 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2021-resit/9/iii/b/solution

Write

Δ = ⟨ u, v, w ⟩ = u \cdot (v \times w)

. The displayed definitions require

Δ \neq = 0

V \times W = \frac{( w \times u ) \times ( u \times v )}{Δ ^{2}} = \frac{u ( ( w \times u ) \cdot v ) - v ( ( w \times u ) \cdot u )}{Δ ^{2}} = \frac{u}{Δ} .

Δ

by cyclic permutation, and the second is zero because a cross product is orthogonal to both its factors. Thus

⟨ U, V, W ⟩ = \frac{( v \times w ) \cdot u}{Δ ^{2}} = \frac{1}{Δ} .

This is the reciprocal basis involution identity. When

Δ = 0

, the proposed reciprocal basis is undefined; for example, three vectors in the

x y

plane cannot be inserted into the printed quotients.

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