AC 1 = AB + BC + CC 1, DD 1 = AA 1. D 1 C 1 = 1 D 1 F = 1. AF = 1 a + b + ( ( (((

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1 ? / / / o/ / / / o/ / / / , D D 1, E F 1 D 1. = a, D = b, 1 = c. a, b, c : #$ #$ #$ 1) 1 ; : 1)!" ) D 1 ; ) F ; = D, )!" D 1 = D + DD 1, % ) F = D + DD 1 + D 1 F, % 4) EF. 1 = 1, 1 = a + b + c. D = D = b a, DD 1 = 1, D 1 = b a + c. 1 = + + 1, DD 1 = 1. D 1 F = 1 D 1 1 = 1, F = 1 a + b + c. 4) EF = F E = F + E) = F + E) = b + c a 1 b = 1 a + 1 F uuu 6 /. v '? / } ? > D 6 6? D uuu 1 6 > ' uuu > >? > ? > D c b P P P P P P P P E :!F a D &')*+, 1) a + b = a + b ; ) a + b = a b ; ) a b = a b ; 4) a b = a + b. b + c. a, b,-./1? F? D / v u o o b uuuuuuuu o ;: EF a 1) + 1 ) = 1 a + gc b a : 1) :;<=48 >: a / b.?&' a + b = a + b : a b, D a, b EFGH". 1

2 ) I c = a + b, J a = c b, K)L": c b = c + b. #$ b / c?m. N O>P: a / b, M,? a b, D b =. ) I c = a b, J a = b + c, K)L": b + c = b + c. N 1) : b / c?. #$ a / b?. Q! a b, #$ a b, D b =. 4) I c = a b, J a = b + c, K)L": b + c = c b. N ) : c / b?m, D b =, R, c b. #$ a / b?m, D b = D a =.. STUV), WXTVY/RZ*+. 1) b a a b ; ) a + b + c a + b + c. : 1), [\<=48 >P]SUV). V)*+1>^_` a ): a / b,,? a b, D b =. ) I d = b + c. J: a + b + c = a + d a + d = a + b + c a + b + c. VY*+b?cb i) a, b, c de?, D ii) a, b, c EF7f" g>h* i) i j). 4. kl86 D, = a + b, = 4 a b, D = 5 a b a, b mo ). ST D "p6. : D = + + D = 8 a b D p6. =, D /. q D =, #$ 5. DEF "r86, s + + D + E + F. : D = + F = E +, + + D + E + F = D. F E = uuuup mmm m m m m uuuuuuuurp RF m SRSR ;: EF m S RSRS m R SRSR m S m RSRR m S c RSRS m u e K] L[ % PuuuuRZ 5 F D D N GH UZ% V MNMMMNMMMNMMMNMMMNMM << / M uuuuuuuu : o L 6 6. L, M, N ,,. ST4 $u*hf456. :!" L, M, N >$u*hf456, b?cbvw4f L = 1 + ), >P: L + M + N =. 7. k456 sh O, ' M = 1 + ), O + O + O =. L, M, N > 9:"o. N N = 1 + ),

3 : L, J L = 1 + ). kxlyzo, ' O = L = 1 + ), J O k {. }S: O + O + O = O + O + ) + O + ) = O + + = + ) + + =. 8.,,, D Hfllf~, M, N 8, D. ST: MN = 1 D + ). #$ :, M = 1 + ), N = 1 D, MN = N M = 1 D 1 + ) = 1 D + ). X QQRQQ Q QQQQ M \& \&? \& ]& s ssss \' M M M M L M M M M L M M M M L M M D \& Q\& a P b N ` 8 O D7uuuuuuP '^\]\ \ \\ \ ]\ \ M \\ \ ]\ 7 \ \ Puuuuuu 9 #$ 9. M l86 D, O H. ST: O + O + O + OD = 4 OM. :,!" OM = 1 O + O), OM = 1 O + OD), O + O + O + OD = 4 OM. 1. D l86, P, Q 8, D. ST P, Q 5 D e< E, F, %v D 4V. : = a, D = b, J D = D = b a, P = + 1 = a + 1 b, Q = D + 1 D = b + 1 a. Q J q E = k P k > ), E = k a + k b, F = m Q m > ), m F = m b + a. E = + D = a + b a ) = 1 ) a + b > ).

