Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 1 / 5B / a / iii / Solution

LetLinear independence implies that the scalar triple product $Delta$ is nonzero. Applying the vector triple-product formulas to $(A\times B)\times (C\times D)$ recovers the coefficients of $D$ in the basis $A,B,C$. Equivalently, the reciprocal basis to $A,B,C$ isTaking scalar products with $D$ therefore yields