What exactly is the difference between right and left contraction in Geometric algebra?

Question

We motivated the original definition of $A\rfloor B$ in (3.6) in terms of geometrical subspaces as “A taken out of B”. This is clearly asymmetrical in A and B, and we could also have used the same geometrical intuition to define an operation $ B \lfloor A$, interpreted as “take B and remove A from it.” The two are so closely related that we really only need one to set up our algebra, but occasionally formulas get simpler when we switch over to this other contraction. Let us briefly study their relationship

PG-79 , Leo Dorst Geometric Algebra book

I don't quite get what is the difference is between "A taken out of B" and "take B and remove A from it".. in English isn't "a ball taken out of the bag " and "take a bag and then take the ball out" the same procedure..?

Definition of right contraction:

$$ B \star (A \wedge X) = (B \lfloor A) \star X$$

Left contraction is defined axiomatically in the book on page 74. Also wiki link.

Answer 1

Your analogy is not quite adequate because the ball is contained wholly within the bag, while A is not necessarily included within B. Apart from that, you are correct that $A ⌋ B$ is very similar to $B ⌊ A$ since they "differ only by a grade-dependent sign" (p. 78).

As for the reason why they bother to define two operations, I guess it's because of the non-commutativity of the contraction ($A ⌋ B ≠ B ⌋ A$),which makes it stand apart from the other products... or maybe the right contraction is really useful at times, and I just can't figure when...

Answer 2

I think the difference is so obvious, that you're missing it by looking for something deeper in the English expressions.

Working through the math makes it clear why the grade-dependent sign is there, depending on whether the contraction applied from the left or right. Whether you take the geometric product as fundamental or built from e.g. dot and wedge, it has symmetric and asymmetric parts. The asymmetric part tends to produce order dependence in secondary convenience products like the left and right contractions here. The authors even state in your quote, that "...this is clearly asymmetrical in A and B...".

The English expressions, on the other hand, are merely finding ways to concisely suggest - but not define - the intuitive meanings while using the variable labels in the order that they appear in the mathematical expressions while speaking or reading them. I don't think there's really anything there beyond that. The surrounding exposition in the book provides more mathematical detail and context, so the short English expressions are merely suggestive for one who's worked through the previous material.