4 #$ : k a + k b = 1 ) a + b, k + 1) a = k ) b, N< a b U, #$, N >P: P: XT E, F x D 4V. { k + 1 = k = { k = = 1. F = + s D = 1 s) b + s b s > ), { m + s 1 = s m =, F = D, : { m = s =. E = 1 D, 11. O "r ST: O 1 + O + + O =. : "6. OR. G: O O =. h "6: H~ 1,, 'Kr 86 1 *": , whf 8 6. #$ O 1 + O 1 + O + O + + O + O =. O i N O i Hf5 θ %P < θ < π), : p = O1 + + O, q = O1 + + O, #$ q N p θ 5%P. N< < θ < π, q p U,! p + q = b?cb p = q =. 1. O "r 86 1, P H. ST: P 1 + P + + P = P O. :!" = 11 a"#). P O = P i + i O i = 1,,, ), P O = P P + 1 O + + O) = P P l86 D 5" : D. = a, D = b. s, D, D. :, = O + O = D = 1 a b ),

5 #$ D = = 1 a b ), D = D = O + OD) = 1 a + b ). D7uuuuuuP SZ% fc D ) '^\]\ 77 \ \\ \ ]\ \ O \\ \ ]\ 7 \ \ Puuuuuu 1 D uuuuuuuu o M. k456, M, N 8y4V. = a, = b. s M, N. :,!" M = 1, N =, M = + M = + 1 = + 1 ) = 1 b + 1 b + N = + N = + = + ) =. kl O, G 456 $. O, O, :, N OG = O + G, G = D, D = 1 + ), >P G = 1 + ). % = O O, = O O, #$ G = 1 O + O O), N OG = 1 O + O + O). a, a. O OG. k = q ql l ll l ll l q%y qqqq ll l+ N N N N M N QRRR M N N N N M! N Q! N S Q" Q" Q" Q" Q" pq" Q" Q" R Q" QRRR G D Q RR O D o o <; G/ ot o o f i uuuuuuuu o 4 4. T 456 < T ), v T, %. : T = k, J T = 1 k). N5&'>, : = k : 1 k),!o +. < T = + T = + k = 1 k) + k = 5. a, b U, J ). c = a + b d = a b )&e*? 5

6 + " : G k m ': k c + m d =, N< a, b U,! a, b &,*, #$ { k + m = c, d &,*. k a + k b + m a m b =, k + m) a + k m) b =. k m = -P k = m =, 6. ST4f k 1 a k b, k b k c, k c k1 a. : NV) >w f k 1 a k b ) + k b k c ) + k c k1 a ) =, &e*, #$. 7. O Hf, ST: <UkH.y f,,, M /< y& 11k k 1, k, k, 'P OM = k 1 O + k O + k O,? k1 + k + k = 1. : 4U,!, &,*. M /< yb?cb M,,, : M,, &e*, b?cb1ku" m 1, m, m, ' m 1M + m + m =, b?cb< O G: b?cb m 1 OM O) + m O O) + m O O) =, m 1OM = m1 + m + m ) O m O mo. m 1, U, &,*45.!O: J? k 1 + k + k = 1. k 1 = 1 m 1 m 1 + m + m ), k = m m 1, k = m m 1, OM = k 1O + ko + ko, 8. O Hf, ST: M /< y 6788) &11k9 k 1, k, k, 'P OM = k 1O + ko + ko,? k1 + k + k = 1. : :; M, > < D.!O M = l D, < l 1. N< D kx y, => j.1, 1k m 1, m, 'P OD = m 1 O + m O, m1 + m = 1, m 1, m. < OM = O + M = 1 l) O + l OD = 1 l) O + lm 1O + lmo. 6

