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‰ 7q qM‚KZÔTUV1`KLaRQTULOF g T@Œ@v'H Š„ TUYaR[yPRSN gp| HURýRQLONS[Y k FOLORSL k L h z k nUTßnUT%THKN k FOHKNSHKFOv=nRS— h TV kXk FWLŒ-v=n nUFâLWZ k ‚€[P(RQN g | H k FWZiHKJ=n gih nUT k FjP[n+[L Š L ŠU„ LOF g T{n€[yP k H0u g Ty[yPRSN gp|k HUNQn gih nUT k FjPU[yPj`nUZÚb – YaRr[yPRQN gp|€k FWHKNXFjnRQP h nUTyn h'h…„‡gÔk HKNQnc`Un h Œ7[ g\h RýnUF‘LOZ k ‚cRQY h%„ LWFO‚ h LWTY k n h7h%„‡g H k FOZ\HKJ=nd[ gih R n{[P k H0u g TRQY h%„ LWFO‚ h LWT-Œ¶Y k PUZpRÃnUZ\P h H k }iRS`n€TUV h7h%„ L h ZžPRSv=nR w ŒÔv'H Š„ L Š„ nUT(PUNXv=P h…„ RSY h…„ LOFW‚ k k FWHUNSFjn(R+L k LORQY h n k FWHKNXFjnRS[P(RQN g |0h nUTùn h'h…„‡grk HKNSn©`n h Œ.[ g\h R+nUFLOZ k ‚©RQY h%„ LWFO‚ h LWTY k n h'h…„‡g H k FOZ\HKJ=n0Œ7[ g\h R¼nUFâ—7RSHKZ kXw RSY h…„ LOFW‚ h LWT-b kXk FOLOx?H Š Z\nUZi`n ´HKNXRSH h HKNX[PUZpRáq'PUF gik q=n h n+[P(RQN g | HKTyTVKFWVRSR g [Ù^7`KLOZ\LORSLWTßnUZ\nUTWoSn_} ŠU„ f

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(A + B)ij = Aij + Bij , (λA)ij = λAij , (AB)ij = Aik Bkj .

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Aij = λδij = λEij ,

`Un ŠU„‡g\k n h…„0† oˆRÃP k HKJ@LWNQP(RQHKNSn+nUFIL ŠU„‡k Y Š HKJ@LWNSPRQHKN λæ k FWHKNXH k n7b  > @"‡#"0)O&(" M"…/Cµ9/ !<¸
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xvi = (vx)vi = vj xj vi = vi vj xj ,

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v12 v1 v2 v1 v3 V = v2 v1 v22 v2 v3  . v3 v1 v3 v2 v32

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T x = xt

T g x?LoXLWFWY kSk LWZ%YWNˆRQLWZi[yLOFWv'LaRQ‚'bBEF e q=PF g\k LWZiLW[ L ŠU„ L h L k NSL j

T ej = ej t = 2(vej )v − ej = 2vj v − ej .

E F]L ŠU„ L h ZiLORSLORánUF]LWZ\‚UFW‚IL k LORQv7LWFv'n k H h Z w n h€k TnZžPUN g\k n hk FWHKNXHKFO`ncnUF e q=PF g\k `KLOT%RSHKNSNSnUZڌ%[L Š æ TnJ‡oX—'Tn v g NQP h…„ `KLOT…RQHKN † L Š„ L h L k NXLd`nUZ w RQ'T%NSVUFWY k HKJ@LWNSPRQHKNSP h nUTn+[PRSN g | P(R i

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Tij = ei T ej = ei 2vj v − ei ej = 2vi vj − δij .

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Tij2 = 4vi vk vk vj − 2vi vk δkj − 2δik 2vk vj + δik δkj ,

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Vij = ei V ej = ei (ej − nj n) = δij − ni nj .

