rotation of spherical coordinate system











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If I rotate the coordinate system by spherical angles θ and ϕ, and the vector in the new system is v'=(x′,y′,z′), what is its coordinate (x,y,z) in the original coordinate system? Thank you in advance!










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    If I rotate the coordinate system by spherical angles θ and ϕ, and the vector in the new system is v'=(x′,y′,z′), what is its coordinate (x,y,z) in the original coordinate system? Thank you in advance!










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      If I rotate the coordinate system by spherical angles θ and ϕ, and the vector in the new system is v'=(x′,y′,z′), what is its coordinate (x,y,z) in the original coordinate system? Thank you in advance!










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      If I rotate the coordinate system by spherical angles θ and ϕ, and the vector in the new system is v'=(x′,y′,z′), what is its coordinate (x,y,z) in the original coordinate system? Thank you in advance!







      matrices coordinate-systems






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      asked Nov 19 at 3:42









      Alicia

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      82






















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          It's not immediately clear what is your coordinate system, and how you define rotations. Let's assume that you have first a rotation around $x$ axis by angle $theta$, then a rotation around $z$ axis by angle $phi$. You can then write $$textrm v'=R_z(phi)R_x(theta)textrm v$$
          Note the order of rotations in the above equation. You can find the matrix representation of this rotation on wikipedia for example.



          Then just multiplying with the inverses, in reverse order, you get
          $$R_x^{-1}(theta)R_z^{-1}(phi)textrm v'=R_x^{-1}(theta)R_z^{-1}(phi)R_z(phi)R_x(theta)textrm v=textrm v$$



          Note that the inverse of the simple rotation matrices can be written as the transpose or rotation by negative angle:
          $R^{-1}(alpha)=R^T(alpha)=R(-alpha)$






          share|cite|improve this answer





















          • Thank you so much for your reply! It helps! I just have one more question about the order of rotations. The new coordinate system is created as following: there is a point on the sphere with coordinate (θ, ϕ); The line passing through the origin and this point is my new z axis (the orientation of x and y axes don't matter). In this case, should I first rotate around x axis by angle θ, or rotate aroung z axis by angle ϕ?
            – Alicia
            Nov 19 at 15:58










          • You need to rotate first around your new $z$ axis by $-phi$. That way your $x'$ axis rotates first to $x$.
            – Andrei
            Nov 19 at 16:57










          • I see, thank you a lot!
            – Alicia
            Nov 19 at 17:25











          Your Answer





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          1 Answer
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          1 Answer
          1






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          active

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          up vote
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          down vote



          accepted










          It's not immediately clear what is your coordinate system, and how you define rotations. Let's assume that you have first a rotation around $x$ axis by angle $theta$, then a rotation around $z$ axis by angle $phi$. You can then write $$textrm v'=R_z(phi)R_x(theta)textrm v$$
          Note the order of rotations in the above equation. You can find the matrix representation of this rotation on wikipedia for example.



          Then just multiplying with the inverses, in reverse order, you get
          $$R_x^{-1}(theta)R_z^{-1}(phi)textrm v'=R_x^{-1}(theta)R_z^{-1}(phi)R_z(phi)R_x(theta)textrm v=textrm v$$



          Note that the inverse of the simple rotation matrices can be written as the transpose or rotation by negative angle:
          $R^{-1}(alpha)=R^T(alpha)=R(-alpha)$






          share|cite|improve this answer





















          • Thank you so much for your reply! It helps! I just have one more question about the order of rotations. The new coordinate system is created as following: there is a point on the sphere with coordinate (θ, ϕ); The line passing through the origin and this point is my new z axis (the orientation of x and y axes don't matter). In this case, should I first rotate around x axis by angle θ, or rotate aroung z axis by angle ϕ?
            – Alicia
            Nov 19 at 15:58










          • You need to rotate first around your new $z$ axis by $-phi$. That way your $x'$ axis rotates first to $x$.
            – Andrei
            Nov 19 at 16:57










          • I see, thank you a lot!
            – Alicia
            Nov 19 at 17:25















          up vote
          0
          down vote



          accepted










          It's not immediately clear what is your coordinate system, and how you define rotations. Let's assume that you have first a rotation around $x$ axis by angle $theta$, then a rotation around $z$ axis by angle $phi$. You can then write $$textrm v'=R_z(phi)R_x(theta)textrm v$$
          Note the order of rotations in the above equation. You can find the matrix representation of this rotation on wikipedia for example.



