1. The problem statement, all variables and given/known data This isn't really a problem so much as me not being able to see how a proof has proceeded. I've only just today learned about Dirac notation so I'm not too good at actually working with it. The proof in the book is: |Z> = |V> - <W|V>/|W|^2|W> <Z|Z> = <V - ( <W|V>/|W|^2 ) W| V - ( <W|V>/|W|^2 ) W> = <V|V> - ( <W|V><V|W>/|W|^2 ) - ( <W|V>*<W|V>/|W|^2 ) + ( <W|V>*<W|V><W|W>/|W|^4| ) >= 0 This last step has me completely lost, and if someone could explain it I'd really appreciate it. It looks like he's pulled out the <V| and |V> and combined them to get <V|V>, but I don't get where the minus sign comes from, or how he gets any of the rest of the equation, so I'm assuming I'm wrong about where the <V|V> comes from. I'm deeply sorry about the notation here, I typed this on a phone or I would've used LaTeX. 2. Relevant equations <Z|Z> >= 0 3. The attempt at a solution Mentioned in the first section.