3-D  Rotational Matrix

 

               
    ROTATION AROUND Z-axis in X-Y plane, from positive x to y          
    Angle α (in Degree ) Rz:        
    matrix X1(anti-clockwise)        
    x11 x12 x13
    x21 x22 x23
    x31 x32 x33
    matrix X2(clockwise)        
    ax11 ax12 ax13
    ax21 ax22 ax23
    ax31 ax32 ax33
    ( for reflection)          
    ROTATION AROUND X-axis in Y-Z plane, from positive y to z          
    Angle β(in Degree )Rx :        
    matrix Y1(anti-clockwise)        
    y11 y12 y13
    y21 y22 y23
    y31 y32 y33
    matrix Y2(clockwise)        
    ay11 ay12 ay13
    ay21 ay22 ay23
    ay31 ay32 ay33
               
    ROTATION AROUND Y-axis in Z-X plane, from positive z to x          
    Angle  γ (in Degree )Ry :        
    matrix Z1(anti-clockwise)        
    z11 z12 z13
    z21 z22 z23
    z31 z32 z33
    matrix Z2(clockwise)        
    az11 az12 az13
    az21 az22 az23
    az31 az32 az33
               
    General 3x3 matrix            
   

(for reflecion)

         
    (Put Value of A either x1.x2,y1,y2,z1,z2 or 0. If 0 is put , write down the value of matrix.)A        
    a11 a12 a13
    a21 a22 a23
    a31 a32 a33
    det A        
    (Put Value of B either x1.x2,y1,y2,z1,z2 or 0. If 0 is put , write down the value of matrix)B        
    b11 b12 b13
    b21 b22 b23
    b31 b32 b33
    det B        
    (Put Value of C either x1.x2,y1,y2,z1,z2 or 0.If 0 is put , write down the value of matrix )C        
    c11 c12 c13
    c21 c22 c23
    c31 c32 c33
    det C

       
   

       ABC

         
    d11 d12 d13
    d21 d22 d23
    d31 d32 d33
    Det ABC        
    Trace ABC (d11+d22+d33)        
    Σii≠jjdii*djj(d11*d22+d22*d33+d33*d11)        
    Σi≠jdij*dji(d12*d21+d23*d32+d13*d31)        
    Point P represented by vector before rotation of axis  i  j k
    Point P' represented by vector after rotation of axis  i  j k
               
    Point Q represented by vector before rotation of axis  i 

j

k

    Point Q' represented by vector after rotation of axis  i 

j

k

    Length of vector OP        
    Length of Vector OP'        
    Length of vector OQ        
    Length of Vector OQ'        
    Angle between OP & OQ in degree        
    Angle between OP' & OQ' in degree        
    Unit vector along OP  i j k
    Unit vector along OP'  i j k
    D1=(d11)2+(d21)2+(d31)2 norm    
    D2=(d12)2+(d22)2+(d32)2 norm    
    D3=(d13)2+(d23)2+(d33)2 norm    
    Euclidian distance between D1 & D2:d(D1,D2)        
    Euclidian distance between D2 & D3:d(D2,D3)        
    Euclidian distance between D1 & D3:d(D1,D3)        
    Dot Product of the column vectors column-1 & 2 of matrix ABC :d11*d12+d21*d22+d31*d32

       
    Dot Product of the column vectors column-2 & 3 of matrix ABC :d12*d13+d22*d23+d32*d33        
    Dot Product of the column vectors column-1 & 3 of matrix ABC :d11*d13+d21*d23+d31*d33        
             
             
             
   

the following 48 3-D rotational matrices can be tried:- (x1y1z1,x1y1z2,x1y2z1,x1y2z2),(x2y1z1,x2y1z2,x2y2z1,x2y2z2)

                                                                                       (y1x1z1,y1x1z2,y1x2z1,y1x2z2),(y2x1z1,y2x1z2,y2x2z1,y2x2z2)

                                                                                       (z1x1y1,z1x1y2,z1x2y1,z1x2y2),(z2x1y1,z2x1y2,z2x2y1,z2x2y2)

                                                                                       (z1y1x1,z1y1x2,z1y2x1,z1y2x2),(z2y1x1,z2y1x2,z2y2x1,z2y2x2)

                                                                                       (y1z1x1,y1z1x2,y1z2x1,y1z2x2),(y2z1x1,y2z1x2,y2z2x1,y2z2x2)

                                                                                       (x1z1y1,x1z1y2,x1z2y1,x1z2y2),(x2z1y1,x2z1y2,x2z2y1,x2z2y2)

         
   

How to know that there are 48 rotational matrices  ? The following example shall illustrate the thing. Suppose there is 1 coin with 2 numbers. How many possible outcomes due to a toss ? It is 21 * 1! =2.

Suppose there are 2 coins with 4 different numbers. How many possible outcomes due to a toss ? It is 22 * 2! =8

Suppose there are 3 coins with 6 different numbers. How many possible outcomes due to a toss ? It is 23 * 3! =48

Suppose there are 4 coins with 8 different numbers. How many possible outcomes due to a toss ? It is 24 * 4! =384

         
   

el1(sinα):   el12(sinαsinβ)  :     el15(sinαcosβ)  :  el4(23)=(cosαsinβsinγ):

         
    el2(sinβ):   el23(sinβsinγ)  :     el26(sinβcosγ)  :  el5(13)=(sinαcosβsinγ):          
    el3(sinγ):   el13(sinαsinγ)  :     el34(cosαsinγ)  :  el6(12)=(sinαsinβcosγ):          
   

el4(cosα):  el45(cosαcosβ) :    el24(cosαsinβ)  :  el1(56)=(sinαcosβcosγ):

         
    el5(cosβ):  el56(cosβcosγ) :    el35(cosβsinγ)  :  el2(46)=(cosαsinβcosγ):          
    el6(cosγ):  el46(cosαcosγ) :    el16(sinαcosγ)  :  el3(45)=(cosαcosβsinγ):          
                                       --------                                    ---------       el(12)3=(sinαsinβsinγ ):          
                                       ---------                                   ---------      el(45)6=(cosαcosβcosγ):          
    no. -6                                                    no.-12                                                            no.-8     TOTAL COMBINATION: 26          
               
    (matrix-1)aR11 aR12 aR13
    aR21 aR22 aR23
    aR31 aR32 aR33
    trace det    
    Σi=1,2,3(aRij)2    
               
    (matrix-10)ar11 ar12 ar13
    ar21 ar22 ar23
    ar31 ar32 ar33
    trace det    
    Σi=1,2,3(arij)2    
               
