Definition
We can define a cross product on a Euclidean space V as a bilinear map from V × V to V mapping vectors x and y in V to another vector x × y also in V, where x × y has the properties
- orthogonality:
-
- ,
- magnitude:
where (x·y) is the Euclidean dot product and |x| is the vector norm. The first property states that the cross product is perpendicular to its arguments, while the second property gives the magnitude of the cross product. An equivalent expression in terms of the angle θ between the vectors is
or the area of the parallelogram in the plane of x and y with the two vectors as sides. As a third alternative the following can be shown to be equivalent to either expression for the magnitude:
Read more about this topic: Seven-dimensional Cross Product
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