7 I k 1 = 1 l, k = lm 1, k = lm, P JG OM = k 1O + ko + ko, k1 + k + k = 1, k 1, k, k. M9, U? k 1 1, - 1 l = k 1 lm 1 = k lm = k >P l = 1 k 1, m 1 = k 1 k 1, m = k 1 k 1, m 1 + m = 1, m 1, m, < l 1. I OD = m 1O + mo, J D kx y. N OM = 1 l) O + l OD >$P% M = l D,!O M kx D y, %k y. 9. ST: U4,, &11kU"o k 1, k, k, 'P = k 1 O + k O + k O,? k1 + k + k =. :, b?cb l + m = l, m mu"o), b?cb l O O) + m O O) =, b?cb l + m) O + l O + m O =. I k 1 = l + m), k = l, k = m, 8U"o,?: k 1 O + k O + k O =, k1 + k + k =. 1. ST: Ul,,, D &11klfU"o, 'P = k 1 O + k O + k O + k4 OD,? k1 + k + k + k 4 =. : D b?cb,, D &e*, b?cbgu"o l, m, ': l + m + D =, b?cb b?cb l O O) + m O O) + OD O) =, l + m + ) O + l O + m O + OD =. k 1 = l + m + ), k = l, k = m, k 4 =, 8U"o, 'P k 1 O + k O + k O + k4 OD =, k1 + k + k + k 4 =. 7

8 D o oo K o o Do E V o o o uuuuuuuu M o o 5 KLKLKKLKLKLKLKKLKLKLK F DSTEF: ,, GHIJ* F : F = k 1 : k, D : D = k : k 1, E : E = k : k, <, k 1, k.k K"r. J456 ~8 8L<H M,?< H O G 1 OM = k O + k1o + ko). k 1 + k + k < : => D E M>P D = k k 1 + k, D = + D = + = D E < M, JG k 1 k 1 + k + k k 1 + k, NOPP)QR$V): l E = E = M = l D, M = E = k k + k. k k 1 + k = + k k 1 + k ) k k + k. M = m E. + M, P ) k1 k + = + m k 1 + k k 1 + k ) k. k + k N< &,*, NySV)P: lk = mk l = k 1 + k k 1 + k k + k k -P 1 + k + k lk 1 = 1 m m = k + k. k 1 + k k 1 + k + k M = Q D F e< M, >P M = M M $T,!O D, E, F <H M. 8 O, G OM = O + M = O + = O + = k 1 + k k 1 + k + k D. k 1 + k k 1 + k + k D, k 1 + k k 1 + k + k k 1 O O) + k 1 + k + k 1 k O + k1o + ko). k 1 + k + k ) k k + k 1 + k k 1 + k k k 1 + k + k O O)

9 1 1. P, Q 7kUV [O; e 1, e, e ] WU,, 1), 1, 1, ). X% P, Q /Y. : _Z. e.l r~ ~ xxr x P P xx.u ;: EF P,,.l 1) & xx O e 5. 5./.... e1.../ uur.l Q 1, 1, ).l.l.l.l.l.l.l.l.l.l P x xx G G G G D7uuuuuuP '^\]\ \ \\ \ ]\ \ O \\ \ ]\ 7 \ \ Puuuuuu 1. <l86d, s, D, D, DkUV[;, D] WU. : = = 1) + D, WU" 1, ); D WU" 1, 1 ) ; D WU" 1, 1 ) ; D WU", 1). WU. D = D = D, D = D, D = D = + 1) D,. a, b, c WU 1, 5, ),,, 4),,, 1). s a + c, a + b + 4 c : a + c = 1, 5, ) +,, 1) =, 1, 4) +,, 1) =, 1, ). a + b + 4 c = 1, 5, ) +,, 4) + 4,, 1) 4. ST45641<H $). =, 15, 6) +, 6, 8) + 8, 1, 4) = 11, 9, ). : D, E, F 8,, y. D E < G, D F < G. J G = k 1 D = k + 1 ) = k + k. [+\]UV [;, ], J G WU" k, k ). Q G = m 1 E = m + 1 ) [ = m ] ) = m m, #$ G WU" m, m ). q G = + G, #$, k, k ) = 1, ) + m, m ), 9