    (matrix-2)bR11 bR12 bR13
    bR21 bR22 bR23
    bR31 bR32 bR33
    trace det    
               
    (matrix-9)br11 br12 br13
    br21 br22 br23
    br31 br32 br33
    trace det    
               
    (matrix-3)cR11 cR12 cR13
    cR21 cR22 cR23
    cR31 cR32 cR33
    trace det    
               
    (matrix-8)cr11 cr12 cr13
    cr21 cr22 cr23
    cr31 cr32 cr33
    trace det    
               
    (matrix-4)dR11 dR12 dR13
    dR21 dR22 dR23
    dR31 dR32 dR33
    trace det    
               
    (matrix-7)dr11 dr12 dr13
    dr21 dr22 dr23
    dr31 dr32 dr33
    trace det    
               
    (matrix-5)eR11 eR12 eR13
    eR21 eR22 eR23
    eR31 eR32 eR33
    trace det    
               
    (matrix-12)er11 er12 er13
    er21 er22 er23
    er31 er32 er33
    trace det    
               
    (matrix-6)fR11 fR12 fR13
    fR21 fR22 fR23
    fR31 fR32 fR33
    trace det    
               
    (matrix-11)fr11 fr12 fr13
    fr21 fr22 fr23
    fr31 fr32 fr33
    trace det    
               
    (matrix-13)gR11 gR12 gR13
    gR21 gR22 gR23
    gR31 gR32 gR33
    trace det    
               
    (matrix-16)gr11 gr12 gr13
    gr21 gr22 gr23
    gr31 gr32 gr33
    trace det    
               
    (matrix-14)hR11 hR12 hR13
    hR21 hR22 hR23
    hR31 hR32 hR33
    trace det    
               
    (matrix-15)hr11 hr12 hr13
    hr21 hr22 hr23
    hr31 hr32 hr33
    trace det    
               
    (matrix-17)iR11 iR12 iR13
    iR21 iR22 iR23
    iR31 iR32 iR33
    trace det    
               
    (matrix-20)ir11 ir12 ir13
    ir21 ir22 ir23
    ir31 ir32 ir33
    trace det    
               
    (matrix-18)jR11 jR12 jR13
    jR21 jR22 jR23
    jR31 jR32 jR33
    trace det    
               
    (matrix-19)jr11 jr12 jr13
    jr21 jr22 jr23
    jr31 jr32 jr33
    trace det    
               
    (matrix-21)kR11 kR12 kR13
    kR21 kR22 kR23
    kR31 kR32 kR33
    trace det    
               
    (matrix-23)kr11 kr12 kr13
    kr21 kr22 kr23
    kr31 kr32 kr33
    trace det    
               
    (matrix-22)lR11 lR12 lR13
    lR21 lR22 lR23
    lR31 lR32 lR33
    trace det    
               
    (matrix-24)lr11 lr12 lr13
    lr21 lr22 lr23
    lr31 lr32 lr33
    trace det    
               
    (matrix-25)mR11 mR12 mR13
    mR21 mR22 mR23
    mR31 mR32 mR33
    trace det    
               
    (matrix-26)mr11 mr12 mr13
    mr21 mr22 mr23
    mr31 mr32 mr33
    trace det    
               
    (matrix-27)nR11 nR12 nR13
    nR21 nR22 nR23
    nR31 nR32 nR33
    trace det    
               
    (matrix-28)nr11 nr12 nr13
    nr21 nr22 nr23
    nr31 nr32 nr33
    trace det    
               
    (matrix-29)oR11 oR12 oR13
    oR21 oR22 oR23
    oR31 oR32 oR33
    trace det    
               
    (matrix-30)or11 or12 or13
    or21 or22 or23
    or31 or32 or33
    trace det    
               
    (matrix-31)pR11 pR12 pR13
    pR21 pR22 pR23
    pR31 pR32 pR33
    trace det    
               
    (matrix-32)pr11 pr12 pr13
    pr21 pr22 pr23
    pr31 pr32 pr33
    trace det    
               
    (matrix-33)qR11 qR12 qR13
    qR21 qR22 qR23
    qR31 qR32 qR33
    trace det    
               
    (matrix-34)qr11 qr12 qr13
    qr21 qr22 qr23
    qr31 qr32 qr33
    trace det    
               
    (matrix-35)rR11 rR12 rR13
    rR21 rR22 rR23
    rR31 rR32 rR33
    trace det    
               
    (matrix-36)rr11 rr12 rr13
    rr21 rr22 rr23
    rr31 rr32 rr33
    trace det    
               
    (matrix-37)sR11 sR12 sR13
    sR21 sR22 sR23
    sR31 sR32 sR33
    trace det    
               
    (matrix-38)sr11 sr12 sr13
    sr21 sr22 sr23
    sr31 sr32 sr33
    trace det    
               
    (matrix-39)tR11 tR12 tR13
    tR21 tR22 tR23
    tR31 tR32 tR33
    trace det    
               
    (matrix-40)tr11 tr12 tr13
    tr21 tr22 tr23
    tr31 tr32 tr33
    trace det    
               
    (matrix-41)uR11 uR12 uR13
    uR21 uR22 uR23
    uR31 uR32 uR33
    trace det    
               
    (matrix-42)ur11 ur12 ur13
    ur21 ur22 ur23
    ur31 ur32 ur33
    trace det    
               
    (matrix-43)vR11 vR12 vR13
    vR21 vR22 vR23
    vR31 vR32 vR33
    trace det    
               
    (matrix-44)vr11 vr12 vr13
    vr21 vr22 vr23
    vr31 vr32 vr33
    trace det    
               
    (matrix-45)wR11 wR12 wR13
    wR21 wR22 wR23
    wR31 wR32 wR33
    trace det    
               
    (matrix-46)wr11 wr12 wr13
    wr21 wr22 wr23
    wr31 wr32 wr33
    trace det    
               
    (matrix-47)xR11 xR12 xR13
    xR21 xR22 xR23
    xR31 xR32 xR33
    trace det    
               
    (matrix-48)xr11 xr12 xr13
    xr21 xr22 xr23
    xr31 xr32 xr33
    trace det    
               
    (matrix-1a)raR11 raR12 raR13
    raR21 raR22 raR23
    raR31 raR32 raR33
    trace det    
               
    (matrix-10a)rar11 rar12 rar13
    rar21 rar22 rar23
    rar31 rar32 rar33
    trace det    
               
    (matrix-2a)rbR11 rbR12 rbR13
    rbR21 rbR22 rbR23
    rbR31 rbR32 rbR33
    trace det    
               
    (matrix-9a)rbr11 rbr12 rbr13
    rbr21 rbr22 rbr23
    rbr31 rbr32 rbr33
    trace det    
               
    (matrix-3a)rcR11 rcR12 rcR13
    rcR21 rcR22 rcR23
    rcR31 rcR32 rcR33
    trace det    
               
    (matrix-8a)rcr11 rcr12 rcr13
    rcr21 rcr22 rcr23
    rcr31 rcr32 rcr33
    trace det    
               
    (matrix-4a)rdR11 rdR12 rdR13
    rdR21 rdR22 rdR23
    rdR31 rdR32 rdR33
    trace det    
               
    (matrix-7a)rdr11 rdr12 rdr13
    rdr21 rdr22 rdr23
    rdr31 rdr32 rdr33
    trace det    
               
    (matrix-5a)reR11 reR12 reR13
    reR21 reR22 reR23
    reR31 reR32 reR33
    trace det    
               
    (matrix-12a)rer11 rer12 rer13
    rer21 rer22 rer23
    rer31 rer32 rer33
    trace det    
               
    (matrix-6a)rfR11 rfR12 rfR13
    rfR21 rfR22 rfR23
    rfR31 rfR32 rfR33
    trace det    
               
    (matrix-11a)rfr11 rfr12 rfr13
    rfr21 rfr22 rfr23
    rfr31 rfr32 rfr33
    trace det    
               
    (matrix-13a)rgR11 rgR12 rgR13
    rgR21 rgR22 rgR23
    rgR31 rgR32 rgR33
    trace det    
               
    (matrix-16a)rgr11 rgr12 rgr13
    rgr21 rgr22 rgr23
    rgr31 rgr32 rgr33
    trace det    
               
    (matrix-14a)rhR11 rhR12 rhR13
    rhR21 rhR22 rhR23
    rhR31 rhR32 rhR33
    trace det    
               
    (matrix-15a)rhr11 rhr12 rhr13
    rhr21 rhr22 rhr23
    rhr31 rhr32 rhr33
    trace det    
               
    (matrix-17a)riR11 riR12 riR13
    riR21 riR22 riR23
    riR31 riR32 riR33
    trace det    
               
    (matrix-20a)rir11 rir12 rir13
    rir21 rir22 rir23
    rir31 rir32 rir33
    trace det    
               
    (matrix-18a)rjR11 rjR12 rjR13
    rjR21 rjR22 rjR23
    rjR31 rjR32 rjR33
    trace det    
               
    (matrix-19a)rjr11 rjr12 rjr13
    rjr21 rjr22 rjr23
    rjr31 rjr32 rjr33
    trace det    
               
    (matrix-21a)rkR11 rkR12 rkR13
    rkR21 rkR22 rkR23
    rkR31 rkR32 rkR33
    trace det    
               
    (matrix-23a)rkr11 rkr12 rkr13
    rkr21 rkr22 rkr23
    rkr31 rkr32 rkr33
    trace det    
               
