Conventions

As GA is still in evolution, people around the world hasn’t settled down to a common way of indicating elements and operations. Here are the ones that we believe are the most common and least likely to cause confusion.

Symbol	Meaning
	orthonormal basis vectors
	vector (bold lowercase letter)
	bivector
	unit pseudoscalar
	inverse of a multivector M (in euclidean space )
	conjugate (reverse) of a multivector M; it is also indicated with tilde above , especially in spacetime algebra
	dual of a multivector, i.e. its representation in the orthogonal complement
	rotor of amplitude on B plane (bivector) coherent with the orientation of B
	rotor of amplitude around the axis (3D only)
	generalized rotation of multivector M by the rotor R

The dual of a multivector $A$ is defined by multiplying it by the pseudoscalar $I$. But “multiplying by $I$” hides three ambiguities, and for this reason a convention must be fixed:

1. On the right or on the left? $AI$ or $IA$. In 3D the pseudoscalar commutes with everything, so $AI = IA$ and this choice does not change the sign. But in 2D and in even dimensions $I$ anticommutes with vectors, so $vI = -Iv$: there, the right/left choice changes the sign. This is the first source of ambiguity.
2. I or $AI^{-1}$? Some authors define the dual as $AI$, others as $AI^{-1}$. And $I^{-1} = \pm I$ depending on the dimension: in 3D $I^2 = -1$, so $I^{-1} = -I$. Thus those who use $I$ and those who use $I^{-1}$ get opposite signs already in 3D. This is the most subtle source of sign, because both conventions are widespread in the literature (Hestenes often uses $I^{-1}$ precisely to stabilize the signs).
3. The orientation of II itself. $I = e_1 e_2 e_3$​ or $I = e_3 e_2 e_1$​? They differ by a sign. And in 4D the ordering $e_0 e_1 e_2 e_3$ versus other permutations comes into play as well.

We fix here, once and for all, our convention, valid throughout the site:

• pseudoscalar in increasing order: $I = e_1 e_2 e_3$​ (in 3D);
• dual defined as multiplication on the right by $I^{-1}$: $A^* = A\,I^{-1}$;
• in 3D, $I^2 = -1$I, hence $I^{-1} = -I$.

With this convention, the relationship between the outer product and the cross product is:

(1)