> Forget about angles and geometry unless you are comfortable with N-dimensional space with N>>3. Most people aren't.
The whole point of measuring similarity this way is that any two vectors exist in a two-dimensional space, which is where you measure the angle between them. Why would you need to be comfortable with high-dimensional spaces?
No, I'm talking about the fact that the space spanned by two vectors is sufficient to contain those vectors. All of the analysis you could ever theoretically want to do on them can be done within that space. If you only have two vectors, you never need to consider a space with higher dimensionality than 2. Each vector is a dimension of the space, and that's it.
(A) Look at this space. Every point within it can be reached by combining these two vectors.
(B) Look at this space. No point outside it can be reached by combining these two vectors.
Saying that two vectors span a space is claim (A). Saying that the space they span contains them is... much weaker than claim (B), but it's related to claim (B) and not to claim (A).
for one reason, if you're just thinking about it as fancy 2d, you will miss a lot of phenomena that occur in higher dimensional spaces. for example, almost all vectors are almost completely orthogonal which isn't true at all in low dimensional spaces
Phrased like that it sounds like a qualitative difference between "low" and "high" dimensional spaces. But isn't it simply a consequence of the fact that the more dimensions you have, the less likely that randomly distributed, sparse non-zeros will end up in the same positions?
The whole point of measuring similarity this way is that any two vectors exist in a two-dimensional space, which is where you measure the angle between them. Why would you need to be comfortable with high-dimensional spaces?