Expand the dot product and cross product in Cartesian components:

u \cdot (v \times w) = u_{1} v_{2} w_{3} - u_{1} v_{3} w_{2} + u_{2} v_{3} w_{1} - u_{2} v_{1} w_{3} + u_{3} v_{1} w_{2} - u_{3} v_{2} w_{1} = (u_{2} v_{3} - u_{3} v_{2}) w_{1} + (u_{3} v_{1} - u_{1} v_{3}) w_{2} + (u_{1} v_{2} - u_{2} v_{1}) w_{3} = (u \times v) \cdot w .

The two expressions are the same scalar triple product, or determinant with columns

u, v, w

u \cdot (v \times w) = (u \times v) \cdot w .

The equality also holds when this determinant is zero.

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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