Axial Vector: Small Angular Displacement

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SUMMARY

Angular displacement is classified as an axial vector only for small values due to the non-commutative nature of rotations. When combining two rotations, R1 and R2, the order of application affects the final result, which deviates from standard vector addition. For larger angular displacements, the representation of rotations cannot be uniquely defined as vectors, leading to ambiguity in their combination.

PREREQUISITES
  • Understanding of vector addition and properties
  • Familiarity with rotational mechanics
  • Knowledge of axial vectors and their characteristics
  • Basic concepts of angular displacement and rotation
NEXT STEPS
  • Research the mathematical representation of rotations in 3D space
  • Explore the implications of non-commutative operations in physics
  • Study the differences between axial vectors and polar vectors
  • Learn about quaternion representation of rotations for larger angles
USEFUL FOR

Physics students, mechanical engineers, and anyone interested in the mathematical foundations of rotational dynamics and vector analysis.

Sakshi Negi
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why is angular displacement considered an axial vector only for small values?
 
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Sakshi, It's because rotations don't add the way vectors are supposed to add. If you had one rotation R1 and followed it with another one: R2, that will give you a different result from rotating R2 and then following that with R1. (Try it.) But if you tried to represent rotations Ri with vectors Ri, the result would be the same in both cases: R1 + R2.
 
why is angular displacement considered an axial vector only for small values

Angular displacement can be many revolutions.

This cannot be represented uniquely by vector, axial or regular.

go well
 

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