10 - { k = 1 m k = m P k = m =. #$ G WU" 1, 1 )., >$^P G WU" D o oo o o EV o o D o o uuuuuuuu M o o MNMMMNMMMNMMMNMMMNMM F 4 5. ST456415<H. 1, 1 ). SP G = G. D o oo o E %Y o SSSS od o S SSSS o o uuuuuuuu M S SSSS o o S SS F 5 : 415" D, E : F.? D E < T. I T = k k D = + + ). [+\]UV [;, ],?I = a, = b. J T WU ` #$ E = T = m E = ), = m m a a + a + b. b a N< T = + T, #$ k ) b a + b, k a a + = 1, ) + b : -P: Q D F < T, P T WU" 1 T = F = m a + b a b a a + a b a )) k b a + b k a a + b k = T = s D = s b a + b, s a a + b = 1 m = m, m a a + b a a + b a + b + b a. k b a + b, k ) a a +. _ b m a a + b a s b a + s a a + b a + b. b ). + + b ) = b + a a b, b ),

11 N T = + T, P: s ) b a + b, s a a + ) b =, 1) + b b + a b,, : -P: NO>_ s = k, T = T. s b a + b s a a + b s = = b b + a b = 1 a + b a + b + a b. 6. x I,, ) : D5,, ) 4V, s%wx7fa, WU. : U?,,, D l #., 7WU" x, y, z ) x, y, z ), J = D, D = D. #$ :, x x, y y, z z ) = x D x, y D y, z D z ), x = x x D = 1 y = y y D = z = z z D = 4. x = x D x = 8 y = y D y = 4 z = z D z =.!O, 7WU" 1,, 4) 8, 4, ) 7b>c). 7. 7WU" 1,, ), 4, 1, ). 1) d P WU, ' P x *e : ; ) d P WU, ' P x *e :. : 1) N P : P = : >P P = WU 14 5, 1 5, 1 ). 5 ) N1>P P = uuuuuuuu D 6 P. j.1 ef), z k =, >P P P, ef)gp P WU 1, 7, ). uuuuuuuurp D E F 9 8. D "l86., h5wu", 1, 5),,, 4), 1, 1, ). d, D WU. : 5" M,, D WU" x, y, z ), x D, y D, z D ). N< M,,! O 1 + x ) = y ) = z ) =, -P WU" 5,, 1). N< M g, D, >P D WU", 1, ). 11

12 p p p iv. 9., 7WU" 6, 5, 8), 4,, 7), d, D, E, F, ', D, E, F vx : U?,,, D, E, F. #$ = 1, 4 E = E, D = D, F = 4 F. ef)gp WU" 4, 4, 5), D WU",, ), E WU",, 1), F WU ", 1, 4). 1., O O , P k 1 y,? P = P 1, S k 1 y,? S = 1 S 1, Q, R O 1 1,. ss:. P Q. RS. : [+WUj [O; O, O, OO 1 ].!" P = P 1, #$ P = P 1 = 1, % klm, #$ OP = O + P = O + 1 = O + OO 1, OQ = OO 1 + O 1 Q = OO O, P Q = OQ OP = O + OO O. OR = O + R = + 1 O, OS = O + S = O + OO 1, RS = OS OR = O + OO O. wog P Q = RS, < P Q / RS. l86. J Q Puuuuuuu uuuuuuupq q qq qq O 1 G G G G G G G G G G G G G G G 1 1 P P P P P P P P P P P P P P P P P P P uuuuuuupq qq G G G G G G G G G G G G G G G S P O R 1 1 E ) ))m m lmm ) )*) m ml ) ))) m m P ) f ))* PPPPPPPPPPPPPPPP H G pql86 D, v, D, D h $ee9, STwlfu*Hf :. " E, F, G, H. E = k E = 1, 1 + k EF = F E = 1 klm, N H = k $h G = k D >$P l86. 1 F = k F D = 1 D, 1 + k D. 1 + k HG = 1 D.!O EF = HG, STr EF GH F D

