Doesn't the dot product in Figure 16 on p. 44 equal a(b cos theta) + b(a cos theta), i.e. isn't the dot product equal to the sum of the dot products in each direction?
No, I think that Fig.16 has you confused -- it really has too much going on. There is only one dot product but Fig.16 tries to discuss two equivalent ways to calculate it. See the proceeding posting for a somewhat simpler diagram.