    (matrix-22a)rlR11 rlR12 rlR13
    rlR21 rlR22 rlR23
    rlR31 rlR32 rlR33
    trace det    
               
    (matrix-24a)rlr11 rlr12 rlr13
    rlr21 rlr22 rlr23
    rlr31 rlr32 rlr33
    trace det    
               
    (matrix-25a)rmR11 rmR12 rmR13
    rmR21 rmR22 rmR23
    rmR31 rmR32 rmR33
    trace det    
               
    (matrix-26a)rmr11 rmr12 rmr13
    rmr21 rmr22 rmr23
    rmr31 rmr32 rmr33
    trace det    
               
    (matrix-27a)rnR11 rnR12 rnR13
    rnR21 rnR22 rnR23
    rnR31 rnR32 rnR33
    trace det    
               
    (matrix-28a)rnr11 rnr12 rnr13
    rnr21 rnr22 rnr23
    rnr31 rnr32 rnr33
    trace det    
               
    (matrix-29a)roR11 roR12 roR13
    roR21 roR22 roR23
    roR31 roR32 roR33
    trace det    
               
    (matrix-30a)ror11 ror12 ror13
    ror21 ror22 ror23
    ror31 ror32 ror33
    trace det    
               
    (matrix-31a)rpR11 rpR12 rpR13
    rpR21 rpR22 rpR23
    rpR31 rpR32 rpR33
    trace det    
               
    (matrix-32a)rpr11 rpr12 rpr13
    rpr21 rpr22 rpr23
    rpr31 rpr32 rpr33
    trace det    
               
    (matrix-33a)rqR11 rqR12 rqR13
    rqR21 rqR22 rqR23
    rqR31 rqR32 rqR33
    trace det    
               
    (matrix-34a)rqr11 rqr12 rqr13
    rqr21 rqr22 rqr23
    rqr31 rqr32 rqr33
    trace det    
               
    (matrix-35a)rrR11 rrR12 rrR13
    rrR21 rrR22 rrR23
    rrR31 rrR32 rrR33
    trace det    
               
    (matrix-36a)rrr11 rrr12 rrr13
    rrr21 rrr22 rrr23
    rrr31 rrr32 rrr33
    trace det    
               
    (matrix-37a)rsR11 rsR12 rsR13
    rsR21 rsR22 rsR23
    rsR31 rsR32 rsR33
    trace det    
               
    (matrix-38a)rsr11 rsr12 rsr13
    rsr21 rsr22 rsr23
    rsr31 rsr32 rsr33
    trace det    
               
    (matrix-39a)rtR11 rtR12 rtR13
    rtR21 rtR22 rtR23
    rtR31 rtR32 rtR33
    trace det    
               
    (matrix-40a)rtr11 rtr12 rtr13
    rtr21 rtr22 rtr23
    rtr31 rtr32 rtr33
    trace det    
               
    (matrix-41a)ruR11 ruR12 ruR13
    ruR21 ruR22 ruR23
    ruR31 ruR32 ruR33
    trace det    
               
    (matrix-42a)rur11 rur12 rur13
    rur21 rur22 rur23
    rur31 rur32 rur33
    trace det    
               
    (matrix-43a)rvR11 rvR12 rvR13
    rvR21 rvR22 rvR23
    rvR31 rvR32 rvR33
    trace det    
               
    (matrix-44a)rvr11 rvr12 rvr13
    rvr21 rvr22 rvr23
    rvr31 rvr32 rvr33
    trace det    
               
    (matrix-45a)rwR11 rwR12 rwR13
    rwR21 rwR22 rwR23
    rwR31 rwR32 rwR33
    trace det    
               
    (matrix-46a)rwr11 rwr12 rwr13
    rwr21 rwr22 rwr23
    rwr31 rwr32 rwr33
    trace det    
               
    (matrix-47a)rxR11 rxR12 rxR13
    rxR21 rxR22 rxR23
    rxR31 rxR32 rxR33
    trace det    
               
    (matrix-48a)rxr11 rxr12 rxr13
    rxr21 rxr22 rxr23
    rxr31 rxr32 rxr33
    trace det    
               
    (matrix-1u)uaR11 uaR12 uaR13
    uaR21 uaR22 uaR23
    uaR31 uaR32 uaR33
    trace det    
    Σi=1,2,3(uaRij)2    
               
    (matrix-10u)uar11 uar12 uar13
    uar21 uar22 uar23
    uar31 uar32 uar33
    trace det    
    Σi=1,2,3(uarij)2    
               
    (matrix-2u)ubR11 ubR12 ubR13
    ubR21 ubR22 ubR23
    ubR31 ubR32 ubR33
    trace det    
    Σi=1,2,3(ubRij)2    
               
    (matrix-9u)ubr11 ubr12 ubr13
    ubr21 ubr22 ubr23
    ubr31 ubr32 ubr33
    trace det    
    Σi=1,2,3(ubrij)2    
               
    (matrix-3u)ucR11 ucR12 ucR13
    ucR21 ucR22 ucR23
    ucR31 ucR32 ucR33
    trace det    
    Σi=1,2,3(ucRij)2    
               
    (matrix-8u)ucr11 ucr12 ucr13
    ucr21 ucr22 ucr23
    ucr31 ucr32 ucr33
    trace det    
    Σi=1,2,3(ucrij)2    
               
               
               
               
               

 

 

48 3-D Rotational Matrices

  Remarks
1 x2(α)y2(β)z2(γ) cosαcosγ +sinαsinβsinγ -sinαcosβ -cosαsinγ+sinαsinβcosγ 01=(10)T
    sinαcosγ-cosαsinβsinγ cosαcosβ -sinαsinγ-cosαsinβcosγ  
    cosβsinγ sinβ cosβcosγ  
           
2 x1(α)y1(β)z1(γ) cosαcosγ -sinαsinβsinγ sinαcosβ cosαsinγ+sinαsinβcosγ 02=(09)T
    -sinαcosγ-cosαsinβsinγ cosαcosβ -sinαsinγ+cosαsinβcosγ  
    -cosβsinγ -sinβ cosβcosγ  
           
3 y2(β)z2(γ)x2(α) cosαcosγ -sinαcosγ -sinγ 03=(08)T
    -cosαsinβsinγ+sinαcosβ sinαsinβsinγ+cosαcosβ -sinβcosγ  
    cosαcosβsinγ+sinαsinβ -sinαcosβsinγ+cosαsinβ cosβcosγ  
           
4 y1z1x1 cosαcosγ sinαcosγ sinγ 04=(07)T
    -cosαsinβsinγ-sinαcosβ -sinαsinβsinγ+cosαcosβ sinβcosγ  
    -cosαcosβsinγ+sinαsinβ -sinαcosβsinγ-cosαsinβ cosβcosγ  
           
5 z2x2y2 cosαcosγ -sinαcosβcosγ-sinβsinγ sinαsinβcosγ-cosβsinγ 05=(12)T
    sinα cosαcosβ -cosαsinβ  
    cosαsinγ -sinαcosβsinγ+sinβcosγ sinαsinβsinγ+cosβcosγ  
           
6 z1x1y1 cosαcosγ sinαcosβcosγ-sinβsinγ sinαsinβcosγ+cosβsinγ 06=(11)T
    -sinα cosαcosβ cosαsinβ  
    -cosαsinγ -sinαcosβsinγ-sinβcosγ -sinαsinβsinγ+cosβcosγ  
           
7 x2z2y2 cosαcosγ -sinαcosβ-cosαsinβsinγ sinαsinβ-cosαcosβsinγ x2z2y2=(y1z1x1)T
    sinαcosγ cosαcosβ-sinαsinβsinγ -cosαsinβ-sinαcosβsinγ 07=(04)T
    sinγ sinβcosγ cosβcosγ

 

 
           
8 x1z1y1 cosαcosγ sinαcosβ-cosαsinβsinγ sinαsinβ+cosαcosβsinγ x1z1y1=(y2z2x2)T
    -sinαcosγ cosαcosβ+sinαsinβsinγ cosαsinβ-sinαcosβsinγ 08=(03)T
    -sinγ -sinβcosγ cosβcosγ  
           
