Let be an arbitrary fixed unit vector and show that any vector satisfies

Explain this result in words, with the help of a picture.

So I got to the point where I can see the end in sight:

Obviously the 's will all go away since u is a unit vector. My only problem is that I have some leftover terms from the dotting of the cross products which I don't see how to get rid of since u and b are so generalized. Since the first part seems to be right, I have to assume that I either did something wrong in the cross product arithmetic or else I'm missing some property of dot products that will make that last term go away. Any thoughts?