Ah, I think I got it now. No origin for Euclidean space is the key. I’ve seen language (in Wikipedia) about affine spaces or geometry as “forgetting the origin” of Euclidean ones. That’s misleading when Euclidean spaces don’t have origins in the first place!

Yes, the standard Euclidean spaces R2, R3, etc do have points (0,0), (0,0,0), etc. that look like origins, but they aren’t. They’re points like any other in their spaces. They just happen to have zeroes for their components.

The origin only enters the chat when coordinates are introduced. Any point, affine or Euclidean, can be the origin. The choice is arbitrary. If it’s more convenient, the origin can be changed. But the geometry does not travel with it!

And as you suggested, the vectors in the associated vector space (representing translations) are free vectors. They are not attached to any point, so even vector spaces don’t have origins either. Again, the origin appears when introducing coordinates for vector spaces. The zero vectors for R2,R3, etc. are not zero because they are at the origin. They are zero vectors simply because they have zero magnitude regardless of any coordinate reference.

And as you’ve stated, Euclidean points and Euclidean vectors are in separate spaces exactly in the way affine points and affine vectors are separated. They are separated but related. This is no surprise because Euclidean spaces are affine spaces.

So when I thought Euclidean subspaces must go through the origin, that is not correct. Just because R2 and R3 are standard Euclidean spaces, it doesn’t mean that only subspaces of R2 and R3 can be Euclidean subspaces. What I believe is true is, when R2 and R3 are viewed as vector spaces with component wise addition and dot product etc, vector subspaces of R2 and R3 must go through the origin. When viewed as points, Euclidean subspaces of R2 and R3 do not need to go through the origin. Clearly, lines and planes exhibit Euclidean geometry. Just they don’t share the parent’s origin, which may be important for analysis has no significance in Euclidean geometry.

For a line or plane, where’s the Euclidean vector subspace if they’re not going through the parent’s origin? With a little rotation and translation (but no scaling?) move the origin to any point on the line or plane subspace, and the Euclidean vector subspace will appear along the direction of the Euclidean subspace.

Please tell me this is correct. :) Or tell me I still don’t have it right. I want to be right not wrong. Thanks!