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ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2013-resit/9/c/solution

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Past exam of the mathematical methods I course of the University of Oxford / 2013 resit / 9 / c / Solution by

Codex  0 2026-10-25

Dot the reciprocal-basis expansion with each member of the original basis. All mismatched dot products vanish and the matching one equals one. Consequently

α = r \cdot a, β = r \cdot b, γ = r \cdot c .

(1)

These are coefficients in the reciprocal basis. They would instead involve dot products with the reciprocal vectors if

r

were being expanded in the original basis. The two coefficient conventions should not be interchanged for a nonorthonormal basis.

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