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nU[ g u'L h }\z giw¨k FWLON g\h Rþn T g x?LoXLWFWY kSk LWZÙL ŠU„ LOF g T [L Š Œ k F g\h RQY h TUVUFO`KLaRQZiL h 'Z [yL Š T(nUJ'v=nRXRQ—'T`KHKZ h n€n+[(VPRSx)N g | T =g x?LoXVLWFOY xk YaRjb EF n h HKNS[yPUZ gikS†k }\T%NSn`UnZ w `ULORQ}pRQY k [yPRSN gp| P h nUTßTUHK[JMH h L h'k L g RInUF+HKNˆRQH h HUNS[yPUZiRGq=PUF g\k q'n h NSY k FWZiLORQL k L h x?LWZi}\Nˆ`Un   A

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1 − n21 −n1 n2 −n1 n3 V = −n1 n2 1 − n22 −n2 n3  . −n1 n3 −n2 n3 1 − n23 ;

‰‹Š„ RQLaR k FO‚KZ\L Š L k x `KLOT%RSHKN k }iT‡NSn€`nUZ w RQ'T%NXVKFWVURXR¸oXL

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T x = xt = x − 2(nx)n.

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T ej = ej t = ej − 2(nej )n = ej − 2nj n .

EFLWZ\‚UFW‚ßL k LORQLOT%v'LWFv=n k H h Z w n h nUF€L ŠU„ L h ZiLORSLOR ig k [YOR k T(nUZ\PUN gik n h¡k FWHKNXHKFO`n©nF q=PUF gik `ULWTKæ RQHKNXNQnUZ!Œ7[L Š T(nUJ‡oX—7T{nUF n h HKNX[PZ g\kX†{k }iT%NQn_`nUZ w SR 'T%NSVKFOY k HKJMLONQPRSHKNQP h nUT{n€[yPRQN egp| PR i

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Tij = ei T ej = ei ej − ei 2nj n = δij − 2ni nj .

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nU[ g u'L h }iz g\wdk FOLWN gih R“n T g x?LÌoXLWFOY kXk LWZÔL ŠU„ LWF g Ty[L Š Œ k F g\h RSY h TUVUFO`KLaRQZiL h 'Z [L Š æ TnJ'v=nRXRQ—'T`KHKZ h nyn k }\T%NQ(Tn_`x)nUZ w =RS'TT%NSVKxFOY k [P(RQN g | P h nUTT g x?LÌoXLOFWY k YORWb EF n h HKNS[yPUZ g\kS†€k }\T%NQnâ`nUZ w RQ7T‡NXVKFWY k [yPRQN gp| P h nTÙTUHU[yJ@H h L h'k L g RánUFGHUNXRQH h HKNS[yPUZpR.q=PUF g\k q'n h NSY k FWZiLORQL k L h x?LWZi}\Nˆ`Un   W

i

ij j

1 − 2n21 −2n1 n2 −2n1 n3 T = −2n2 n1 1 − 2n22 −2n2 n3  . −2n3 n1 −2n3 n2 1 − 2n23

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Vij2 = Vik Vkj = (δik − ni nk )(δkj − nk nj )

Yk nUv'H h'h n h

Vij2 = δik δkj − δik nk nj − ni nk δkj + ni nk nk nj = δij − ni nj − ni nj + ni nj = δij − ni nj = Vij Tij2 = Tik Tkj = (δik − 2ni nk )(δkj − 2nk nj )

Tij2 = δik δkj − 2δik nk nj − 2ni nk δkj + 4ni nk nk nj = δij − 2ni nj − 2ni nj + 4ni nj = δij = Eij ,