13 1. ST: ll1<h l$),?ovsh1*e" : 1 N~g) 7. : l l~<$l) : [+\]WUj [V ; V, V, V ]. G " $, G 1 " V $, N`a 1 = a: V G = 1 V + V + 1 V ) =, 1, 1 ), V ) = 1 V V + G 1 = zn V G * : 1 M, R, = 1 V + V + V ) = V M = V G = 4 1, 1, 1 V V V ) ). 1 4, 1 4, 1 ), 4 V + 1 G 1 = 4 4, 1 4, 1 ) = V M. 4 #$ M k V G G 1 y., >SP: G, G V : V $. J G G g< M,? V M = 1 4, 1 4, 1 4 ),!O M = M,?sH*e : 1. VX QQRQQ Q QQQQ? s ssss M M M M L M M M M L M M M M L M M G 1 G ` VX QQRQQ QE QQQQ? M M M M L M M M M L M M M M L M M s ssss D ` lue71u", shlu"lv. ST: l 41v<8$,?O$NsH1v*;weV7. : ya[+\]wuj [V ; V, V, V ], M l$, J V M = 1 4, 1 4, 1 ). 4 D, E ", V. J #$ N< D = 1 = 1 V 1 V D = 1 V + 1 V E = 1 V = V, 1 V =, 1 ),.,, 1 ), ED = V D 1 V E =, 1, 1 ). V E ED = 4, 1 4, 1 4 XT$ M k ED y?v ED. >S<8. ) = V M, 1 4 1

14 1. xg y y): : 1) 1 1 = 5. ) cos θ si θ si θ cos θ = ) 1 = 1. 4) =. 1 1 x y z 5) =. 6) z x y y z x = x + y + z xyz.. yd y)-&: { x y + z = 9, x y = 5, 1) ) x 5y + z =, x + y = 1; x + y z = : 1) x = = 1 1 = 1, y = 1 = 1 1 = ) x = = 4 = , y = 1 6 = = 6 7, z = 1 6 = = e 1, e, e "z. 1) ST: a = e1 + e e, b = e 1 e 1 e, c = e 1 + e + 6 e &,*; ) s d = a b + c kz e1, e, e WU; ) s f, ' a + b c + f =. : 1) G x 1, x, x -.&*j) x 1 a + x b + x c =, P*WU6)o x 1 + x x = x 1 x + x = x 1 1x + 6x =, 8j)!Owf{Go-, a, b, c &,*. 1 1 = 5, ) d = a b + c = e 1 + e e ) e 1 e 1 e ) + e 1 + e + 6 e ) = e e + e,! d WU, 17, ). ) f = 1 a b + c ) = 1 6 e e + 7 e ) = e e + 7 e. 4. }s4f a, b, c )? c)v c *<87f &T? c, ~%). 14 1) a 5,, 1), b 1, 4, ), c 1, 1, 5); ) a,, ), b 6, 6, 4), c 1, 1, );

15 ) a 1,, ), b, 4, 6), c 1,, 5). : a""s- x 1 a + x b + x c =. 1) H& <j) {Go-,!K ) H& <j) Go-,!K U. 5x 1 x x = x 1 + 4x x = x 1 + x + 5x =, = 11, 1 5 x 1 + 6x + x = x 1 + 6x x = x 1 + 4x =, =,. "v c * a, b &T, >z x = 1 QR, P x 1 + 6x = 1 x 1 + 6x = 1 x 1 + 4x =, w45,!o c Uc* a, b &T. ) H& <j) Go-,!K x 1 x + x = x 1 4x = x 1 + 6x + 5x =, =,. "v c * a, b &T, >z x = 1 QR, P x 1 x = 1 x 1 4x = x 1 + 6x = 5, w45,!o c Uc* a, b &T. 5. a, b, c WU 1, 1, ),, k, 1), 1, 1 k, k). : b k z/ R, a, b, c? im, k z/ R, a, c? : w f &1<WU)V<, 1 1 k k k = k k + =. 15

