Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 1 / 5B / iii / Solution

Dotting $c={\sum }_j{\gamma }_j{a}_j$ with $e_i$ gives $e_i\cdot c={\gamma }_i{\lambda }_i$, proving ${\gamma }_i=(e_i\cdot c)/{\lambda }_i$. AlsoCombining these with $V=|\Delta |/6$ gives the displayed face formula, with the orientation chosen so its signed right-hand side is positive.