Un v'HKZ@T g v=n k F h PZiRQ—7T@Œ0v'H ŠU„ [ g `ULWZMnUF n L ŠU„‡k Y Š `KLOT…RQHKN]L Š„0k Y ŠKh…„‡g v'H kXk F †'k P ŠU† Œ'} Š„ n n = 1 b E TUYaRIY k v=PNSHK[u g [L h F g\w q=n h T(nUJ@HURSRJMH h RSNQn7ŒÔL ŠU„ L h L k NSL g Z\Z\LaRS`UL k }iT%NQnRQ7T‡NXVKFW‚7ŒÔ`nUZžnU[ g\h R n k }\T%NSn_Y k L ŠU„ L h L k NXLc`ULORQ}pRQ‚€[yPRQN gp| HKT(nRán+RS—'Zžn1oXu'H h'k P Š n g TUH h TLWNXL k FaRQ'Z;RSLOR k FW‚KZiL Š L k u g [L h æ F g\w NQn+nUFcnUZ\PUq'q g u'L h }\z giw T%T(nUZÔRQLON?oXL k FORSv'LOR?oX'TßT g 4²"…³4Bµ!·…?¹©º T ÕÈU˸ÀaØ 0Ê(Ë˸Ð%Ä(ÀQÝW»(¿€ÕÈU˸ÀÍØ KÉ@Å(ĀÉÔ½aÒU½S»Ã»WÐ%Ä1δ×7Å k k

A

g

QP

QP

T 2 = E.

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P

ÕÈU˸ÀÍØ 0Ê(Ë#Æ@ÀQÊÌÇj½ÃÄ(˝Ê(ÀÍÕÈUË!ÀÍØ KÉ@Å(Ä+ÉÔ½aÒU½S»Ã»WЅÄ1Î4×7Å QP

QP

P 2 = P.

 #*-/W/ @"%2Ô/1*-&C&C:=< 7:=
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Ax = xa = (a × x) .

œh u'L | L k nUZ\nUT%q=n h x?LWZi}\Nˆ`Un nUv'H h'h n h n+`ULWT…RQHUN g PZ g\k“k FWHKNXFjP k AR¼Z\LWx}iN w =[y(aPRQN×gp| x)L ŠU„ =L k TUHKa[J@xH ,h L h7k L gih LWTßnZžnUTWoSn ij j

i

ikj k j

Aij = ikj ak .

E[PRSN g | THK[yJ@H h L h'k L g nUFâHKNXRSH h HKNX[PZiRrq'PUF gik q=n h NSY k FOZ\LORSL k L h x?LOZ\}iNX`n 

 0 −a3 a2 A =  a3 0 −a1  . −a2 a1 0

 ;*-&C>-:7/1$=)€*-,."‡&C$7/j*@&C: › nRQPUNSHKFOFW—'Ty[yL Š nÙx?HKN Š nRÃP k HKJMLONQPRSHKNQP h nUT[PRSN g | P(RrLWZi‚ k FOVKN“TUYaR]u g [L h F g\w q=n h Œ0[nCoXu{RSLWNÌæ oXL kSk FW'TT g LOFOR]v=PNSHK[òu g [L h F g\w NSn7b – YaR]u g [L h F g\w q'n h nUF x `ULWT…RQHKN ϕ k FOV ŠKŠ LOZÔ`nUZ w LOZix?HKN Š n(æ RQHURXR¸oSn ¸u7L h }iz giwk FWLON g\h RGnUF w NSnU[_—7RQnR w oSPNQP k P1`nUZÔLWZ\ZiL h RQYaRQL k¼g NQP h…„ R“RQLOT g\h R¸oX'TJMHKF g RS}i` h nUT b •’

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Gc

A

d

y

xϕ

ϕ

x x

?N oX—'T x?LWZ g\k [yYaRßLWZi‚ k FOVKNßnUFäL Š„ L k q=PUF g\k `KLWT…RSHKNSHKT-ŒInUF x Y k nUF y g NQP h…„‡† L ŠU„‡k Y Š `ULWT…RQHKNXHKT LWZix?HUN Š nRSHURSR?oSn g R i

Cg

y

y

ϕ ϕ x

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e1 ϕ = cos ϕ e1 + sin ϕ e2 ,

e2 ϕ = cos ϕ e2 − sin ϕ e1 ,

[nCoXußx?LÌoXLOFWFO'TßT g LOFWLWT%TLWZ.nUF x `KLWT…RSHKNLOZix?HKN Š nRQHRSR¸oSP(Rjb ™g `ULWZBnx?HKN Š nRQP k Z g\h LjPUN g\k [_^0`KLWZiLORWŒ } ŠU„ nUF `KLWT…RSHKN ϕ k FWV ŠKŠ LWZ `nUZ w LOZix?HKN Š nRSHURSR?oSn x = x e + x e 1 1



