Question about the group SO(3)

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
-1
down vote

favorite












How can I show that any element of SO(3) can be written in the form $ Z_Phi X_theta Z_Psi $ ?



Where, $$ Z_theta =
beginpmatrix
cos(theta) & -sin(theta) & 0 \
sin(theta) & cos(theta) & 0 \
0 & 0 & 1 \
endpmatrix
$$



$$ X_theta =
beginpmatrix
1 & 0 & 0 \
0 & cos(theta) & -sin(theta) \
0 & sin(theta) & cos(theta) \
endpmatrix
$$



Any hints/suggestions will be greatly appreciated ?




linear-algebra geometry lie-groups lie-algebras




share|cite|improve this question





















  • Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
    – José Carlos Santos
    2 days ago














up vote
-1
down vote

favorite












How can I show that any element of SO(3) can be written in the form $ Z_Phi X_theta Z_Psi $ ?



Where, $$ Z_theta =
beginpmatrix
cos(theta) & -sin(theta) & 0 \
sin(theta) & cos(theta) & 0 \
0 & 0 & 1 \
endpmatrix
$$



$$ X_theta =
beginpmatrix
1 & 0 & 0 \
0 & cos(theta) & -sin(theta) \
0 & sin(theta) & cos(theta) \
endpmatrix
$$



Any hints/suggestions will be greatly appreciated ?




linear-algebra geometry lie-groups lie-algebras




share|cite|improve this question





















  • Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
    – José Carlos Santos
    2 days ago












up vote
-1
down vote

favorite









up vote
-1
down vote

favorite











How can I show that any element of SO(3) can be written in the form $ Z_Phi X_theta Z_Psi $ ?



Where, $$ Z_theta =
beginpmatrix
cos(theta) & -sin(theta) & 0 \
sin(theta) & cos(theta) & 0 \
0 & 0 & 1 \
endpmatrix
$$



$$ X_theta =
beginpmatrix
1 & 0 & 0 \
0 & cos(theta) & -sin(theta) \
0 & sin(theta) & cos(theta) \
endpmatrix
$$



Any hints/suggestions will be greatly appreciated ?




linear-algebra geometry lie-groups lie-algebras




share|cite|improve this question













How can I show that any element of SO(3) can be written in the form $ Z_Phi X_theta Z_Psi $ ?



Where, $$ Z_theta =
beginpmatrix
cos(theta) & -sin(theta) & 0 \
sin(theta) & cos(theta) & 0 \
0 & 0 & 1 \
endpmatrix
$$



$$ X_theta =
beginpmatrix
1 & 0 & 0 \
0 & cos(theta) & -sin(theta) \
0 & sin(theta) & cos(theta) \
endpmatrix
$$



Any hints/suggestions will be greatly appreciated ?





linear-algebra geometry lie-groups lie-algebras





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited 2 days ago









md2perpe

5,5691821




5,5691821









asked 2 days ago









Algebraicwonder

1




1











  • Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
    – José Carlos Santos
    2 days ago
















  • Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
    – José Carlos Santos
    2 days ago















Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
– José Carlos Santos
2 days ago




Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
– José Carlos Santos
2 days ago










1 Answer
1






active

oldest

votes

















up vote
0
down vote













Look at any book on classical mechanics, I think this goes under the name of Euler's Theorem and the angles are called Euler Angles.



Basically the idea is the following imagine the sphere with some fixed z direction. Then if I did some $SO(3)$ transformation I'm going to map the sphere back to itself, so let's see what can possibly happen to the sphere. First of all, lets see where the z axis went to under this transformation, it can be anywhere on the sphere so you need two angles to tell me where on the sphere you rotated your z-axis to. Next, even if I know where the z-axis went I can still rotate the sphere keeping the z-axis fixed in its new place, so I need another angle to fix this degree of freedom.



These matrices are sort of carrying out that procedure






share|cite|improve this answer





















    Your Answer




    StackExchange.ifUsing("editor", function ()
    return StackExchange.using("mathjaxEditing", function ()
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    );
    );
    , "mathjax-editing");

    StackExchange.ready(function()
    var channelOptions =
    tags: "".split(" "),
    id: "69"
    ;
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function()
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled)
    StackExchange.using("snippets", function()
    createEditor();
    );

    else
    createEditor();

    );

    function createEditor()
    StackExchange.prepareEditor(
    heartbeatType: 'answer',
    convertImagesToLinks: true,
    noModals: false,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    );



    );








     

    draft saved


    draft discarded


















    StackExchange.ready(
    function ()
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2872205%2fquestion-about-the-group-so3%23new-answer', 'question_page');

    );

    Post as a guest

















    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    0
    down vote













    Look at any book on classical mechanics, I think this goes under the name of Euler's Theorem and the angles are called Euler Angles.



    Basically the idea is the following imagine the sphere with some fixed z direction. Then if I did some $SO(3)$ transformation I'm going to map the sphere back to itself, so let's see what can possibly happen to the sphere. First of all, lets see where the z axis went to under this transformation, it can be anywhere on the sphere so you need two angles to tell me where on the sphere you rotated your z-axis to. Next, even if I know where the z-axis went I can still rotate the sphere keeping the z-axis fixed in its new place, so I need another angle to fix this degree of freedom.



