Homework SourseLet S2 R3 be the sphere of radius one Prove than SO3 acts tr
Let S2 R3 be the sphere of radius one.
Prove than SO(3) acts transitively on S2.
Solution
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ANSWER IS GIVEN BELOW
All rotations of S2R3 when written as linear maps T: R3R3 are elements of SO(3). Therefore it is enough to show that for any two points pp, qS2there is a rotation T with T(p)=q.
When p=q the identity will do. When p and q are antipodal points of S2 then a 180-rotation about any axis orthogonal to pq will do.
When p and q are linearly independent the line spanned by p×qis the axis of a rotation T with T(p)=q.
hence proved
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| down voteD | ANSWER IS GIVEN BELOW All rotations of S2R3 when written as linear maps T: R3R3 are elements of SO(3). Therefore it is enough to show that for any two points pp, qS2there is a rotation T with T(p)=q. When p=q the identity will do. When p and q are antipodal points of S2 then a 180-rotation about any axis orthogonal to pq will do. When p and q are linearly independent the line spanned by p×qis the axis of a rotation T with T(p)=q. hence proved |
