LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The vector triple product identity $u \times (v \times w) = ⟨ u, w ⟩ v - ⟨ u, v ⟩ w$

Statement

For all $u, v, w \in R^{3}$ , $u \times (v \times w) = ⟨ u, w ⟩ v - ⟨ u, v ⟩ w .$

Facts & Assumptions

Given: Vectors $u, v, w \in R^{3}$ with coordinates $u = (u_{1}, u_{2}, u_{3})$ , $v = (v_{1}, v_{2}, v_{3})$ , $w = (w_{1}, w_{2}, w_{3})$ .

[L1]

The cross product is defined coordinatewise by $u \times v = (u_{2} v_{3} - u_{3} v_{2}, u_{3} v_{1} - u_{1} v_{3}, u_{1} v_{2} - u_{2} v_{1})$ (The cross product in $R^{3}$ ).

[L2]

The Euclidean inner product is $⟨ u, w ⟩ = \sum_{k = 1}^{3} u_{k} w_{k}$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[L3]

Finite sums may be expanded, reordered and combined (Laws of finite sums and finite products).

Proof

technique · direct

1.1L1L2L3algebra

First component. By [L1], $(u \times (v \times w))_{1} = u_{2} (v \times w)_{3} - u_{3} (v \times w)_{2} = u_{2} (v_{1} w_{2} - v_{2} w_{1}) - u_{3} (v_{3} w_{1} - v_{1} w_{3}),$ which expands to $v_{1} (u_{2} w_{2} + u_{3} w_{3}) - w_{1} (u_{2} v_{2} + u_{3} v_{3})$ ; since $u_{2} w_{2} + u_{3} w_{3} = ⟨ u, w ⟩ - u_{1} w_{1}$ and $u_{2} v_{2} + u_{3} v_{3} = ⟨ u, v ⟩ - u_{1} v_{1}$ by [L2], this equals $v_{1} (⟨ u, w ⟩ - u_{1} w_{1}) - w_{1} (⟨ u, v ⟩ - u_{1} v_{1}) = (⟨ u, w ⟩ v - ⟨ u, v ⟩ w)_{1}$ , which is the $1$ -component of the asserted identity.

2.1L1L2L3algebra

Second and third components. The same computation, with the coordinate labels cyclically relabelled and the components of the cross product read off in the corresponding cyclic order, gives $(u \times (v \times w))_{2} = (⟨ u, w ⟩ v - ⟨ u, v ⟩ w)_{2}$ and $(u \times (v \times w))_{3} = (⟨ u, w ⟩ v - ⟨ u, v ⟩ w)_{3}$ : each is the same finite expansion of [L1] as in step 1.1 with $(1, 2, 3)$ cyclically replaced, and [L2], [L3] are unaffected by the relabelling.

3.1step 1.1step 2.1L1algebra∎

Two vectors of $R^{3}$ with equal components are equal, so steps 1.1 and 2.1 prove the identity in all three coordinates.

The vector triple product identity $u \times (v \times w) = ⟨ u, w ⟩ v - ⟨ u, v ⟩ w$

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

The vector triple product identity u×(v×w)=⟨u,w⟩v−⟨u,v⟩wu×(v×w)=⟨u,w⟩v−⟨u,v⟩w

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

The vector triple product identity $u \times (v \times w) = ⟨ u, w ⟩ v - ⟨ u, v ⟩ w$