** **Let us consider a vector space *V* with
dimension *n*. The *outer* (or *exterior*) *product*
of two vectors *a* and *b* is defined as the geometric element
obtained by parallel transport of the vector *a* following the way
indicated by *b*. The outer product is indicated by the circumflex
accent

*a* ^ *b*

If *a* and *b* are geometric segments, then *a* ^ *b*
is the parallelogram formed by both vectors. By repeating this operation
of parallel transport we create geometric elements with higher dimension.
When we use the components of the vectors in a base of V, the outer product is always
a linear combination of the determinants multiplied by the corresponding
outer products of the base vectors. For example, in a three-dimensional
space, we have:

*a* = *a*_{1}*e*_{1}
+ *a*_{2} *e*_{2} + *a*_{3}*e*_{3}
*b* = *b*_{1} *e*_{1} + *b*_{2}*e*_{2}
+ *b*_{3} *e*_{3}

*a* ^ *b* = ( *a*_{2}*b*_{3}
- *a*_{3} *b*_{2} ) *e*_{2} ^ *e*_{3}
+ ( *a*_{3} *b*_{1} - *a*_{1} *b*_{3}
) *e*_{3} ^ *e*_{1} + ( *a*_{1}*b*_{2}
- *a*_{2} *b*_{1} ) *e*_{1} ^ *e*_{2}

For example if *a*, *b* and *c* are segments, then *a*
^ *b* ^ *c* is the parallelepiped whose three concurrent sides
are the three given segments. The outer product is always anticommutative:
any permutation of two vectors changes the sign of the outer product (you
can read that the permutation of any two rows or columns changes the sign
of the determinant). The dimension of the outer products of *m* vectors
is always the combinatorial number *C _{m, n }*.

The Grassmann-Clifford Geometric Algebra

The CGGA has a strong analogy with the Boole's Algebra

*a* **· ***b* = *b* **·
***a*

whose random analogy is the intersection of events. The *geometric
product* (also called the *Clifford product*) is defined as:

*a b* = *a ***·** *b* +
*a*
^ *b*

and it is represented with no special symbol. The geometric product
is associative (also the outer product is associative but not the inner
product). The *Cl*(*V*) is a closed set with regard to the addition
and product of vectors, that is, the result of any sum or product (inner,
outer or geometric one) of elements of the geometric algebra is also an element of
the geometric algebra. The geometric product is associative and has the unity as
the neutral element. Then *Cl*(*V*) is an associative algebra
with unity element, a type of algebras always having isomorphic matrix
representation. Then, the geometric product is isomorphic to the matrix
product, a very important result, especially for the programming of computer applications.

You can find more technical
explanations on the web What
ARE Clifford Algebras and Spinors?