Editing Rodrigues' rotation formula (section)

==Statement==
If {{math|'''v'''}} is a vector in {{math|ℝ<sup>3</sup>}} and {{math|'''k'''}} is a [[unit vector]] describing an axis of rotation about which {{math|'''v'''}} rotates by an angle {{mvar|θ}} according to the [[Right hand rule#Direction associated with a rotation|right hand rule]], the Rodrigues formula for the rotated vector {{math|'''v'''<sub>rot</sub>}} is
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|equation =<math>\mathbf{v}_\mathrm{rot} = \mathbf{v} \cos\theta + (\mathbf{k} \times \mathbf{v})\sin\theta + \mathbf{k} ~(\mathbf{k} \cdot \mathbf{v}) (1 - \cos\theta)\,.</math>
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The intuition of the above formula is that the first term scales the vector down, while the second skews it (via [[Euclidean vector#Addition and subtraction|vector addition]]) toward the new rotational position. The third term re-adds the height (relative to <math>\textbf{k}</math>) that was lost by the first term.

An alternative statement is to write the axis vector as a [[cross product]] {{math|'''a''' × '''b'''}} of any two nonzero vectors {{math|'''a'''}} and {{math|'''b'''}} which define the plane of rotation, and the sense of the angle {{math|''θ''}} is measured away from {{math|'''a'''}} and towards {{math|'''b'''}}. Letting {{math|''α''}} denote the angle between these vectors, the two angles {{math|''θ''}} and {{math|''α''}} are not necessarily equal, but they are measured in the same sense. Then the unit axis vector can be written

:<math>\mathbf{k} = \frac{\mathbf{a}\times\mathbf{b}}{|\mathbf{a}\times\mathbf{b}|} = \frac{\mathbf{a}\times\mathbf{b}}{|\mathbf{a}||\mathbf{b}|\sin\alpha}\,. </math>

This form may be more useful when two vectors defining a plane are involved. An example in physics is the [[Thomas precession]] which includes the rotation given by Rodrigues' formula, in terms of two non-collinear boost velocities, and the axis of rotation is perpendicular to their plane.