Cross product

The cross product is defined by both direction and magnitude.

• If k = 0, then k × v = 0; otherwise write k = |k| k̂
• The direction k̂ determines the perpendicular plane k̂⊥ and the 90° rotation in that plane; the length |k| scales the result
• For unit k̂, [k̂]× projects onto k̂⊥, rotates by 90° in that plane and returns the result to ℝ³
• The matrix [k̂]× is skew-symmetric, and the three independent entries are the components of k̂
• The entries are derived three ways: from geometric action, by a Householder change of basis and by a Givens-rotation change of basis
• Householder reflection reverses orientation, so the middle 90° rotation carries a minus sign: [k̂]× = −H [e⃗₁]× H
• With an orientation-preserving Givens change of basis G, the formula becomes [k̂]× = Gᵀ [e⃗₁]× G
• The construction gives |k̂ × v| = |v_⊥| = |v| sin(θ), and [j]× = |j|[k̂]× for non-unit j

The chapter now emphasizes why different orthogonal changes of basis give the same final cross-product matrix, while orientation determines the sign of the 90° rotation.

For a nonzero first vector, write k = |k| k̂. The unit direction k̂ determines the plane k̂⊥ and the 90° rotation inside that plane. The matrix [k̂]× performs the unit construction, and multiplying by |k| gives the general matrix [k]× = |k|[k̂]×.

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A Householder reflection changes basis by reversing orientation, so the 90° rotation in the reflected plane turns the opposite way and the formula is [k̂]× = −H [e⃗₁]× H. The Givens construction uses an orthogonal matrix G with det(G) = 1, so it preserves orientation and the corresponding formula is [k̂]× = Gᵀ [e⃗₁]× G. Both derivations simplify to the same skew-symmetric cross-product matrix.

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