ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-16

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.

Projection onto a plane in $R^{3}$ and the nearest-point calculation

Example

Let $W = {(x, y, z) \in R^{3} : x + y + z = 0}$ and $v = (1, 2, 3)$ . Then

$P_{W} v = (- 1, 0, 1),$

and for every $q \in W$ ,

$∥ v - q ∥^{2} = 12 + ∥ (- 1, 0, 1) - q ∥^{2} .$

Thus $(- 1, 0, 1)$ is the unique nearest point of $W$ to $v$ .

Facts & Assumptions

Given: The displayed plane $W$ and vector $v$ .

[L1]

An orthonormal basis $(u_{i})$ of $W$ gives $P_{W} v = \sum_{i} ⟨ v, u_{i} ⟩ u_{i}$ (Orthogonal projection is linear, and an orthonormal basis $(e_{i})$ of $W$ gives $P_{W} v = \sum_{i} ⟨ v, e_{i} ⟩ e_{i}$ ).

[L2]

Orthogonal projection is the unique nearest point in the subspace (The orthogonal projection is the unique nearest point in the subspace).

[L3]

The standard real coordinate inner product is the dot product (The standard formulas $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and $\sum_{k < n} x_{k} \overline{y_{k}}$ on $C^{n}$ are inner products).

Verification

technique · computation

1.1L3algebra

The vectors $u_{0} = (1, - 1, 0) / \sqrt{2}$ and $u_{1} = (1, 1, - 2) / \sqrt{6}$ form an orthonormal basis of $W$ by [L3]. Their coefficients against $v$ are $- 1 / \sqrt{2}$ and $- 3 / \sqrt{6}$ .

2.1step 1.1L1L3

Substitution in [L1] gives $P_{W} v = (- 1, 0, 1)$ . The residual is $(2, 2, 2)$ , which is orthogonal to $W$ and has squared norm $12$ .

3.1step 2.1L2∎

For $q \in W$ , the residual and $P_{W} v - q$ are orthogonal, so Pythagoras gives the displayed equality. It is minimised uniquely at $q = P_{W} v$ , in agreement with [L2].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Projection onto a plane in R3R3 and the nearest-point calculation