Why a finite angular displacement is not a vector?

In summary, the conversation discusses why angular displacement cannot be considered a vector due to the non-commutative nature of addition. The example of rotating the Earth is used to illustrate this concept, with the idea that even small rotations can lead to significant differences unless taken to the limit of infinitesimal rotations. The conversation also touches on the idea of vector spaces and how finite rotations of the sphere do not form a vector space. The final point is that reversing the order of rotations does not result in the same outcome, as the turn direction must also be reversed.
  • #1
gracy
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83
One of my friend has answered this question in this way.

Angular displacement can't be a vector because addition is not commutative. Say we are looking at the Earth with North America facing us and the North Pole facing up:if we rotate the Earth so that we move 90 degrees north, now the NP is facing us and NA faces down. If we the rotate the Earth to move 90 degrees east, we end up with NA facing down and the NP facing to the left. But, if we reverse the order, we end up facing the NP while NA faces to the the left.

For very small displacements, however, the surface of the Earth is basically flat, so "rotating" the Earth is simply moving in a given direction without swapping any axis. So if we are facing some point A on the equator, either way we do the rotations we end up facing a point just NE of A, with the NP facing up.
But I am not getting this.Please help me out.
 
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  • #2
Your "very small" rotations give rise to differences that are also very small, but they are there unless you go to the limit of infinitesimal rotations. If you do that, you essentially get the tangent vector spaces of the sphere. But finite rotations of the sphere do not form a vector space.
 
  • #3
gracy said:
Angular displacement can't be a vector because addition is not commutative. S

Vector addition is commutative. When you make several different rotations N- 90 and E 90 if reversing the order is E 90 and N 90 in that order that is not reversing the original rotations. You must us the opposite rotations in the reverse sequence i.e., west 90 and south 90 which will bring you to your initial orientation. Take a displacement on a plane use the sequence move x turn Rt move y turn Lt/ reversing the sequence will not bring you to your initial position without reversing the turn direction too. If B = A + ΔA then A ≠ B + ΔA but A = B - ΔA. - ΔA being the reverse of ΔA. Or have I missed the point?
 

1. Why can't a finite angular displacement be represented as a vector?

A vector is a mathematical quantity that has both magnitude and direction. However, a finite angular displacement only has magnitude (usually measured in radians or degrees) and does not have a specific direction. This is because it is a measure of the change in angle from one position to another, rather than a specific line or direction in space.

2. What is the difference between a finite angular displacement and a vector quantity?

A vector quantity has both magnitude and direction, while a finite angular displacement only has magnitude. Vectors can also be added or subtracted using mathematical operations, while angular displacements cannot be combined in the same way.

3. Can a finite angular displacement be converted into a vector?

No, a finite angular displacement cannot be converted into a vector because it is a different type of quantity. However, it can be represented graphically as an arc or a rotation, which can be useful for visualizing the change in angle.

4. Is there any situation where a finite angular displacement can be treated as a vector?

In some cases, a finite angular displacement can be approximated as a vector for certain calculations or simplifications. This is often seen in physics problems where the displacement is small and the direction is not important, such as in circular motion.

5. How is a finite angular displacement different from a vector displacement?

A vector displacement is a measure of the change in position from one point to another, while a finite angular displacement is a measure of the change in angle. They are fundamentally different quantities and cannot be equated or compared in the same way. Additionally, a vector displacement has both magnitude and direction, while a finite angular displacement only has magnitude.

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