In this section you have the statement ⟨𝑥⃗ ⋅𝑦⃗ ⟩=|𝑥⃗ |⋅|𝑦⃗ |⋅𝑐𝑜𝑠𝛼. Should that really be ⟨𝑥⃗ ⋅𝑦⃗ ⟩=||𝑥⃗ ||⋅||𝑦⃗ ||⋅𝑐𝑜𝑠𝛼 ? Earlier in the chapter you had introduced the concept of ||𝑥⃗||, but then in this section you only seem to be using single vertical bars, not double. I think in this whole section you really intended to use double vertical bars, but wanted to be sure that this actually wasn’t something different, or somehow the absolute values (which is what I think of with single vertical bars).
If you did intend double vertical bars, perhaps someone could fix the multiple typos in this section so the notation is consistent.