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For this formal definition and for the formal definition of norms, will any function that satisfies the given constraints also capture the idea of "how close" and "how big"?


For the last rule, I am not able to visualize geometrically. Is this related to the difference of angles that w makes with u, v, and u + v?


For the last rule, I tried sketching out the LHS and RHS with parallelogram method of vector addition, and by drawing vector v's tail to vector u's tip, we extend the portion where u+v is along the direction of vector w, by exactly the amount of <v, w>. Is this the correct intuition for a geometric proof?


Agreed with corgo. I proved the last rule similarly. From my understanding, the larger the inner product is, the closer the two vectors are.


I also proved the last rule using the parallelogram method, similar to what corgo explained above.


I'm a little confused about the intuition for rule 4


What is the relationship between the inner product and the norm? If we specify an inner product in some vector space, does that induce a norm? If we have a norm in some vector space, will the inner product be unique?


When we talk about an inner product, we need to first specify a vector space. Is that right?


I think the geometrical explanation with the last rule is that <u, w> and <v,w> are two parts that make up the projection of u + v to w, which is .


Usually when we think inner product we usually just think of the dot product (or a derivative of one). Are there any other inner products that we might find ourselves using during this semester?


To answer @niyiqiul's question, I do think there are inner product vector spaces, where you have a vector space and then specify that inner product + they're useful for generalizing vector spaces. However I think I've only seen the Euclidean vector space used in this class--are there other vector spaces used in graphics?