I know this question has been asked many times before, but I want to narrow it down into some more specific questions I have regarding this topic.
Many descriptions online say that an inner product is the dot product for a real vector space, but then why is something like $\langle u, v \rangle = 3u_1v_1 + 4u_2v_2$ considered an inner product as well? What would be the purpose of such an inner product?
I think geometrically it’s a bit difficult to understand why the kind of inner product I mentioned earlier is using the same concept of vector projections as a standard dot product.
When I learned about dot products, I always thought of $u \cdot v = |u||v|\cos(\theta)$ as the main definition and the fact that this is also equal to $u_1v_1 + u_2v_2$ was just a nice bonus to make calculations easier. With inner products, though, it seems we only learn about them in terms of the second definition but not the first, and so I’m struggling to understand them geometrically.