What are good examples of applications where product or quotient of vector spaces is used?

[vivid hound]

Products: The product of two Vectorspaces can be used to form a new vector space. If you want that the new VS has the structure of the 2 original VS then you will come across the Tensor product. So why we need this?
Consider finitly many VS V_i, lets say we have n VS and we call the product of them V. And now we have a multilinear map f: V -> W where W is just any arbitrary VS. Multilinear means that our f is linear in each of the n components. Now we can take the tensor product V' of the V_i and we can construct a unique map g: V' -> W which will be a linear map!

Quotients are (to my knowledge) often used in group theory to show some nice isomorphisms to work with or to show nice theorems (e.g. Chinese remander theorem)

I never worked with them in application but some PHD told me that they need them for some science stuff