Relationship between translation and rotation

[kolua]

Homework Statement

Prove or disprove: Every translation is a product of two non-involutory rotations.

Homework Equations

The Attempt at a Solution

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I am not sure if I got the right proof for the special situation: A translation is the product of two reflections with parallel reflections lines. And the reflections lines can be written as the product of two non-involutory rotations of 90 degrees, which means that the translation is a product of two non-involutory rotations under this specific condition.

I don't know how to deal with the general conclusion. What happens when the two rotations have different centers? How should I prove the statement for a general case?

 

[fresh_42]

Do you consider rotations only with the origin as fixed point or around any axis? Do translations have fixed points?

 

[kolua]

Do you consider rotations only with the origin as fixed point or around any axis? Do translations have fixed points?

no, i don't think there fixed pionts should be taken into consideration. should they?

 

[fresh_42]

A rotation is usually a linear map with $0$ as center. A translation maps this point to another one, so how can they be related?

 

[kolua]

A rotation is usually a linear map with $0$ as center. A translation maps this point to another one, so how can they be related?

by reflection?

 

[fresh_42]

There is a connection between reflections and rotations, yes, but a translation moves everything, a rotation has a fixed center point and a reflection even a fixed axis. Have you tried to draw the different mappings? You should do it.

 

[kolua]

There is a connection between reflections and rotations, yes, but a translation moves everything, a rotation has a fixed center point and a reflection even a fixed axis. Have you tried to draw the different mappings? You should do it.

so there is no such relationship and I should disprove statement?

 

[fresh_42]

Yes. Take any point and see what a translation does and where you can get by rotations.