9 z2y2x2 cosαcosγ-sinαsinβsinγ -sinαcosγ-cosαsinβsinγ -cosβsinγ z2y2x2=(x1y1z1)T
    sinαcosβ cosαcosβ -sinβ 09=(02)T
    cosαsinγ+sinαsinβcosγ -sinαsinγ+cosαsinβcosγ cosβcosγ  
           
10 z1y1x1 cosαcosγ+sinαsinβsinγ sinαcosγ-cosαsinβsinγ cosβsinγ z1y1x1=(x2y2z2)T
    -sinαcosβ cosαcosβ sinβ 10=(01)T
    -cosαsinγ+sinαsinβcosγ -sinαsinγ-cosαsinβcosγ cosβcosγ  
           
11 y2x2z2 cosαcosγ -sinα -cosαsinγ y2x2z2=(z1x1y1)T
    sinαcosβcosγ-sinβsinγ cosαcosβ -sinαcosβsinγ-sinβcosγ 11=(06)T
    sinαsinβcosγ+cosβsinγ cosαsinβ -sinαsinβsinγ+cosβcosγ  
           
12 y1x1z1 cosαcosγ sinα cosαsinγ y1x1z1=(z2x2y2)T
    -sinαcosβcosγ-sinβsinγ cosαcosβ -sinαcosβsinγ+sinβcosγ 12=(05)T
    sinαsinβcosγ-cosβsinγ -cosαsinβ sinαsinβsinγ+cosβcosγ  
           
13 x1(α)y1(β)z2(γ) cosαcosγ+sinαsinβsinγ sinαcosβ -cosαsinγ+sinαsinβcosγ 13=(16)T
    -sinαcosγ+cosαsinβsinγ cosαcosβ sinαsinγ+cosαsinβcosγ  
    cosβsinγ -sinβ cosβcosγ  
           
14 x2(α)y2(β)z1(γ) cosαcosγ-sinαsinβsinγ -sinαcosβ cosαsinγ+sinαsinβcosγ 14=(15)T
    sinαcosγ+cosαsinβsinγ cosαcosβ sinαsinγ-cosαsinβcosγ  
    -cosβsinγ sinβ cosβcosγ  
           
15 z2y1x1 cosαcosγ-sinαsinβsinγ sinαcosγ+cosαsinβsinγ -cosβsinγ z2y1x1=(x2y2z1)T
    -sinαcosβ cosαcosβ sinβ 15=(14)T
    cosαsinγ+sinαsinβcosγ sinαsinγ-cosαsinβcosγ cosβcosγ  
           
16 z1y2x2 cosαcosγ+sinαsinβsinγ -sinαcosγ+cosαsinβsinγ cosβsinγ z1y2x2=(x1y1z2)T
    sinαcosβ cosαcosβ -sinβ 16=(13)T
    -cosαsinγ+sinαsinβcosγ sinαsinγ+cosαsinβcosγ cosβcosγ  
           
17 z2x1y1 cosαcosγ sinαcosβcosγ+sinβsinγ -cosβsinγ+sinαsinβcosγ 17=(20)T
    -sinα cosαcosβ cosαsinβ  
    cosαsinγ sinαcosβsinγ-sinβcosγ sinαsinβsinγ+cosβcosγ  
           
18 z1x2y2 cosαcosγ -sinαcosβcosγ+sinβsinγ cosβsinγ+sinαsinβcosγ 18=(19)T
    sinα cosαcosβ -cosαsinβ  
    -cosαsinγ sinαcosβsinγ+sinβcosγ -sinαsinβsinγ+cosβcosγ  
           
19 y1x1z2 cosαcosγ sinα -cosαsinγ y1x1z2=(z1x2y2)T
    -sinαcosβcosγ+sinβsinγ cosαcosβ sinαcosβsinγ+sinβcosγ 19=(18)T
    cosβsinγ+sinαsinβcosγ -cosαsinβ -sinαsinβsinγ+cosβcosγ  
           
20. y2x2z1 cosαcosγ -sinα cosαsinγ y2x2z1=(z2x1y1)T
    sinαcosβcosγ+sinβsinγ cosαcosβ sinαcosβsinγ-sinβcosγ 20=(17)T
    -cosβsinγ+sinαsinβcosγ cosαsinβ sinαsinβsinγ+cosβcosγ  
           
21 z1x1y2 cosαcosγ sinαcosβcosγ+sinβsinγ cosβsinγ-sinαsinβcosγ 21=(23)T
    -sinα cosαcosβ -cosαsinβ  
    -cosαsinγ sinβcosγ-sinαcosβsinγ sinαsinβsinγ+cosβcosγ  
           
22 z2x2y1 cosαcosγ -sinαcosβcosγ+sinβsinγ -sinαsinβcosγ-cosβsinγ 22=(24)T
    sinα cosαcosβ cosαsinβ  
    cosαsinγ -sinβcosγ-sinαcosβsinγ -sinαsinβsinγ+cosβcosγ  
           
23 y1x2z2 cosαcosγ -sinα -cosαsinγ y1x2z2=(z1x1y2)T
    sinαcosβcosγ+sinβsinγ cosαcosβ sinβcosγ-sinαcosβsinγ 23=(21)T
    cosβsinγ-sinαsinβcosγ -cosαsinβ sinαsinβsinγ+cosβcosγ  
           
24 y2x1z1 cosαcosγ sinα cosαsinγ y2x1z1=(z2x2y1)T
    -sinαcosβcosγ+sinβsinγ cosαcosβ -sinβcosγ-sinαcosβsinγ 24=(22)T
    -sinαsinβcosγ-cosβsinγ cosαsinβ -sinαsinβsinγ+cosβcosγ  
           
25 x1(α)y2(β)z2(γ) cosαcosγ-sinαsinβsinγ sinαcosβ -cosαsinγ-sinαsinβcosγ 25=(26)T
    -sinαcosγ-cosαsinβsinγ cosαcosβ sinαsinγ-cosαsinβcosγ  
    cosβsinγ sinβ cosβcosγ  
           
26 z1y1x2 cosαcosγ-sinαsinβsinγ -sinαcosγ-cosαsinβsinγ cosβsinγ z1y1x2=(x1y2z2)T
    sinαcosβ cosαcosβ sinβ 26=(25)T
    -cosαsinγ-sinαsinβcosγ sinαsinγ-cosαsinβcosγ cosβcosγ  
           
27 z2y2x1 cosαcosγ+sinαsinβsinγ sinαcosγ-cosαsinβsinγ -cosβsinγ 27=(28)T
    -sinαcosβ cosαcosβ -sinβ  
    cosαsinγ-sinαsinβcosγ sinαsinγ+cosαsinβcosγ cosβcosγ  
           
28 x2y1z1 cosαcosγ+sinαsinβsinγ -sinαcosβ cosαsinγ-sinαsinβcosγ x2y1z1=(z2y2x1)T
    sinαcosγ-cosαsinβsinγ cosαcosβ sinαsinγ+cosαsinβcosγ 28=(27)T
    -cosβsinγ -sinβ cosβcosγ  
           
29 y1z1x2 cosαcosγ -sinαcosγ sinγ 29=(30)T
    sinαcosβ-cosαsinβsinγ cosαcosβ+sinαsinβsinγ sinβcosγ  
    -cosαcosβsinγ-sinαsinβ sinαcosβsinγ-cosαsinβ cosβcosγ  
           
30 x1z2y2 cosαcosγ sinαcosβ-cosαsinβsinγ -cosαcosβsinγ-sinαsinβ x2z2y2=(y1z1x2)T
    -sinαcosγ cosαcosβ+sinαsinβsinγ sinαcosβsinγ-cosαsinβ 30=(29)T
    sinγ sinβcosγ cosβcosγ  
           
31 x2z1y1 cosαcosγ -sinαcosβ-cosαsinβsinγ cosαcosβsinγ-sinαsinβ 31=(32)T
    sinαcosγ cosαcosβ-sinαsinβsinγ sinαcosβsinγ+cosαsinβ  
    -sinγ -sinβcosγ cosβcosγ  
           
