I am studying linear algebra and I am trying to get a better intuition on how I should think about finite dimensional real inner product spaces.(I am doing this since for example in Analysis it is often my "intuition" for what it means for something to be continuous or have a certain derivative e.t.c that helps me understand theorems and solve problems.)
Currently I try to think of a real finite dimensional inner product space as a set of "effects"(vectors), each effect having a type and a magnitude(the norm)(two vectors having the same type iff one is a scalar multiple of the other). The set of these effects(vectors) also includes a basis consisting of effects which are independent in the following sense; if one of these effects occur it does not cause any of the other type of effects to occur .
Since we are talking about an n dimensional vector space each "effect" can be seen as a linear combination of these independent effects.
Then the inner product of two vectors(effects)a and b can be regarded as the product of their magnitudes times the magnitude of of the effect of type b that occurs when an effect of type a with magnitude 1 occurs. I am pretty satisfied with this intuition and it seems to help give me a better intuition for some of the basic theorems like for example gram Schmidt and why Orthogonal basis are so useful (since they break down an effect into "independent" effects).
My question is if I am missing something or is this a "correct" way to think about inner products?