How does this vector formula work?

[keen fjord]

well they defined $P$ such that [ \vv{AP} = \frac{\lambda}{\lambda + \mu} \vv{AB} \qtext{and} \vv{PB} = \frac{\mu}{\lambda + \mu} ]
which ensures both that $AP/AB = \lambda/\mu$ and $\vv{AP} + \vv{PB} = \vv{AB}$