It's not immediately clear to me why this definition satisfies the "geometric" interpretation mentioned earlier in the slides -- namely, that <u, v> represents the length of the projection of u onto v, or vice versa, at least when the vectors are unit length. Is there a nice way to see this correspondence?
Although the definition meets the specified rules, the geometric meaning behind this is kind of vague. Are there any specific reasons for defining the inner product like this?
The lecture has talk a lot about inner product, I heard that there is also a outer product, will we learn and use outer product in this course?