32 y2z2x1 cosαcosγ sinαcosγ -sinγ y2z2x1=(x2z1y1)T
    -sinαcosβ-cosαsinβsinγ cosαcosβ-sinαsinβsinγ -sinβcosγ 32=(31)T
    cosαcosβsinγ-sinαsinβ sinαcosβsinγ+cosαsinβ cosβcosγ  
           
33 x1z1y2 cosαcosγ sinαcosβ+cosαsinβsinγ cosαcosβsinγ-sinαsinβ 33=(34)T
    -sinαcosγ cosαcosβ-sinαsinβsinγ -sinαcosβsinγ-cosαsinβ  
    -sinγ sinβcosγ cosβcosγ  
           
34 y1z2x2 cosαcosγ -sinαcosγ -sinγ y1z2x2=(x1z1y2)T
    sinαcosβ+cosαsinβsinγ cosαcosβ-sinαsinβsinγ sinβcosγ 34=(33)T
    cosαcosβsinγ-sinαsinβ -sinαcosβsinγ-cosαsinβ cosβcosγ  
           
35 y2z1x1 cosαcosγ sinαcosγ sinγ 35=(36)T
    cosαsinβsinγ-sinαcosβ sinαsinβsinγ+cosαcosβ -sinβcosγ  
    -cosαcosβsinγ-sinαsinβ -sinαcosβsinγ+cosαsinβ cosβcosγ  
           
36 x2z2y1 cosαcosγ cosαsinβsinγ-sinαcosβ -cosαcosβsinγ-sinαsinβ x2z2y1=(y2z1x1)T
    sinαcosγ sinαsinβsinγ+cosαcosβ -sinαcosβsinγ+cosαsinβ 36=(35)T
    sinγ -sinβcosγ cosβcosγ  
           
37 x1(α)y2(β)z1(γ) cosαcosγ+sinαsinβsinγ sinαcosβ cosαsinγ-sinαsinβcosγ 37=(38)T
    -sinαcosγ+cosαsinβsinγ cosαcosβ -sinαsinγ-cosαsinβcosγ  
    -cosβsinγ sinβ cosβcosγ  
           
38 z2y1x2 cosαcosγ+sinαsinβsinγ -sinαcosγ+cosαsinβsinγ -cosβsinγ z2y1x2=(x1y2z1)T
    sinαcosβ cosαcosβ sinβ 38=(37)T
    cosαsinγ-sinαsinβcosγ -sinαsinγ-cosαsinβcosγ cosβcosγ  
           
39 x2(α)y1(β)z2(γ) cosαcosγ-sinαsinβsinγ -sinαcosβ -cosαsinγ-sinαsinβcosγ 39=(40)T
    sinαcosγ+cosαsinβsinγ cosαcosβ -sinαsinγ+cosαsinβcosγ  
    cosβsinγ -sinβ cosβcosγ  
           
40 z1y2x1 cosαcosγ-sinαsinβsinγ sinαcosγ+cosαsinβsinγ cosβsinγ z1y2x1=(x2y1z2)T
    -sinαcosβ cosαcosβ -sinβ 40=(39)T
    -cosαsinγ-sinαsinβcosγ -sinαsinγ+cosαsinβcosγ cosβcosγ  
           
41 y1x2z1 cosαcosγ -sinα cosαsinγ 41=(42)T
    sinαcosβcosγ-sinβsinγ cosαcosβ sinαcosβsinγ+snβcosγ  
    -sinαsinβcosγ-cosβsinγ -cosαsinβ -sinαsinβsinγ+cosβcosγ  
           
42 z2x1y2 cosαcosγ sinαcosβcosγ-sinβsinγ -sinαsinβcosγ-cosβsinγ z2x1y2=(y1x2z1)T
    -sinα cosαcosβ -cosαsinβ 42=(41)T
    cosαsinγ sinαcosβsinγ+sinβcosγ -sinαsinβsinγ+cosβcosγ  
           
43 z1x2y1 cosαcosγ -sinαcosβcosγ-sinβsinγ -sinαsinβcosγ+cosβsinγ 43=(44)T
    sinα cosαcosβ cosαsinβ  
    -cosαsinγ sinαcosβsinγ-sinβcosγ sinαsinβsinγ+cosβcosγ  
           
44 y2x1z2 cosαcosγ sinα -cosαsinγ y2x1z2=(z12x2y1)T
    -sinαcosβcosγ-sinβsinγ cosαcosβ sinαcosβsinγ-sinβcosγ 44=(43)T
    -sinαsinβcosγ+cosβsinγ cosαsinβ sinαsinβsinγ+cosβcosγ  
           
45 y1z2x1 cosαcosγ sinαcosγ -sinγ 45=(46)T
    cosαsinβsinγ-sinαcosβ sinαsinβsinγ+cosαcosβ sinβcosγ  
    cosαcosβsinγ+sinαsinβ sinαcosβsinγ-cosαsinβ cosβcosγ  
           
46 x2z1y2 cosαcosγ cosαsinβsinγ-sinαcosβ cosαcosβsinγ+sinαsinβ x2z1y2=(y1z2x1)T
    sinαcosγ sinαsinβsinγ+cosαcosβ sinαcosβsinγ-cosαsinβ 46=(45)T
    -sinγ sinβcosγ cosβcosγ  
           
47 x1z2y1 cosαcosγ sinαcosβ+cosαsinβsinγ -cosαcosβsinγ+sinαsinβ 47=(48)T
    -sinαcosγ -sinαsinβsinγ+cosαcosβ sinαcosβsinγ+cosαsinβ  
    sinγ -sinβcosγ cosβcosγ  
           
48 y2z1x2 cosαcosγ -sinαcosγ sinγ y2z1x2=(x1z2y1)T
    sinαcosβ+cosαsinβsinγ -sinαsinβsinγ+cosαcosβ -sinβcosγ 48=(47)T
    -cosαcosβsinγ+sinαsinβ sinαcosβsinγ+cosαsinβ cosβcosγ  
    interchangeSin&Cosof matrices      
1a x'2y'2z'2 sinαsinγ +cosαcosβcosγ -cosαsinβ -sinαcosγ+cosαcosβsinγ 01a=(10a)T
    cosαsinγ-sinαcosβcosγ sinαsinβ -cosαcosγ-sinαcosβsinγ  
    sinβcosγ cosβ sinβsinγ  
           
2a x'1y'1z'1 sinαsinγ -cosαcosβcosγ cosαsinβ sinαcosγ+cosαcosβsinγ 02a=(09a)T
    -cosαsinγ-sinαcosβcosγ sinαsinβ -cosαcosγ+sinαcosβsinγ  
    -sinβcosγ -cosβ sinβsinγ  
           
3a y'2z'2x'2 sinαsinγ -cosαsinγ -cosγ 03a=(08a)T
    -sinαcosβcosγ+cosαsinβ cosαcosβcosγ+sinαsinβ -cosβsinγ  
    sinαsinβcosγ+cosαcosβ -cosαsinβcosγ+sinαcosβ sinβsinγ  
           
4a y'1z'1x'1 sinαsinγ cosαsinγ cosγ 04a=(07a)T
    -sinαcosβcosγ-cosαsinβ -cosαcosβcosγ+sinαsinβ cosβsinγ  
    -sinαsinβcosγ+cosαcosβ -cosαsinβcosγ-sinαcosβ sinβsinγ  
           
5a z'2x'2y'2 sinαsinγ -cosαsinβsinγ-cosβcosγ cosαcosβsinγ-sinβcosγ 05a=(12a)T
    cosα sinαsinβ -sinαcosβ  
    sinαcosγ -cosαsinβcosγ+cosβsinγ cosαcosβcosγ+sinβsinγ  
           
6a z'1x'1y'1 sinαsinγ cosαsinβsinγ-cosβcosγ cosαcosβsinγ+sinβcosγ 06a=(11a)T
    -cosα sinαsinβ sinαcosβ  
    -sinαcosγ -cosαsinβcosγ-cosβsinγ -cosαcosβcosγ+sinβsinγ  
           
