It's been a little under two decades since I did geometry, but it came up again because I'm learning about cosine similarity between two vectors (useful when looking at word embeddings in machine learning).
Cosine similarity has this definition: $$\frac{U^{T}V}{||U||_2||V||_2}$$
However the professor in my online course noted that this would also just be equal to the cosine of the angle between the two vectors. So it shows the difference in orientation (not magnitude) between the two vectors. I haven't calculated this to verify.
In any case, I realized I was confused about more basic things:
Why isn't cos(90 degrees) undefined? The answer is 0, but I think it should be undefined. In general with cosine we are talking about right triangles. So when you say cos(30 degrees), the other angles are 60 and 90. But when you say cos(90 degrees), now you have two 90 degree angles and can't form a triangle at all! So is it just a convention that the answer is 0?
When you say cos(100 degrees), the online visualizations I saw here basically just make the triangle face "backwards" so now the right angle is on the left instead of the right. Apparently here we have negative values for cosine(angle). But I'm confused because both the "near side" and the hypotenuse have positive lengths, so the ratio should be positive. So again is it just a convention that the value is now negative?
How can we even identify the vertices of the triangle? Are we just saying that the origin is one vertex, and the other two vertices are the points that you'd get if you plotted the vector in (say) 300 dimensional space? Actually technically he never said it was a triangle, he talked about the angle between two vectors, but I assume it's more or less equivalent to what I'm saying here. Correct me if I'm wrong.
Thanks in advance!