LWv'LOZ „ LORSRSL k }iRS`UL

e1 ϕ

Yk

e2 ϕ

nUZ\nUTWoSPR

2 2

xϕ = x 1 e 1 ϕ + x 2 e 2 ϕ .

xϕ = x1 (cos ϕ e1 + sin ϕ e2 ) + x2 (cos ϕ e2 − sin ϕ e1 ),

[nCoXußP(RQNSL h u'LOFO`KLÙn+TnJMHURXR]V kSk FWLax? ŠKŠ Y k R

xϕ = (cos ϕ x1 − sin ϕ x2 )e1 + (sin ϕ x1 + cos ϕ x2 )e2 .

RQ}\Nˆ`n_[PRSN g | nZžnUT%q=n ”

   ϕ  cos ϕ − sin ϕ x1 x1 = , sin ϕ cos ϕ x2 x2

} ŠU„ n ϕ k FWV ŠKŠ LWZ `nUZ w x?HUN Š nRQP k [yPRQN gp| n ϕ

O =



 cos ϕ − sin ϕ . sin ϕ cos ϕ

Eòv=PNSHK[®u g [yL h F giw qMLOZ g x?HKN Š nRQP k L h'h YOZáZ\Y h…„ L Š L k L h qMH h…„ HKZ\—'ZpRÃnUq7q b ‰‹Š„ nUu7HURSR n RSL h'Š LWZ „ TUVKNX'Z g !nùRSL h'Š LWZ „¨g NSP h…„ PUq w Z h YWFa`KLäu'L h }\z g\wÿk FWLON g\h RŽnUF w NQn[_—7RÃn(R w oSPUNSP k Pj`nUZ¼LWZiZ\L h RSYORQL k c

W

z

ϑn

n

y

ϕn x

g NQP h%„ RâRSLWT g\h R¸oX7TþJMHUF g RS}i` h nUT k FOV ŠKŠ LOZ.`UnZ w LWZpx?HKN Š nRÃP k RâZ\L Š L Š„0k FWLONS^'q'q@L h6†'ŠU„ }iNSv=nRS— h T Z\LUŒMv'H ŠU„ nUF n `ULWT…RQHUNXRcLWZi‚ k FOVKN‘ϕq@LOx?HKN Š nR¸oX—7TŽ`nUZ\nU[yLOZ „‡g T-Œ@JMYOZ\u=P—'Z4nUF x RSL h'Š LOZ „©g NQP h%„ PUq=n7Œ LWZi`UY Š LOFWFO'Tßn ϕ k FWV Š ^ßx?HKN Š nRÃP k RjŒ0[n1oXuß` gikSk FWnx?HKN Š nR¸oX—7TnUF n g NSP h…„ PUq'n7b E x?LWZ\nUu=nRÙqMH h%„ HKZ\—7ZiR k P Š n©[ g n(RSR h YWFOFW h T6LWZ\‚Uq'q LWNXNSL{L ŠU„ ` g\k FWH h…„ Zžn Š L ŠU„‡k FWLWNX^¨J@YWZiu=PR •v=nRÃPNSHKFOFW—'Tä[L nUF Š n = ( , 0, ) RSL h'Š LWZ „ TVKNS7Z 60 æT(nUZ4`nUZ w LOZix?HKN Š nRÃP k [PRSN g | P(RjbEGF d

