Introduction to Electrodynamics · Problem 1.9

Vector Analysis — David J. Griffiths

Physics · how-vectors-transform

Problem

Find the transformation matrix RR that describes a rotation by 120∘120^\circ about an axis from the origin through the point (1,1,1)(1, 1, 1). The rotation is clockwise as you look down the axis toward the origin.
Intuitive explanation

Rotations in 3D are easiest when you align your axis with a coordinate direction. The key insight is to use Rodrigues' rotation formula, which lets you build a rotation matrix directly from an axis direction and angle—no need to change coordinates. First, normalize your axis vector, then apply the formula: R=I+sin⁡(θ)[u]×+(1−cos⁡θ)[u]×2R = I + \sin(\theta)[u]_\times + (1-\cos\theta)[u]_\times^2, where [u]×[u]_\times is the skew-symmetric cross-product matrix. This gives you the full rotation in the original frame.

Intuitive explanation

Rotations in 3D are easiest when you align your axis with a coordinate direction. The key insight is to use Rodrigues' rotation formula, which lets you build a rotation matrix directly from an axis direction and angle—no need to change coordinates. First, normalize your axis vector, then apply the formula: R=I+sin⁡(θ)[u]×+(1−cos⁡θ)[u]×2R = I + \sin(\theta)[u]_\times + (1-\cos\theta)[u]_\times^2, where [u]×[u]_\times is the skew-symmetric cross-product matrix. This gives you the full rotation in the original frame.

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