7a x'2z'2y'2 sinαsinγ -cosαsinβ-sinαcosβcosγ cosαcosβ-sinαsinβcosγ x'2z'2y'2=(y'1z'1x'1)T
    cosαsinγ sinαsinβ-cosαcosβcosγ -sinαcosβ-cosαsinβcosγ 07a=(04a)T
    cosγ cosβsinγ sinβsinγ

 

 
           
8a x'1z'1y'1 sinαsinγ cosαsinβ-sinαcosβcosγ cosαcosβ+sinαsinβcosγ x'1z'1y'1=(y'2z'2x'2)T
    -cosαsinγ sinαsinβ+cosαcosβcosγ sinαcosβ-cosαsinβcosγ 08a=(03a)T
    -cosγ -cosβsinγ sinβsinγ  
           
9a z'2y'2x'2 sinαsinγ-cosαcosβcosγ -cosαsinγ-sinαcosβcosγ -sinβcosγ z'2y'2x'2=(x'1y'1z'1)T
    cosαsinβ sinαsinβ -cosβ 09a=(02a)T
    sinαcosγ+cosαcosβsinγ -cosαcosγ+sinαcosβsinγ sinβsinγ  
           
10a z'1y'1x'1 sinαsinγ+cosαcosβcosγ cosαsinγ-sinαcosβcosγ sinβcosγ z'1y'1x'1=(x'2y'2z'2)T
    -cosαsinβ sinαsinβ cosβ 10a=(01a)T
    -sinαcosγ+cosαcosβsinγ -cosαcosγ-sinαcosβsinγ sinβsinγ  
           
11a y'2x'2z'2 sinαsinγ -cosα -sinαcosγ y'2x'2z'2=(z'1x'1y'1)T
    cosαsinβsinγ-cosβcosγ sinαsinβ -cosαsinβcosγ-cosβsinγ 11a=(06a)T
    cosαcosβsinγ+sinβcosγ sinαcosβ -cosαcosβcosγ+sinβsinγ  
           
12a y'1x'1z'1 sinαsinγ cosα sinαcosγ y'1x'1z1= (z'2x'2y'2)T
    -cosαsinβsinγ-cosβcosγ sinαsinβ -cosαsinβcosγ+cosβsinγ 12a=(05a)T
    cosαcosβsinγ-sinβcosγ -sinαcosβ cosαcosβcosγ+sinβsinγ  
           
13a x'1y'1z'2 sinαsinγ+cosαcosβcosγ cosαsinβ -sinαcosγ+cosαcosβsinγ 13a=(16a)T
    -cosαsinγ+sinαcosβcosγ sinαsinβ cosαcosγ+sinαcosβsinγ  
    sinβcosγ -cosβ sinβsinγ  
           
14a x'2y'2z'1 sinαsinγ-cosαcosβcosγ -cosαsinβ sinαcosγ+cosαcosβsinγ 14a=(15a)T
    cosαsinγ+sinαcosβcosγ sinαsinβ cosαcosγ-sinαcosβsinγ  
    -sinβcosγ cosβ sinβsinγ  
           
15a z'2y'1x'1 sinαsinγ-cosαcosβcosγ cosαsinγ+sinαcosβcosγ -sinβcosγ 15a=(14a)T
    -cosαsinβ sinαsinβ cosβ  
    sinαcosγ+cosαcosβsinγ cosαcosγ-sinαcosβsinγ sinβsinγ  
           
16a z'1y'2x'2 sinαsinγ+cosαcosβcosγ -cosαsinγ+sinαcosβcosγ sinβcosγ 16a=(13a)T
    cosαsinβ sinαsinβ -cosβ  
    -sinαcosγ+cosαcosβsinγ cosαcosγ+sinαcosβsinγ sinβsinγ  
           
17a z'2x'1y'1 sinαsinγ cosαsinβsinγ+cosβcosγ -sinβcosγ+cosαcosβsinγ 17a=(20a)T
    -cosα sinαsinβ sinαcosβ  
    sinαcosγ cosαsinβcosγ-cosβsinγ cosαcosβcosγ+sinβsinγ  
           
18a z'1x'2y'2 sinαsinγ -cosαsinβsinγ+cosβcosγ sinβcosγ+cosαcosβsinγ 18a=(19a)T
    cosα sinαsinβ -sinαcosβ  
    -sinαcosγ cosαsinβcosγ+cosβsinγ -cosαcosβcosγ+sinβsinγ  
           
19a y'1x'1z'2 sinαsinγ cosα -sinαcosγ y'1x'1z'2=(z'1x'2y'2)T
    -cosαsinβsinγ+cosβcosγ sinαsinβ cosαsinβcosγ+cosβsinγ 19a=(18a)T
    sinβcosγ+cosαcosβsinγ -sinαcosβ -cosαcosβcosγ+sinβsinγ  
           
20a y'2x'2z'1 sinαsinγ -cosα sinαcosγ 20a=(17a)T
    cosαsinβsinγ+cosβcosγ sinαsinβ cosαsinβcosγ-cosβsinγ  
    -sinβcosγ+cosαcosβsinγ sinαcosβ cosαcosβcosγ+sinβsinγ  
           
21a z'1x'1y'2 sinαsinγ cosαsinβsinγ+cosβcosγ sinβcosγ-cosαcosβsinγ 21a=(23a)T
    -cosα sinαsinβ -sinαcosβ  
    -sinαcosγ cosβsinγ-cosαsinβcosγ cosαcosβcosγ+sinβsinγ  
           
22a z'2x'2y'1 sinαsinγ -cosαsinβsinγ+cosβcosγ -cosαcosβsinγ-sinβcosγ 22a=(24a)T
    cosα sinαsinβ sinαcosβ  
    sinαcosγ -cosβsinγ-cosαsinβcosγ -cosαcosβcosγ+sinβsinγ  
           
23a y'1x'2z'2 sinαsinγ -cosα -sinαcosγ 23a=(21a)T
    cosαsinβsinγ+cosβcosγ sinαsinβ cosβsinγ-cosαsinβcosγ  
    sinβcosγ-cosαcosβsinγ -sinαcosβ cosαcosβcosγ+sinβsinγ  
           
24a y'2x'1z'1 sinαsinγ cosα sinαcosγ 24a=(22a)T
    -cosαsinβsinγ+cosβcosγ sinαsinβ -cosβsinγ-cosαsinβcosγ  
    -cosαcosβsinγ-sinβcosγ sinαcosβ -cosαcosβcosγ+sinβsinγ  
           
25a x'1y'2z'2 sinαsinγ-cosαcosβcosγ cosαsinβ -sinαcosγ-cosαcosβsinγ 25a=(26a)T
    -cosαsinγ-sinαcosβcosγ sinαsinβ cosαcosγ-sinαcosβsinγ  
    sinβcosγ cosβ sinβsinγ  
           
26a z'1y'1x'2 sinαsinγ-cosαcosβcosγ -cosαsinγ-sinαcosβcosγ sinβcosγ 26a=(25a)T
    cosαsinβ sinαsinβ cosβ  
    -sinαcosγ-cosαcosβsinγ cosαcosγ-sinαcosβsinγ sinβsinγ  
           
27a z'2y'2x'1 sinαsinγ+cosαcosβcosγ cosαsinγ-sinαcosβcosγ -sinβcosγ 27a=(28a)T
    -cosαsinβ sinαsinβ -cosβ  
    sinαcosγ-cosαcosβsinγ cosαcosγ+sinαcosβsinγ sinβsinγ  
           
28a x'2y'1z'1 sinαsinγ+cosαcosβcosγ -cosαsinβ sinαcosγ-cosαcosβsinγ 28a=(27a)T
    cosαsinγ-sinαcosβcosγ sinαsinβ cosαcosγ+sinαcosβsinγ  
    -sinβcosγ -cosβ sinβsinγ  
           
