3D rotation

A rotation around an axis keeps one component fixed and rotates the orthogonal component inside a plane. This chapter builds that structure step by step.

• The matrix [ k̂ ]× acts as a 90° rotation inside the plane orthogonal to k̂ and sends k̂ to 0
• The skew-symmetric form of [ k̂ ]× follows from those geometric actions and from the one-dimensional null space
• Rodrigues algorithm starts by decomposing v into the part along k̂ and the part in the orthogonal plane
• The orthogonal plane is represented by an orthonormal basis Q = [ q̂₁ | q̂₂ ], rotated in 2D and lifted back to ℝ³
• Expanding that sequence gives A = cosθ I + (1 − cosθ) k̂k̂ᵀ + sinθ [ k̂ ]×
• The rotation matrix keeps k̂ fixed and its remaining action is a 2D rotation inside the orthogonal plane

The chapter treats 3D rotation as a composition of projection, coordinate change, 2D rotation and reconstruction.

Because a rotation around k̂ does not act the same way on every part of a vector. The component along k̂ stays fixed, while the component in the plane orthogonal to k̂ is the part that actually rotates.

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One term keeps the component along the axis, one term rotates the orthogonal component without changing its length and one skew-symmetric term gives the quarter-turn direction inside the orthogonal plane. The formula is compact, but each term comes from a separate geometric action.

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