16 !Obk = 1DRwf. &' a, c '8e,WU*ej, 1 1 = 1 k = k 1, -P k =.!Ob k = R a, c. 6. e 1, e, e "z. v c)" a, b, c &T? c, J~%P). 1) a = e 1 + e + 4 e, b = e 1 e + e, c = e 1 + e + e, v = 6 e 1 + e + 15 e ; ) a = e 1 e + e, b = e 1 + e + e, c = e e + 5 e, v = e 1 + e. : v = x 1 a + x b + x c, a""-&. 1) " x 1 + x + x = 6 x 1 x + x = 4x 1 + x + x = 15, j) 1 1 =, Uc})G D-P x = k >$P&) v = k) a + k) b + k c, < k " x 1 + x x = ) " x 1 + x + 6x = 1 j) x 1 + x + 5x =, = 6, G-: x 1 = = , x 5 = 6 &)" v = 11 6 a + 16 b c. 7. b a " R, l: = 16 9, x = M 1 1, a, a ), M 1, 1, 1), M, 1, ), M 4 1,, ). : =>^# 4.5, O 4 & a 1 a a 1 a a a = 7a 5a + =, -P a = 1 D 7. { x1 = x x = 1 x, = I => F MST: F α, G 1) α = ; ) 1)α = α; ) k = k); 4) kα = ^% k = D α =. : α = a 1,, a ), J: 16

17 1) α = a 1,, a ) =,, ) =. ) 1)α = 1)a 1,, 1)a ) = a 1,, a ) = α. ) k = k,, k) =,, ) =. 4) kα = ka 1,, ka ) =,, ). α, J1k a i, N ka i = >P k =.. ST: H m6g. : K "Hf, J 1 K. #$ r+ G = K, W? 9 }{{} K.!O K G +. Q! +, K, 1 ", J 1 K, #$ G m < m, +), G m = m K,!G Q K.. ST: 6 *Tu*Hf. a + b, a, b Q : NwfT" K, a 1 + b 1, a + b K, J!"G:[G); a 1 + b 1 ) ± a + b ) = a1 ± a ) + b 1 ± b ) K a 1 + b 1 )a + b ) = a1 a + b 1 b ) + a 1 b + a b 1 ) K!"G: [G); b a + b R, a 1 + b 1 = a 1 + b 1 )a b ) a + b a b = a 1a b 1 b a b + a b 1 a 1 b a b K G*<Dg).!OT K *<Dm, *"Hf. 4. K ", V " K y F pq. ST: #G k K, α, β V, G 1) kα β) = kα kβ; ) α + α + + α = α; }{{} ) α + β = α + γ, J β = γ. : α = a 1,, a ), β = b 1,, b ) a i, b i K). J k K, 1) kα β) = ka 1 b 1,, a b ) = ka 1 b 1 ),, ka b )) = ka 1 kb 1,, ka kb ) = ka 1,, a ) kb 1,, b ) = kα kβ. ) α + + α = a }{{} 1 + a 1, a }{{} + + a,, a }{{} + + a ) = a }{{} 1,, a ) = α. ) γ = c 1,, c ),? α + β = α + γ, : a 1 + b 1,, a + b ) = a 1 + c 1,, a + c ), J G a i + b i = a i + c i, b i = c i, #$ β = γ. 1. v /L: 1 6 1) a = 5 i 6 j + k ; ) b = 1 i 1 k. : 1) a = a = 7 a 7 5 i 6 j + k ). 17

18 ) b = b = 1 b 1 i k ).. xg 5: 1) a = 1,, ), b =, 1, ); ) a =, 1, 1), b = 1, 1, 4). : 1) a = 14, b =, a b = 6, #$ cos a, b = 6 14 = 14 7, a, b = π arc cos ) a = 6, b =, a b = 7, #$ cos a, b = 7 6 = 7 18, a, b = π arc cos s a k e y: 1) a = 1, 1, ), e = 1, 1, 1); ) a =, 1, ), e = 1,, ). : 1) e = e = 1, 1, 1), J e pr e a = Πe a ) e = a e ) e = )! a e =, #$ pr e a =. e = 1, 1, 1) = 1, 1, 1). 4. ST: $, 1, ),, 4, ),,, 1) "~456V456. : =,, ), = 6,, 1), =, 6, 1), =, = = 46, #$ V ST: $,, 1), 7, 6, 9), 9, 1, 5) "~ : = 4, 8, 8), = 6,, 6), = 1 + 4) =, #$ a b ) c = a b c ) H*+? % V)*+1. : V)a c, a a. 7amUV<, J a c, 1k k, ' a = k c. M9 a = k c, 8 = k c b ) c = k c ) b c ) = 8. V)78mV<, JD a, b, c EFGHfo ; D 4f muv<, q a b = b c =, b a, c Kr. 7. s- a x = b x. :!" a b ) x =, 7b!: a) a b, J- x " a b ". ; b) a b =, J x m-. 8. G4f a, b, c 77u* 6 5,? a = 4, b =, c = 6. s a + b + c ;w. : a + b + c = a + b + c ) a + b + c ) = a + b + c + a b + a c + b c = ) cos 6 = 1. #$ a + b + c = a =, b =, a, b = π 6. s a + b a 5 b {. : a + b ) a 5 b ) = 6 a 1 b 11 a b = = Hfl8689; a, b, c, d,?<5de".. ss89;g a, b, c, d Hl86715ged".. 18

