TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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The squared cross-product norm is the Gram determinant of two vectors

Statement

For $u, v \in R^{3}$ , $∥ u \times v ∥_{2}^{2} = \det G (u, v)$ , and this value is positive exactly when $u$ and $v$ are linearly independent.

Equivalently, $∥ u \times v ∥_{2}^{2} = ∥ u ∥_{2}^{2} ∥ v ∥_{2}^{2} - ⟨ u, v ⟩^{2} .$

Facts & Assumptions

Given: Vectors $u, v \in R^{3}$ .

[L1]

The cross product has the displayed coordinate formula, and it is bilinear, alternating, and orthogonal to its factors (The cross product in $R^{3}$ , The cross product is bilinear, alternating, and orthogonal to both factors).

[L2]

The two-vector Gram matrix has entries $⟨ u_{i}, u_{j} ⟩$ , and its determinant is positive exactly for a linearly independent pair and zero exactly for a dependent pair (The Gram matrix $G (v_{0}, \dots, v_{r - 1}) = (⟨ v_{i}, v_{j} ⟩)_{i, j < r}$ and Gram determinant, with empty value $1$ , A Gram determinant is nonnegative and is positive exactly when the vector list is linearly independent).

Proof

technique · direct

1.1

Expanding the three squared coordinates of $u \times v$ and collecting terms gives $∥ u \times v ∥_{2}^{2} = (u_{x}^{2} + u_{y}^{2} + u_{z}^{2}) (v_{x}^{2} + v_{y}^{2} + v_{z}^{2}) - (u_{x} v_{x} + u_{y} v_{y} + u_{z} v_{z})^{2}$ .

givenL1algebra

2.1

The right side of step 1.1 is $∥ u ∥_{2}^{2} ∥ v ∥_{2}^{2} - ⟨ u, v ⟩^{2} = \det (\begin{array}{cc} ⟨ u, u ⟩ & ⟨ u, v ⟩ \\ ⟨ v, u ⟩ & ⟨ v, v ⟩ \end{array})$ , which is $\det G (u, v)$ by [L2].

step 1.1L2algebra

3.1

The positivity and vanishing assertions follow from [L2], including the cases in which either vector is zero.

step 2.1L2∎

Depends on

Used by

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

M. E. Taylor, Introduction to Analysis in Several Variables, formulas 3.2.18-3.2.20 (standard reference, not scraped)
R. Sjamaar, Manifolds and Differential Forms, Theorem 8.4 (standard reference, not scraped)