29a y'1z'1x'2 sinαsinγ -cosαsinγ cosγ 29a=(30a)T
    cosαsinβ-sinαcosβcosγ sinαsinβ+cosαcosβcosγ cosβsinγ  
    -sinαsinβcosγ-cosαcosβ cosαsinβcosγ-sinαcosβ sinβsinγ  
           
30a x'1z'2y'2 sinαsinγ cosαsinβ-sinαcosβcosγ -sinαsinβcosγ-cosαcosβ 30a=(29a)T
    -cosαsinγ sinαsinβ+cosαcosβcosγ cosαsinβcosγ-sinαcosβ  
    cosγ cosβsinγ sinβsinγ  
           
31a x2z1y1 sinαsinγ -cosαsinβ-sinαcosβcosγ sinαsinβcosγ-cosαcosβ 31a=(32a)T
    cosαsinγ sinαsinβ-cosαcosβcosγ cosαsinβcosγ+sinαcosβ  
    -cosγ -cosβsinγ sinβsinγ  
           
32a y'2z'2x'1 sinαsinγ cosαsinγ -cosγ 32a=(31a)T
    -cosαsinβ-sinαcosβcosγ sinαsinβ-cosαcosβcosγ -cosβsinγ  
    sinαsinβcosγ-cosαcosβ cosαsinβcosγ+sinαcosβ sinβsinγ  
           
33a x'1z'1y'2 sinαsinγ cosαsinβ+sinαcosβcosγ sinαsinβcosγ-cosαcosβ 33a=(34a)T
    -cosαsinγ sinαsinβ-cosαcosβcosγ -cosαsinβcosγ-sinαcosβ  
    -cosγ cosβsinγ sinβsinγ  
           
34a y'1z'2x'2 sinαsinγ -cosαsinγ -cosγ 34a=(33a)T
    cosαsinβ+sinαcosβcosγ sinαsinβ-cosαcosβcosγ cosβsinγ  
    sinαsinβcosγ-cosαcosβ -cosαsinβcosγ-sinαcosβ sinβsinγ  
           
35a y'2z'1x'1 sinαsinγ cosαsinγ cosγ 35a=(36a)T
    sinαcosβcosγ-cosαsinβ cosαcosβcosγ+sinαsinβ -cosβsinγ  
    -sinαsinβcosγ-cosαcosβ -cosαsinβcosγ+sinαcosβ sinβsinγ  
           
36a x'2z'2y'1 sinαsinγ sinαcosβcosγ-cosαsinβ -sinαsinβcosγ-cosαcosβ 36a=(35a)T
    cosαsinγ cosαcosβcosγ+sinαsinβ -cosαsinβcosγ+sinαcosβ  
    cosγ -cosβsinγ sinβsinγ  
           
37a x'1y'2z'1 sinαsinγ+cosαcosβcosγ cosαsinβ sinαcosγ-cosαcosβsinγ 37a=(38a)T
    -cosαsinγ+sinαcosβcosγ sinαsinβ -cosαcosγ-sinαcosβsinγ  
    -sinβcosγ cosβ sinβsinγ  
           
38a z'2y'1x'2 sinαsinγ+cosαcosβcosγ -cosαsinγ+sinαcosβcosγ -sinβcosγ 38a=(37a)T
    cosαsinβ sinαsinβ cosβ  
    sinαcosγ-cosαcosβsinγ -cosαcosγ-sinαcosβsinγ sinβsinγ  
           
39a x'2y'1z'2 sinαsinγ-cosαcosβcosγ -cosαsinβ -sinαcosγ-cosαcosβsinγ 39a=(40a)T
    cosαsinγ+sinαcosβcosγ sinαsinβ -cosαcosγ+sinαcosβsinγ  
    sinβcosγ -cosβ sinβsinγ  
           
40a z'1y'2x'1 sinαsinγ-cosαcosβcosγ cosαsinγ+sinαcosβcosγ sinβcosγ 40a=(39a)T
    -cosαsinβ sinαsinβ -cosβ  
    -sinαcosγ-cosαcosβsinγ -cosαcosγ+sinαcosβsinγ sinβsinγ  
           
41a y'1x'2z'1 sinαsinγ -cosα sinαcosγ 41a=(42a)T
    cosαsinβsinγ-cosβcosγ sinαsinβ cosαsinβcosγ+cosβsinγ  
    -cosαcosβsinγ-sinβcosγ -sinαcosβ -cosαcosβcosγ+sinβsinγ  
           
42a z'2x'1y'2 sinαsinγ cosαsinβsinγ-cosβcosγ -cosαcosβsinγ-sinβcosγ 42a=(41a)T
    -cosα sinαsinβ -sinαcosβ  
    sinαcosγ cosαsinβcosγ+cosβsinγ -cosαcosβcosγ+sinβsinγ  
           
43a z'1x'2y'1 sinαsinγ -cosβcosγ-cosαsinβsinγ sinβcosγ-cosαcosβsinγ 43a=(44a)T
    cosα sinαsinβ sinαcosβ  
    -sinαcosγ -cosβsinγ+cosαsinβcosγ sinβsinγ+cosαcosβcosγ  
           
44a y'2x'1z'2 sinαsinγ cosα -sinαcosγ 44a=(43a)T
    -cosβcosγ-cosαsinβsinγ sinαsinβ -cosβsinγ+cosαsinβcosγ  
    sinβcosγ-cosαcosβsinγ sinαcosβ sinβsinγ+cosαcosβcosγ  
           
45a y'1z'2x'1 sinαsinγ cosαsinγ -cosγ 45a=(46a)T
    sinαcosβcosγ-cosαsinβ sinαsinβ+cosαcosβcosγ cosβsinγ  
    sinαsinβcosγ+cosαcosβ -sinαcosβ+cosαsinβcosγ sinβsinγ  
           
46a x'2z'1y'2 sinαsinγ -cosαsinβ+sinαcosβcosγ cosαcosβ+sinαsinβcosγ 46a=(45a)T
    cosαsinγ sinαsinβ+cosαcosβcosγ -sinαcosβ+cosαsinβcosγ  
    -cosγ cosβsinγ sinβsinγ  
           
47a x'1z'2y'1 sinαsinγ cosαsinβ+sinαcosβcosγ cosαcosβ-sinαsinβcosγ 47a=(48a)T
    -cosαsinγ sinαsinβ-cosαcosβcosγ sinαcosβ+cosαsinβcosγ  
    cosγ -cosβsinγ sinβsinγ  
           
48a y'2z'1x'2  sinαsinγ -cosαsinγ cosγ 48a=(47a)T
    cosαsinβ+sinαcosβcosγ sinαsinβ-cosαcosβcosγ -cosβsinγ  
    cosαcosβ-sinαsinβcosγ sinαcosβ+cosαsinβcosγ sinβsinγ  
    EULERANGLES      
1u x2(α)y2(β)x2(γ) cosαcosγ-sinαcosβsinγ -cosαsinγ-sinαcosβcosγ sinαsinβ 1u=(10u)T
    sinαcosγ+cosαcosβsinγ - sinαsinγ+cosαcosβcosγ -cosαsinβ  
    sinβsinγ sinβcosγ cosβ  
           
2u x1(α)y1(β)x1(γ) cosαcosγ-sinαcosβsinγ cosαsinγ+sinαcosβcosγ sinαsinβ 2u=(9u)T
    -sinαcosγ-cosαcosβsinγ -sinαsinγ+cosαcosβcosγ  cosαsinβ  
    sinβsinγ -sinβcosγ cosβ  
           
3u x2(α)y1(β)x2(γ) cosαcosγ-sinαcosβsinγ -cosαsinγ-sinαcosβcosγ -sinαsinβ 3u=(8u)T
    sinαcosγ+cosαcosβsinγ - sinαsinγ+cosαcosβcosγ cosαsinβ  
    -sinβsinγ -sinβcosγ cosβ  
           
4u x1(α)y2(β)x1(γ) cosαcosγ-sinαcosβsinγ cosαsinγ+sinαcosβcosγ -sinαsinβ  
    -sinαcosγ-cosαcosβsinγ - sinαsinγ+cosαcosβcosγ -cosαsinβ  
    -sinβsinγ sinβcosγ cosβ  
           
