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It all seems nice a tidy, but the fact that the vector generated by the cross product isn't the natural result of vector multiplication leaves a sort of smudge on the vectors created by cross products: they transform differently under reflections. This creates the whole confusion over pseudovectors. The product of vectors is naturally a plane, and planes don't switch sign under reflection, while vectors do. For more details see this answer by ACuriousMind to another question.

It all seems nice a tidy, but the fact that the vector generated by the cross product isn't the natural result of vector multiplication leaves a sort of smudge on the vectors created by cross products: they transform differently under reflections. This creates the whole confusion over pseudovectors. The product of vectors is naturally a plane, and planes don't switch sign under reflection, while vectors do. For more details see this answer by ACuriousMind to another question.

an illustration http://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Exterior_calc_cross_product.svg/231px-Exterior_calc_cross_product.svg.png

1 scalar, 4 vectors, 6 plane elements, 4 independent (3D) volumes and only 1 4D volume. It is always true that these sequences are symmetric about the middle. This allows us to define an operator: ($\star$) the hodge dual, that makes from an object on the left half of one of these sequences to the right half and vice versa. In particular, this allows us to associate any plane in 3D with a corresponding vector. This is entirely natural, we do this all of the time in 3D, the natural vector associated with a plane is just the vector perpendicular to the plane. In 4D, every vector is associated with a 3D volume element and vice versa.

1 scalar, 4 vectors, 6 plane elements, 4 independent (3D) volumes and only 1 4D volume. It is always true that these sequences are symmetric about the middle. This allows us to define an operator: ($\star$) the hodge dual, that maps from an object on the left half of one of these sequences to the right half and vice versa. In particular, this allows us to associate any plane in 3D with a corresponding vector. This is entirely natural, we do this all of the time in 3D, the natural vector associated with a plane is just the vector perpendicular to the plane. In 4D, every vector is associated with a 3D volume element and vice versa.