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I am hoping for a meaningful interpretation for the geometric product of two blades of not-necessarily the same grade.

I understand that blades can be expressed as a sum of basis k-vectors, and then I can use the rules of the geometric algebra of basis vectors to compute the resulting product. A description of this is shown here: https://math.stackexchange.com/a/2884286/24205

I’m hoping for an understanding of what the result is geometrically. What does the result of the product of two blades represent? Any thoughts are appreciated.

For a special case, the interpretation is nice. Let e be the standard basis for the underlying vector space. Let $A=e_1 e_2 e_3$ and $B=e_3 e_4 e_5$. Then $AB= e_1 e_2 e_4 e_5$. In this case, then $AB$ is a blade that represents the union of the subspaces minus the intersection. Does something like this hold generally?

NicNic8
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    "union of the subspaces minus the intersection" - That's not quite right. To visualize more easily, let's decrease the dimensions: $A=e_1e_2$, and $B=e_2e_3$, so $AB=e_1e_3$. The set of basis vectors in $AB$ is the union minus the intersection of the sets of basis vectors in $A$ and $B$. But the set of basis vectors is not the subspace itself. It's not clear how to describe $AB$ without referring to a basis. – mr_e_man Nov 28 '23 at 18:30
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    For any two unit blades $A$ and $B$, the magnitudes of the various grades of the product $AB$ encode the angles between the two subspaces. I wrote about this in more detail off-site: http://hi.gher.space/forum/viewtopic.php?f=32&t=2620 (Related on-site: https://math.stackexchange.com/questions/4487760/finding-angles-between-planes-in-4d-space and https://math.stackexchange.com/questions/3123606/principal-angles-between-subspaces ) – mr_e_man Nov 28 '23 at 19:18
  • @mr_e_man Thank you! – NicNic8 Nov 29 '23 at 19:27
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    @mr_e_man André Mandolesi has some work on this as well. – Nicholas Todoroff Dec 01 '23 at 17:23

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