I understand that the inner product measures how well vectors "line up", but how are we defining what magnitudes constitute lining up fairly well versus being quite different? On the next slide, the two functions are obviously quite different and yield an inner product of only 1/30, but would the inner product of 7 on this slide be considered lining up fairly similarly or not?
Please correct me if I'm wrong.
When we calculate <u,u> we get the square of the length (|u|^2). So the <u,v> could be something similar. If u and v totally line up, <u,v>=|u||v|. Thus, how close the value of <u,v> to |u||v| could measure how well u and v line up. In this particular example, |u|*|v|=sqrt(170), which is around 13. Since <u,v> is much smaller than 13, u and v are not well lined up here.
@afm Right. Maybe one thing to consider is normalization: for instance, the inner product of two unit vectors is never greater than one; does that mean they "line up" worse than, say, two vectors of length 1000 that point in very different directions? Perhaps the right thing to do here is to divide by length. Of course, now you're headed toward the relationship between the inner product and angle: if theta is the angle between two (non-unit) vectors u and v, then <u,v> = |u||v|cos(theta), or theta = arccos(<u/|u|,v/|v|>).
@yingxiul Yes, I also like this way of looking at it. You could measure how well you're doing relative to the best you could possibly do.