Some question about the outer product between the vector and the plane

[Net-Maker]

Hello Chen, thanks for your excellent work!
I'm writing to ask some question about the combination of the line vector and the plane mat in density pipeline.
Why this process use outer product, which artfully make the output dims to be 1, and represented to be the density?
What's the relationship between the resolution and component? I found that I can use a different resolution between line vector and plane mat if the component dim is aligned. In my thought, it's just a precision problem, higher resolution means the feature is interpolated from a grid that is smaller, what's your opinion?
By the way, have you clarified why the slim part of the 3D object would disappear? i.e:ship in synthetic_nerf.
I read your new paper Factor Field, which is really amazing, but I'm still confused with this question.
Looking forward to your reply!
Best Wishes!