A

vector product x is a

binary operation on

**R**^{n} 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 **R**^{3} and **R**^{7}. 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

(a_{1}, a_{2}, a_{3}) x (b_{1}, b_{2}, b_{3}) = (a_{2}b_{3}-a_{3}b_{2}, -a_{1}b_{3}+a_{3}b_{1}, a_{1}b_{2}-a_{2}b_{1})

In **R**^{3} 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 **R**^{7} by

(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}) x (b_{1}, b_{2}, b_{3}, b_{4}, b_{5}, b_{6}, b_{7}) =

(a_{2}b_{3}-a_{3}b_{2}+a_{4}b_{5}-a_{5}b_{4}+a_{6}b_{7}-a_{7}b_{6},
-a_{1}b_{3}+a_{3}b_{1}-a_{4}b_{6}+a_{5}b_{7}+a_{6}b_{4}-a_{7}b_{5},
a_{1}b_{2}-a_{2}b_{1}+a_{4}b_{7}+a_{5}b_{6}-a_{6}b_{5}-a_{7}b_{4},
-a_{1}b_{5}+a_{2}b_{6}-a_{3}b_{7}+a_{5}b_{1}-a_{6}b_{2}+a_{7}b_{3},
a_{1}b_{4}-a_{2}b_{7}-a_{3}b_{6}-a_{4}b_{1}+a_{6}b_{3}+a_{7}b_{2},
-a_{1}b_{7}-a_{2}b_{4}+a_{3}b_{5}+a_{4}b_{2}-a_{5}b_{3}+a_{7}b_{1},
a_{1}b_{6}+a_{2}b_{5}+a_{3}b_{4}-a_{4}b_{3}-a_{5}b_{2}-a_{6}b_{1})

The vector product in **R**^{7} 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 G_{2}.

There is a connection between vector product and normed division algebras. The vector product on **R**^{3} can be regarded as the imaginary part of the multiplication of the four-dimensional algerba of quaternions, while the vector product on **R**^{7} is related to the octonions.