Cross product

The cross product is defined by both direction and magnitude.

• k × v is orthogonal to both input vectors
• When the inputs are independent, the direction is chosen so that (k × v, k, v) is a positively oriented triple
• The length is |k × v| = |k| |v| sin(θ), the area of the parallelogram spanned by the inputs
• For a unit vector k̂, the matrix [k̂]_× removes the component parallel to k̂, rotates the remaining plane component by 90° and returns the result to ℝ³
• The cross product matrix is skew-symmetric and has rank 2 when k̂ ≠ 0
• The usual determinant mnemonic is a symbolic cofactor expansion, not an ordinary determinant of a numerical matrix

Several steps in the derivation are special to ℝ³: a plane has a single normal direction and a 3×3 skew-symmetric matrix corresponds to one vector.

It first ignores the part of v parallel to k̂. The remaining perpendicular component lies in the plane orthogonal to k̂. In that plane, [k̂]_× acts like a 90° rotation, then re-embeds the rotated vector in ℝ³. In matrix form this is written as [k̂]_× = QJQᵀ.

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For unit k̂, the cross product matrix only acts on the component of v perpendicular to k̂ and rotates that component without changing its length. Therefore |k̂ × v| = |v_⊥| = |v| sin(θ). For a non-unit vector j, the result is scaled by |j|.

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