cross_cross_eq_smul_sub_smul — Mathlib

English

The triple product equals the determinant of the matrix with the input vectors as rows; equivalently, the cross product appears in the expansion of det.

Русский

Тройное произведение равно детерминанту матрицы с входами как строки; эквивалентно появлению cross в разложении det.

LaTeX

$$$u \\cdot v \\times w = \\det [u; v; w]$$$

Lean4

/-- The scalar triple product expansion of the vector triple product. -/
theorem cross_cross_eq_smul_sub_smul (u v w : Fin 3 → R) : u ⨯₃ v ⨯₃ w = (u ⬝ᵥ w) • v - (v ⬝ᵥ w) • u :=
  by
  simp_rw [cross_apply, vec3_dotProduct]
  ext i
  fin_cases i <;>
    · simp only [Fin.isValue, Nat.succ_eq_add_one, Nat.reduceAdd, Fin.reduceFinMk, cons_val, Pi.sub_apply,
        Pi.smul_apply, smul_eq_mul]
      ring

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