 TOPICS  # Rotation Formula A formula which transforms a given coordinate system by rotating it through a counterclockwise angle about an axis . Referring to the above figure (Goldstein 1980), the equation for the "fixed" vector in the transformed coordinate system (i.e., the above figure corresponds to an alias transformation), is   (1)   (2)   (3)

(Goldstein 1980; Varshalovich et al. 1988, p. 24). The angle and unit normal may also be expressed as Euler angles. In terms of the Euler parameters, (4)

The rotation matrix can be calculated in the Wolfram Language as follows:

```  With[{n = {nx, ny, nz}},
Cos[phi] IdentityMatrix + (1 - Cos[p]) Outer[Times, n, n]
+ Sin[p] {{0, n[], -n[]}, {-n[], 0, n[]}, {n[], -n[], 0}}
]```

Alias Transformation, Alibi Transformation, Euler Angles, Euler Parameters, Rodrigues' Rotation Formula

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## References

Gibbs, J. W. and Wilson, E. B. Vector Analysis: A Text-Book for the Use of Students of Mathematics and Physics, Founded upon the Lectures of J. Willard Gibbs. New York: Dover, p. 338, 1960.Goldstein, H. "Finite Rotations." §4-7 in Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, pp. 164-166, 1980.Grubin, C. "Derivation of the Quaternion Scheme via the Euler Axis and Angle." J. Spacecraft 7, 1251-1263, 1970.Hamel, G. Theoretische Mechanik: Eine Einheitliche Einführung in die Gesamte Mechanik. Berlin: New York: Springer-Verlag, p. 103, 1949.Varshalovich, D. A.; Moskalev, A. N.; and Khersonskii, V. K. "Description of Rotations in Terms of Rotation Axis and Rotation Angle." §1.4.2 in Quantum Theory of Angular Momentum. Singapore: World Scientific, pp. 23-24, 1988.

Rotation Formula

## Cite this as:

Weisstein, Eric W. "Rotation Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RotationFormula.html