 dimensions the Hodge dual of a bivector is two-dimensional – another oriented plane element. So, only in three dimensions is the cross product of *a* and *b* 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.

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