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! =8Suppose there are 3 coins with 6 different numbers. How many possible outcomes due to a toss ? It is 23 * 3! =48Suppose 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 geometry, Euler'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 ê. 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 A†A = 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.