3

Cg

1 2

√ 3 2

0

1” m `KLWT…RSHKNXR y TUVKNX'Z 30 æãTnZ h L Š n(RQ}i` g NQP h%„ q=nx?HKN Š nRS`nq@LOx?HKN Š nR¸oX—7Tän z RQL h7Š LWZ „¡g NQP h%„ PUq=n7Œ n LWZi`UY Š LOFWFO'T¡n 60 x?HUN Š nRQP k RWŒ;[n1oXu¡` gikSk FWnx?HKN Š nRQ— h T-byEGFLOZ\‚KFO‚0LOTänUZ\nUJ‡oSP h O æÚ`UnZ@oXLWZiVKZi`UL nUF x RQL h7Š LWZ „ TUVKNX'Z g α k FWV Š ^LOZix?HKN Š nRQP k RWŒ=n_x?HKN Š n(RÃP k [yPRQN gp| n 
 

 

 

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0

0

α xi

i

EFâL Š„ L k RQL h'Š LWZ „ LOT{TUVKNX'Z g LOZix?HKN Š nRQP k HKT[yPRQN gp| n g 0

0

0

A = Oy30 Oz60 Oy−30 .



•

1 Oxϕ =  0 0

     0 0 cos ϕ 0 sin ϕ cos ϕ − sin ϕ 0 cos ϕ − sin ϕ  , Oyϕ =  0 1 0  , Ozϕ =  sin ϕ cos ϕ 0 . sin ϕ cos ϕ − sin ϕ 0 cos ϕ 0 0 1

Y k FWZiLORSL k L h T g }iNX`n+n k FOHKNSFWnRQHUR

   cos 600 − sin 600 0 cos(−300 ) 0 sin(−300 ) cos 300 0 sin 300 , 0 1 0 0 1 0   sin 600 cos 600 0  A= 0 0 0 0 0 0 1 − sin(−30 ) 0 cos(−30 ) − sin 30 0 cos 30 

qMLOv'LWZ „ LORXRQL k }iRX`KLdnUFâL Š„ L kGk FOV Š x? ŠKŠ `KY h…„ LWTßYWNˆRQYWTYOR

√  1 3 0 12 − 2 2 √ 1 A =  0 1 √0   23 2 0 0 − 12 0 23



√ 3 2

  √3  0 0 − 12 2 0  0 1 √0  , 1 1 0 23 2

LWZi`UY Š LOFO`ULÙn+[yPRQN gp|0k FOHKNSFWP k HKT(nRWŒ'LWZi‚ k FOVKN]n€[P k H0u g TY k v=nUNX[nUu g T[PRSN g | HR    √3 √3 1 − − 0 12 2 √4   4 3 1 A =  0 1 √0   34 − , 2 √4 1 3 1 3 −2 0 2 0 2 2 

√ 3 2

[nCoXu{`UY Š 7Z¶nFcLOZ k ‚[yPRSN gp| HUR“nUFâLWZi‚Kq'qßT(nUJMHRSRÃnZ V kSk FWL k FWHUNSHKFa`Un A=



5  83  √4 3 8

EFIPUZpRÃnUZ\P h H k L k LaRQq@L h ZiL ŠU„ L h nUF n g NSP h…„

− 34

1 √2 3 4

√  3 √8  − 43  . 7 8

n = (sin ϑn cos ϕn , sin ϑn sin ϕn , cos ϑn ).

E v'v'HUFcv7H ŠU„ n+RQL h7Š LWZ „ R¼q@LOx?HKN Š n kXk —'TßnUF z RSL h'Š LWZ „{g NQP h%„ PUq=n7Œ-LOZ\‚ k FWVKN¼n z RSL h'Š LOZ „ TVKNS7ZBLOZ TULWZiZ.x?HKN Š n(R h — h T æ h LWZ!Œ´nß`KLOT%RSHKNÙLOT%TUHKN+nF x, z k }\T%q=nŽTULONS'Z!Œ´[ynCoXu gih'h L h qMLyTULOZ\Z‹[Y Š x?HKN Š n(R h'g n z RSL h'−ϕ Š LOZ „ g NSP h…„ PUq'n7Œ´`n ŠU„‡g\k LWZýTULWZiZx?HUN Š nR h'g nUF y RSL h'Š LOZ „ TUVKNX'Z −ϑ k FOV ŠKŠ LOZ ‰ FOLWTI—7RQP h [yPUN#LWZp`KY Š LWFWv7LOR¸oX7Tdn z RSL h'Š LOZ „ TVKNS'Z g ϕ k FWV Š ^cx?HKN Š nRÃP k RjŒ([n1oXuc` g\kXk FWnx?HKN Š nRQ— h T [ gih u'L h RÙnUF n `KLOT…RQHKN g NQP h…„ PUq=n7ŒB`Un ŠU„‡g\k LWZTULWZiZ`UY Š LOF h  h TÿnUF y TVKNS'Z g ϑ k FWV Š ^ Œ4Y k n z TUVKNX'Z g ϕ x?HKN Š nRÃP k Rjb n Š„0gik n_x?HKN Š n(RÃP k [yPRQN gp| n n

