 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ia/paper-1/5b/a/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Let

Δ = A \cdot (B \times C) .

De lt a

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

\frac{B \times C}{Δ}, \frac{C \times A}{Δ}, \frac{A \times B}{Δ} .

Taking scalar products with

D

therefore yields

D = \frac{D \cdot ( B \times C )}{Δ} A + \frac{D \cdot ( C \times A )}{Δ} B + \frac{D \cdot ( A \times B )}{Δ} C .

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