A vector product
x is a binary operation
with the following properties:
a x b is linear in a and b
a x b is orthogonal to both a and b
|a x b|2 + |a · b|2 = |a|2|b|2
From these properties we can in addition deduce
|a x a|2 = |a|4 - |a · a|2 = 0, so a x a = 0
0 = (a + b) x (a + b) = a x b + b x a, so x is antisymmetric
Vector products only exist in R3 and R7. In 3 dimensions it is uniquely defined, up to sign: if x is a vector product then * is a vector product iff a * b = ±a x b). By convention the sign is chosen so that
(a1, a2, a3) x (b1, b2, b3) = (a2b3-a3b2, -a1b3+a3b1, a1b2-a2b1)
In R3 the vector product (often called cross product because of the symbol used, even though sometimes a wedge is used rather than a cross) is actually quite useful, since the 2-dimensional subspace spanned by a and b is defined by a x b (in English: the vector product gives the normal to the plane containing the two vectors). Also a x b = 0 is a useful necessary and sufficient condition for a and b to be proportional.
A peculiar property of the vector product is that the result is in fact not a vector. If we make a coordinate transformation that is orientation reversing (i.e. we change to a new coordinate frame using a reflection) then the there will be an extra sign change in a x b (since a x b depends on the orientation of a and b) that would not be there if it was a vector. Therefore the vector product is in fact a vector density. The distiction is not a very important one though.
We can define a vector product on R7 by
(a1, a2, a3, a4, a5, a6, a7) x (b1, b2, b3, b4, b5, b6, b7) =
The vector product in R7 is less obviously applicable. It should be emphasized that, unlike in three dimensions, it is not unique: there are many different vector products compatible with the standard inner product. The symmetry group of the vector product is a subgroup of SO(7) called G2.
There is a connection between vector product and normed division algebras. The vector product on R3 can be regarded as the imaginary part of the multiplication of the four-dimensional algerba of quaternions, while the vector product on R7 is related to the octonions.