n

n

€–

n

Onϕ = Ozϕn Oyϑn Ozϕ Oy−ϑn Oz−ϕn .

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Y k FWZiLORSL k L h x?LOZ\}\Nˆ`n_n[yPRSN gp| HKT(nRWŒ 

   cos ϕn − sin ϕn 0 cos ϑn 0 sin ϑn cos ϕ − sin ϕ 0 1 0   sin ϕ cos ϕ 0 A =  sin ϕn cos ϕn 0  0 0 0 1 − sin ϑn 0 cos ϑn 0 0 1    cos(−ϑn ) 0 sin(−ϑn ) cos(−ϕn ) − sin(−ϕn ) 0    sin(−ϕn ) cos(−ϕn ) 0 , 0 1 0 − sin(−ϑn ) 0 cos(−ϑn ) 0 0 1

[nCoXu k HKNXq=n h LWZp`KY Š LWFO`UL¼n‘[PRSN g |7k FWHKNXFjP k HKT(nR¶æ#noXHKq'q€PRXRQLWT gih RSv'LORS‚ k Y Š TULWu0`KYWYONXRq@LO`KLOFWLaRS`KL nUF cos ϑn = ct ,

sin ϑn = st ,

cos ϕn = cf ,

sin ϕn = sf ,

oXLWZ\VUZ\Y k LWTLORræLWZi‚ k FOVKN¼nUFâLWZ k ‚€TUYaRY k nUFI—7RQHKZ kSw TUYaR¼[PRSN g | HR

cos ϕ = c,

sin ϕ = s,



   ct cf −sf st cf c −s 0 ct cf ct sf −st cf st sf   s c 0  −sf cf 0  A = c t s f −st 0 ct 0 0 1 st cf st sf ct

[nCoXu{`UY Š 7Z¶n+T(nUJ@HURSR]v'PUNSHK[ [PRSN g | HRrV kXk FWL k FOHKNSHKFa`n{n_RQLOR k FO‚KZ\L Š L k x?HKN Š nRÃP k [PRSN g | n     2 2 s2t cf sf ct st cf st cf 0 10 0 A = c 0 1 0 + (1 − c) s2t cf sf s2t s2f ct st sf  + s  ct −st sf 00 1 ct st cf ct st sf c2t 

 −ct st sf 0 −st cf  . st cf 0

„ nUTUHKNXZžP k TUYOJ'JML h v=nRÃPNSHKFOFW—'Ty[yL Š nÙx?HKN Š n(RÃP k [yPRSN gp| PRý[yP k x?YOZ\LWTYWJ'J@L hßgik Œ7[yP kXg Tq=PUF gik qMLa`KLWFOLORSY k Ya`KLOZÚb H h R k —7T{x?LWZ;LWZi‚ k FOVKN]nUF x `KLOT%RSHKNXR n g NSP h…„‡† Y k næãNSLâ[LWNS‚UZ\L Š L k NSY k FONSLUb

•—



z

x

x

n

x y

x

nUFâLWZi‚KFW‚‡LWTnUZžnUJ%oSP h

x = xk + x⊥ , xk = (xn) n,

x⊥ = x − (xn) n.