5u y2(α)x2(β)y2(γ) cosβ -sinβcosγ sinβsinγ  
    cosαsinβ -sinαsinγ+cosαcosβcosγ -sinαcosγ-cosαcosβsinγ  
    sinαsinβ cosαsinγ+sinαcosβcosγ cosαcosγ-sinαcosβsinγ  
           
6u y1(α)x1(β)y1(γ) cosβ sinβcosγ sinβsinγ  
    -cosαsinβ -sinαsinγ +cosαcosβcosγ sinαcosγ+cosαcosβsinγ  
    sinαsinβ -cosαsinγ-sinαcosβcosγ cosαcosγ-sinαcosβsinγ  
           
7u y1(α)x2(β)y1(γ) cosβ -sinβcosγ -sinβsinγ  
    cosαsinβ -sinαsinγ +cosαcosβcosγ sinαcosγ+cosαcosβsinγ  
    -sinαsinβ -cosαsinγ-sinαcosβcosγ cosαcosγ-sinαcosβsinγ  
           
8u   cosαcosγ-sinαcosβsinγ sinαcosγ+cosαcosβsinγ -sinβsinγ 8u=(3u)T
    -cosαsinγ-sinαcosβcosγ - sinαsinγ+cosαcosβcosγ -sinβcosγ  
    -sinαsinβ cosαsinβ cosβ  
           
9u   cosαcosγ-sinαcosβsinγ -sinαcosγ-cosαcosβsinγ sinβsinγ 9u=(2u)T
    cosαsinγ+sinαcosβcosγ -sinαsinγ+cosαcosβcosγ  -sinβcosγ  
    sinαsinβ cosαsinβ cosβ  
           
10u   cosαcosγ-sinαcosβsinγ sinαcosγ+cosαcosβsinγ sinβsinγ 10u=(1u)T
    -cosαsinγ-sinαcosβcosγ - sinαsinγ+cosαcosβcosγ sinβcosγ  
    sinαsinβ -cosαsinβ cosβ  
           
           
           
           
           
           
           
           
           
  y2(α)x1(β)y2(γ) cosβ sinβcosγ -sinβsinγ  
    -cosαsinβ -sinαsinγ +cosαcosβcosγ -sinαcosγ-cosαcosβsinγ  
    -sinαsinβ cosαsinγ+sinαcosβcosγ cosαcosγ-sinαcosβsinγ  
           
           
           
           
x1     x2    
cosα sinα 0 cosα -sinα 0
-sinα cosα 0 sinα cosα 0
0 0 1 0 0 1
x'1     x'2    
sinα cosα 0 sinα -cosα 0
-cosα sinα 0 cosα sinα 0
0 0 1 0 0 1
y1     y2    
1 0 0 1 0 0
0 cosβ sinβ 0 cosβ -sinβ
0 -sinβ cosβ 0 sinβ cosβ
y'1     y'2    
1 0 0 1 0 0
0 sinβ cosβ 0 sinβ -cosβ
0 -cosβ sinβ 0 cosβ sinβ
z1     z2    
cosγ 0 sinγ cosγ 0 -sinγ
0 1 0 0 1 0
-sinγ 0 cosγ sinγ 0 cosγ
z'1     z'2    
sinγ 0 cosγ sinγ 0 -cosγ
0 1 0 0 1 0
-cosγ 0 sinγ cosγ 0 sinγ
  Signature + - no. matrix no.
    9 0 nil  
    8 1 nil  
    7 2 16 + 16 (4,7)(14,15)(17,20)(25,26)

(29,30)(43,44)(45,46)(47,48)

(4a,7a)(14a,15a)(17a,20a)(25a,26a)

(29a,30a)(43a,44a)(45a,46a)(47a,48a)

    6 3 16 + 16 (1,10)(6,11)(13,16)(18,19)(22,24)(27,28)(37,38)(41,42)(1a,10a)(6a,11a)(13a,16a)(18a,19a)(22a,24a)(27a,28a)(37a,38a)(41a,42a)
    5 4 16 + 16 (2,9)(3,8)(5,12)(21,23)(31,32)(33,34)(35,36)(39,40)(2a,9a)(3a,8a)(5a,12a)(21a,23a)(31a,32a)(33a,34a)(35a,36a)(39a,40a)
      TOTAL 48 + 48  
           
           
           
           
           
           
* Above are case of Rotational Matrices in 3-D Euclidian Space (R3) which a) preserve origin. b) maintain Euclidian distance isometry c) preserve orientation i.e. handedness in space.

* R=Rz(α)*Ry(γ)*Rx(β)  is an intrinsic rotation where α,γ,β  are called Tait-Bryan Angles about z,y,x axis respectively. Intrinsic rotations occur about the axis of a co-ordinate system attached to a moving body(XYZ). Structure of rotation is

X-Y-Z, Y-Z-X, Z-X-Y, X-Z-Y, Z-Y-X, Y-X-Z

* R=Rz(β)*Ry(γ)*Rx(α)  is an extrinsic rotation where α,γ,β  are called Proper Euler Angles about x,y,z axis respectively. Extrinsic rotations occur about the axis of a co-ordinate system attached to a fixed frame of reference. The Structure of rotation is

x-y-x, x-z-x, y-z-y, y-x-y,z-x-z,z-y-z.

Two frames of reference are connected by Euler Angle provided they possess the same handedness in 3-D space.

In geometryEuler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point. It also means that the composition of two rotations is also a rotation. Therefore the set of rotations has a group structure, known as a rotation group.

The theorem is named after Leonhard Euler, who proved it in 1775 by means of spherical geometry. The axis of rotation is known as an Euler axis, typically represented by a unit vector . Its product by the rotation angle is known as an axis-angle. The extension of the theorem to kinematics yields the concept of instant axis of rotation, a line of fixed points.

In linear algebra terms, the theorem states that, in 3D space, any two Cartesian coordinate systems with a common origin are related by a rotation about some fixed axis. This also means that the product of two rotation matrices is again a rotation matrix and that for a non-identity rotation matrix one eigenvalue is 1 and the other two are both complex, or both equal to −1. The eigenvector corresponding to this eigenvalue is the axis of rotation connecting the two systems.

* The trace of the real 3-D rotation matrix is 1+2cosθ. Trace is invariant under orthogonal matrix similarity transformation. Since the orthogonal matrices are normal matrices and any normal matrix can be diagonalized by a unitary transformation, orthogonal matrices can be diagonalized through a unitary transformation.

* Then, any orthogonal matrix is either a rotation or an improper rotation. A general orthogonal matrix has only one real eigen value, either +1 or −1. When it is +1 the matrix is a rotation. When −1, the matrix is an improper rotation.

* An m  m matrix A has m orthogonal eigenvectors if and only if A is normal, that is, if AA = AA.[b] This result is equivalent to stating that normal matrices can be brought to diagonal form by a unitary similarity transformation:

* These are matrices which form SO(3) group under matrix multiplication where determinant is +1. These are sub-groups of O(3) groups of matrices whose determinant can be +1 or -1.These are sub-groups of GL(3). matrices.

* These are non-abelian group unlike SO(2) which is abelian.

* The group is a manifold and therefore is a Lie Group under the same composition.

* It has dimension 3- 1 co-ordinate for the point and 2 co-ordinates for the axis.

*If  α =180 degree, β =γ= 0 degree, rotation is put at x1,y1,z1, then (x,y) co-ordinates are reflected to (-x.-y) while z co-ordinate remains unchanged. If in addition, z1=0, and c33=-1, then the coordinate becomes (-x.-y.-z). But c33=-1 is not a rotation matrix element but reflection matrix.

Norm and Distance in Vectors :

* The distance is a 2-vector function d(x,y) while the norm is a 1-vector function ||v|| . However, we frequently use the norm to calculate the distance by means of difference of 2 vectors, ||y - x || . A norm always induces a distance , but the reverse is not true. A trivial distance has no equivalent norm as d(x,x)=0 and d(x,y)=1 when x is not equal to y.