# I Controversy about the nature of finite angular displacement

1. Sep 7, 2017

### ovais

Hello all,

I am having hard time to know if the finite angular displacement really a scalar quantity?

In some books they say angular displacement when finite is Scalar and when infinitesimal small is Vector, with direction perpendicular to plane of circle government by right hand rule.

I tried to look for the explanation of the statement of the book over the Internet the explanation given in various blogs or sites is something I fail to understand. They say finite angular displacement doesn't obey the commutation law of vector addition that's why finite angular displacement is not a vector but treated as scalar.

Their attempt to make their point, couldn't explain me how finite angular displacement doesn't obey commutation law of vector addition( I think o should accept that if a quantity does obey commutation law of vector addition we shouldn't treat that quantity as a vector though I didn't find such direct statement in books ).

My questions are two:

1. Is it true that finite angular displacement not a vector in reality?

2. If answer to the 1st question is yes; how can this be explained in a logical and simplified way?

Regards:)

2. Sep 7, 2017

### Staff: Mentor

1. Hold a book in front of you in your left hand, with its cover facing towards you and the text in normal upright orientation.

2. Hold your right hand with its thumb upwards and curl its fingers. Rotate the book around a vertical axis by 90 degrees in the direction shown by the fingers.

3. Now hold your right hand with its thumb towards you and curl its fingers. Starting from where you left it in the preceding step, rotate the book around a horizontal axis by 90 degrees in the direction shown by the fingers.

4. Return the book to its original orientation, facing you. Perform the above two steps in the reverse order, that is:

5. Hold your right hand with its thumb towards you and curl its fingers. Rotate the book around a horizontal axis by 90 degrees in the direction shown by the fingers.

6. Now hold your right hand with its thumb upwards and curl its fingers. Starting from where you left it in the preceding step, rotate the book around a vertical axis by 90 degrees in the direction shown by the fingers.

The book should be oriented in different directions after steps 3 and 6. It makes a difference which order you do the rotations. Therefore the rotations do not commute.

3. Sep 7, 2017

### FactChecker

The rotation operations are not commutative.
1) Take a book lying flat on a table as though you are going to read it.
Rotate it clockwise 90 degrees on the table and then rotate the far side (the bound side) 90 degrees up. That puts the book resting on it's unbound side facing you.

2) Start again with the book flat on the table in reading position.
Rotate the far side (the top) 90 degrees up and then rotate it clockwise 90 degrees. That puts it with the bottom of the pages resting on the table, with its unbound side facing you and the front facing the left.

The two are not equal, so rotation operations are not commutative.
Without the addition of rotations being commutative, they can not be considered a vector space.

4. Sep 7, 2017

### ovais

Thanks both of you, very well explained.

Though this has been explained very well I just need endorsement about the statement, that finite angular displacement is Scalar and infinitesimal small displacement is vector? Can I say it to my students or does this is a valid statement amongst the general physicists?

Regards

Last edited: Sep 7, 2017
5. Sep 7, 2017

### FactChecker

I wouldn't feel comfortable saying that "finite angular displacement is Scalar" unless it was specified that the rotations are around a fixed axis.

6. Sep 7, 2017

### ovais

Yeah that's where I need endorsement I mean I now agree that finite angular displacement doesn't obey commutation law I still don't feel confident in saying that, by not obeying commutation law by the finite displacements they are kept out of the category of vectors.

7. Sep 7, 2017

8. Sep 7, 2017

### jbriggs444

At the risk of stating the obvious, just because they are not vectors, that does not make them scalars.

Scalars are required to form an algebraic field.

9. Sep 7, 2017

### ovais

So should I say that finite angular displacements are neither scalars nor vectors(here risk is that my students will ask in what category are they fall and questions like what is the name of that category etc fall upon me )

Or should I say it is controversial matter just remember that they do not follow vector commutation law but posses direction. You may suggest me some ways to deal with this. Any help will be highly appreciated.

Regards

10. Sep 7, 2017

### jbriggs444

Right, they are in another category altogether.

It is perhaps unfortunate that we describe vectors as "something that has both magnitude and direction" and that we live in a three dimensional world. In three dimensions, it just so happens that there is an easy mapping between rotations and directions, i.e. the "right hand rule". You curl your fingers in the direction of a rotation and you harvest a corresponding direction. The same thing does not work in spaces that are not three dimensional.

I think you just have to suck it up and say that yes, rotations have magnitude and can be assigned a direction or written down as ordered triples (e.g. roll, pitch and yaw). But they do not add like vectors, so they are not actually "vectors".

But then I'm not a teacher and do not know what explanations will resonate best with students.

11. Sep 7, 2017

### Khashishi

As an aside, the space of velocities in special relativity is also not a vector space. That's because velocities don't add like vectors but rather add more like rotations. (Actually, velocity can be thought of as an orientation in hyperbolic space, and acceleration is a hyperbolic rotation.)

12. Sep 7, 2017

### FactChecker

I think it will be enough for you to point out why they are not vectors. You can describe rotations as operations on vectors. Rotations in three dimensions have been a big problem. Unless you want to talk about rotation matrices or operations in quaternians or in a Clifford algebra, your answer will have to remain incomplete.

13. Sep 10, 2017

### ovais

Thank you all :)

14. Sep 10, 2017

### forcefield

Yes (as already pointed out by others). I think the confusion arises from the fact that in reality infinitesimal angular displacement is not a vector either. It is however defined that way to come up with the definitions of torque and angular momentum.

15. Sep 18, 2017

### ovais

What is "that way" I would like if you write it in more open way instead of saying"that way", I mean what will you say how we treat infinitesimal small displacement to come up with definitions of torque and angular momentum as told by you?

Regards

16. Sep 18, 2017

### forcefield

I meant as a vector as you said in your OP.

My information is from http://feynmanlectures.caltech.edu/I_18.html#Ch18-F1 and the text after the figure.