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EF xæãNSLI[yLONS‚KZiL Š L k“g NSP h…„ q'n h n€TYORGL ŠU„‡k Y Š `ULWT…RQHKNGZ\L ŠU„ L h v=PUNSHU[ HKNˆRQH h HUNS[yPUZiRáq=PUF gik `ULWT…RQHKN ˜1 = e

x − (xn) n , |x − (xn) n|

e˜2 =

x⊥

g ZiZ\LaRS`KL

n × (x − (xn) n) , |n × (x − (xn) n)|

ªE `KLWT…RSHKNXR n TVKNS'Z;LOZix?HKN Š nRX`UnÙnUF x SN Y k F h LW[ò`UPZiRQHUF g T-Œ0[y} Š RQYWNXqML h LOZix?HKNXu'—'Z!Œ7} Š„ `UY Š 'Z k

x⊥

g NSP h…„0† b Š„ n i

˜3 = n. e

EF x `KLWT…RSHKNnZžnUTWoSn_LWq7qML h n€q=PUF g\k q'n h T g x?LÌoXLOFO`KL

x = |x⊥ | ˜e1 + |xk | ˜e3 .

n × x⊥

nUF

˜2 } {˜ e1 , e

TUYaRQu g [yL h F giwKk

y

e2

ϕ e1

x

˜1 + sin ϕ e ˜2 ) + |xk | e ˜3 . xϕ = |x⊥ | (cos ϕ e

WL v'LOZ „ LORSRSL k }iRS`ULÙnF x `KLOT%RSHKN¼LWFWL h q=PF g\k qMLOZ g TH0HUNSu gih PRÃP g\h nUT€`nUZžn[ gih R“nUFIL ŠU„‡k Y Š `KLOT…RQHKNXHKT nUZžnUTWoSPR 

ϕ

x = |x − (xn) n|



n × (x − (xn) n) x − (xn) n + sin ϕ cos ϕ |x − (xn) n| |n × (x − (xn) n)|



+ (xn) n,

x?LWZ\q@H h RS`nn k FjPZ\PUZ w q'n h n `ULWT…RQHKN g PUZ g\kk FWHKNXFjP k R©Y k T g v=n k F h PUZi`n7ŒGv'H ŠU„ n h LO`ULWFO‚KqML h n `KLWT…RSHKN g PUZ gikrk FOHKNSFWnRGnUq k FOHKZ † R]YWNXRSYWTL |n × (x − (xn) n)| = |n||x − (xn) n| sin 90 Œ 0

xϕ = cos ϕ (x − (xn) n). + sin ϕ n × x + (xn) n.

RQ}\Nˆ`n_[PRSN g | nZžnUT%q=n7Œ‡[H k R][yPUN]nUFâLONSLWu7LOR g q=PF g\k q=n h ”

xϕi = cos ϕ xi + (1 − cos ϕ) xj nj ni + sin ϕ ikj nk xj .

Eªx?HKN Š n(RÃP k [yPRQN gp| P h nTnUZ\nUTWoSn_} ŠU„ `KY Š 'Z

Aij = cos ϕ δij + (1 − cos ϕ) ni nj + sin ϕ ikj nk ,

nU[ g [yL Š L Š„ LWF g TnFcLOZ\VKq7qŽT(nUJ@HURXRÃnUZ!b

b

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a11 a12 ...a1n a21 a22 ...a2n . |A| = det A = . . an1 an2 ...ann

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  +1 = +1   0

@GB I

>A@

= i1 i2 ... in ai1 1 ai2 2 ... ain n ,

×7Å i i ... i 1Å » 1, 2, ..., n Æ7ÈUÀQÊ1ÂrÆ=½aÀÍÕ ‡ËãȇÞaØ ÌÇOÅ ×7Å i i ... i 1Å » 1, 2, ..., n Æ7ÈUÀQÊ1ÂrÆ=½aÀÍÕ ‡ãË È‡ÞaØ ÌÇOÅ ½ÚÑUӅÁ aąÁaÉ=Ë 1 2

n

1 2

n

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