          Then just multiplying with the inverses, in reverse order, you get
          $$R_x^{-1}(theta)R_z^{-1}(phi)textrm v'=R_x^{-1}(theta)R_z^{-1}(phi)R_z(phi)R_x(theta)textrm v=textrm v$$



          Note that the inverse of the simple rotation matrices can be written as the transpose or rotation by negative angle:
          $R^{-1}(alpha)=R^T(alpha)=R(-alpha)$






          share|cite|improve this answer





















          • Thank you so much for your reply! It helps! I just have one more question about the order of rotations. The new coordinate system is created as following: there is a point on the sphere with coordinate (θ, ϕ); The line passing through the origin and this point is my new z axis (the orientation of x and y axes don't matter). In this case, should I first rotate around x axis by angle θ, or rotate aroung z axis by angle ϕ?
            – Alicia
            Nov 19 at 15:58










          • You need to rotate first around your new $z$ axis by $-phi$. That way your $x'$ axis rotates first to $x$.
            – Andrei
            Nov 19 at 16:57










          • I see, thank you a lot!
            – Alicia
            Nov 19 at 17:25













          up vote
          0
          down vote



          accepted







          up vote
          0
          down vote



          accepted






          It's not immediately clear what is your coordinate system, and how you define rotations. Let's assume that you have first a rotation around $x$ axis by angle $theta$, then a rotation around $z$ axis by angle $phi$. You can then write $$textrm v'=R_z(phi)R_x(theta)textrm v$$
          Note the order of rotations in the above equation. You can find the matrix representation of this rotation on wikipedia for example.



          Then just multiplying with the inverses, in reverse order, you get
          $$R_x^{-1}(theta)R_z^{-1}(phi)textrm v'=R_x^{-1}(theta)R_z^{-1}(phi)R_z(phi)R_x(theta)textrm v=textrm v$$



          Note that the inverse of the simple rotation matrices can be written as the transpose or rotation by negative angle:
          $R^{-1}(alpha)=R^T(alpha)=R(-alpha)$






          share|cite|improve this answer












          It's not immediately clear what is your coordinate system, and how you define rotations. Let's assume that you have first a rotation around $x$ axis by angle $theta$, then a rotation around $z$ axis by angle $phi$. You can then write $$textrm v'=R_z(phi)R_x(theta)textrm v$$
          Note the order of rotations in the above equation. You can find the matrix representation of this rotation on wikipedia for example.



          Then just multiplying with the inverses, in reverse order, you get
          $$R_x^{-1}(theta)R_z^{-1}(phi)textrm v'=R_x^{-1}(theta)R_z^{-1}(phi)R_z(phi)R_x(theta)textrm v=textrm v$$



          Note that the inverse of the simple rotation matrices can be written as the transpose or rotation by negative angle:
          $R^{-1}(alpha)=R^T(alpha)=R(-alpha)$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 19 at 4:34









          Andrei

          10.4k21025




          10.4k21025












          • Thank you so much for your reply! It helps! I just have one more question about the order of rotations. The new coordinate system is created as following: there is a point on the sphere with coordinate (θ, ϕ); The line passing through the origin and this point is my new z axis (the orientation of x and y axes don't matter). In this case, should I first rotate around x axis by angle θ, or rotate aroung z axis by angle ϕ?
            – Alicia
            Nov 19 at 15:58










          • You need to rotate first around your new $z$ axis by $-phi$. That way your $x'$ axis rotates first to $x$.
            – Andrei
            Nov 19 at 16:57










          • I see, thank you a lot!
            – Alicia
            Nov 19 at 17:25


















          • Thank you so much for your reply! It helps! I just have one more question about the order of rotations. The new coordinate system is created as following: there is a point on the sphere with coordinate (θ, ϕ); The line passing through the origin and this point is my new z axis (the orientation of x and y axes don't matter). In this case, should I first rotate around x axis by angle θ, or rotate aroung z axis by angle ϕ?
            – Alicia
            Nov 19 at 15:58










          • You need to rotate first around your new $z$ axis by $-phi$. That way your $x'$ axis rotates first to $x$.
            – Andrei
            Nov 19 at 16:57










          • I see, thank you a lot!
            – Alicia
            Nov 19 at 17:25
















          Thank you so much for your reply! It helps! I just have one more question about the order of rotations. The new coordinate system is created as following: there is a point on the sphere with coordinate (θ, ϕ); The line passing through the origin and this point is my new z axis (the orientation of x and y axes don't matter). In this case, should I first rotate around x axis by angle θ, or rotate aroung z axis by angle ϕ?
          – Alicia
          Nov 19 at 15:58




          Thank you so much for your reply! It helps! I just have one more question about the order of rotations. The new coordinate system is created as following: there is a point on the sphere with coordinate (θ, ϕ); The line passing through the origin and this point is my new z axis (the orientation of x and y axes don't matter). In this case, should I first rotate around x axis by angle θ, or rotate aroung z axis by angle ϕ?
          – Alicia
          Nov 19 at 15:58












          You need to rotate first around your new $z$ axis by $-phi$. That way your $x'$ axis rotates first to $x$.
          – Andrei
          Nov 19 at 16:57




          You need to rotate first around your new $z$ axis by $-phi$. That way your $x'$ axis rotates first to $x$.
          – Andrei
          Nov 19 at 16:57












          I see, thank you a lot!
          – Alicia
          Nov 19 at 17:25




          I see, thank you a lot!
          – Alicia
          Nov 19 at 17:25


















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