    These matrices are sort of carrying out that procedure






    share|cite|improve this answer

























      up vote
      0
      down vote













      Look at any book on classical mechanics, I think this goes under the name of Euler's Theorem and the angles are called Euler Angles.



      Basically the idea is the following imagine the sphere with some fixed z direction. Then if I did some $SO(3)$ transformation I'm going to map the sphere back to itself, so let's see what can possibly happen to the sphere. First of all, lets see where the z axis went to under this transformation, it can be anywhere on the sphere so you need two angles to tell me where on the sphere you rotated your z-axis to. Next, even if I know where the z-axis went I can still rotate the sphere keeping the z-axis fixed in its new place, so I need another angle to fix this degree of freedom.



      These matrices are sort of carrying out that procedure






      share|cite|improve this answer























        up vote
        0
        down vote










        up vote
        0
        down vote









        Look at any book on classical mechanics, I think this goes under the name of Euler's Theorem and the angles are called Euler Angles.



        Basically the idea is the following imagine the sphere with some fixed z direction. Then if I did some $SO(3)$ transformation I'm going to map the sphere back to itself, so let's see what can possibly happen to the sphere. First of all, lets see where the z axis went to under this transformation, it can be anywhere on the sphere so you need two angles to tell me where on the sphere you rotated your z-axis to. Next, even if I know where the z-axis went I can still rotate the sphere keeping the z-axis fixed in its new place, so I need another angle to fix this degree of freedom.



        These matrices are sort of carrying out that procedure






        share|cite|improve this answer













        Look at any book on classical mechanics, I think this goes under the name of Euler's Theorem and the angles are called Euler Angles.



        Basically the idea is the following imagine the sphere with some fixed z direction. Then if I did some $SO(3)$ transformation I'm going to map the sphere back to itself, so let's see what can possibly happen to the sphere. First of all, lets see where the z axis went to under this transformation, it can be anywhere on the sphere so you need two angles to tell me where on the sphere you rotated your z-axis to. Next, even if I know where the z-axis went I can still rotate the sphere keeping the z-axis fixed in its new place, so I need another angle to fix this degree of freedom.



        These matrices are sort of carrying out that procedure







        share|cite|improve this answer













        share|cite|improve this answer



        share|cite|improve this answer











        answered 2 days ago









        yankyl

        62




        62






















             

            draft saved


            draft discarded


























             


            draft saved


            draft discarded














            StackExchange.ready(
            function ()
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2872205%2fquestion-about-the-group-so3%23new-answer', 'question_page');

            );

            Post as a guest
































































            Comments

            Post a Comment

            Popular posts from this blog

            What is the equation of a 3D cone with generalised tilt?

            Image
            Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite What is the equation of a 3D cone with generalized tilt? I've noticed that in most equations given to represent a cone, there is no parameter which defines the tilt of the cone in 3D space and that most of them have their point at the origin $(0,0,0)$ - I was wondering if anyone could give me a more generalized cone equation for a cone in any position in 3D space 3d share | cite | improve this question edited Jul 25 '16 at 0:05 ervx 9,425 3 13 36 asked Jul 24 '16 at 15:38 Charlene 11 2 2 Welcome to Math.SE! Could you please explain a few things to help people give an answer useful to you: 1. What do you mean by "generalized tilt"? 2. Are you asking about a right circular cone? Does the angle at the vertex matter? 3. What form of answer (implicit, parametric...?) are you looking for? 4. If this is homework, what tools do you have available, an...
            Read more

            Color the edges and diagonals of a regular polygon

            Image
            Clash Royale CLAN TAG #URR8PPP up vote 5 down vote favorite 1 Here is the problem: For what $n$ is it possible to color the edges and diagonals of an $n$-side regular polygon with $dfracbinomn23$ colors, such that you use every color exactly three times and for every color the three segments (edges or diagonals) with that color form a triangle? Trivially $nequiv 0 text or 1 pmod3$. I can also prove that if the statement is true for $k$ than it is true for $3k$ as well. How to finish? Please help! Thanks combinatorial-geometry share | cite | improve this question edited Aug 6 at 21:57 alcana 118 4 asked Aug 6 at 21:15 Leo Gardner 356 11 Even assuming "polyhpn" in the title is a typo for polygon , it is unclear what you mean by "the edges and sides". Aren't the side of a regular polygon the same as the edges? A regular $n$-sided polygon has $n$ edges. – hardmath Aug 6 at 21:20 3 $n$ ne...
            Read more

            Relationship between determinant of matrix and determinant of adjoint?

            Image
            Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite We are studying adjoints in class, and I was curious if there is a relationship between the determinant of matrix A, and the determinant of the adjoint of matrix A? I assume there would be a relationship because finding the adjoint requires creating a cofactor matrix and then transposing it. linear-algebra determinant adjoint-operators share | cite | improve this question asked Nov 17 '17 at 17:32 CluelessCoder 32 5 For a square matrix $A$ of order $n$, we have $A(operatornameadj A)=(operatornameadj A)A=|A|I_n$ where $I_n$ is the identity matrix of order $n$. This should tell you about the determinant of the adjoint in terms of that of $A$ (use multiplicative property and other elementary properties of determinants). – Prasun Biswas Nov 17 '17 at 17:35 1 @CluelessCoder: see here: math.stackexchange.com/questions/516127/…...
            Read more