19 :, G = + = D + D, D = + D = + D. ${: D = + ) + D) = + + D + D; D = + ) + D) = + D + D ; D = D + D) + D) = D + D + D D D; D = D + D) + D) = D D + D D D. vys 4 )e, >P: N >P N +P QRy)P 4 D = D + D + D + D. + + D + D = + D = D. + D) = D ) D + D = D + D ). 4 D = D + D ) = b + d a c ). wxt5".&1 a + c = b + d, {l868;g*. P v uu D v Iu Z uu [ZZ v Z uu [ZZ v O [ uu ZZZ v uu [ ZZ[ v uu Z ZZ[ Puuuuuu v u - 1 D uuuuuuuu o H , 8y#" H., H. : v H = + k QR H =, P + k =.!O k =, = QR, + P 1 H = ) [ H = ). )) )) ]. 19

20 p p p p p 1. k.5wuj, a, b, c WU, 5, 7),, 4, ), 1,, 4). s a + 4 b 5 c b + c 5. : p = a + 4 b 5 c = 14, 1, 5), q = b + c = 1, 1, ), J p = 446, q = 15, p q =. #$ 1. s $%: cos p, q = ) a =,, 6); ) b =,, 1). : 1) cos α =, cos β =, cos γ = ) cos α = 11 11, cos β = 11 11, cos γ = xgr58 Hf59q5. : [+.5WUj [;, D, 1 ], $5 1 xgoa. 1 = = + D + 1 = 1, 1, 1). a) 1 = + 1 = + 1 = 1,, 1), #$ cos 1, 1 = , D 1, 1, D 1, 5$%g" b) D = D = 1,, 1), #$ cos 1 1 D, D 1, 1, 1, D 1 5g" π. 1 ruuu q q D ruuu 1 q q 1 uuur q q P P P P P P P P P D , D = 1. = 6. Nu&, =, 1, D = π.. O 1 uuuuu O q qqqq J J J J J P P P P PP P P P P P P P P J J J J J puuuuu q qqqq uuuuu J J q qqqq E E E E E E E E N 1 M , ; O O , O = 8, O = 6, OO 1 = 1. P O y,? P = OP, M y,? M = M, N 1 1. s. 1 P. MN #*5.!O : N O, O, OO 1 / " i, j, k, [+.5WUj [O; i, j, k ]. J O 1 = 8 i + k = 8,, 1); OP = 1 O = j =,, ); OM = 8 i + O = 8 i + 4 j = 8, 4, ); ON = 1 O + 6 j + k = 4, 6, 1); 1 P = OP O 1 = 8,, 1); 1

21 #$ 16. MN = ON OM = 4,, 1). cos 1 P, MN = 5 = a, b, c U, %? a x =, b x =, c x =. J x =. S9. :!" a, b, c U, #$8&,*,? x >N8&, : x = k 1 a +k b + k c. < x = k 1 a x ) + k b x ) + k c x ) =, : x =. 17. ST4f a, b, c &1 a a a b a c b a b b b c c a c b c =. c. : a, b, c b?cbgu"o k 1, k, k ': k 1 a + k b + k c =. % k 1 a + k a b ) + k a c ) = k 1 b a ) + k b + k b c ) = k 1 c a ) + k c b ) + k c =, g H& x a + y a b ) + z a c ) = x b a ) + y b + z b c ) = x c a ) + y c b ) + z c = Go- x = k 1, y = k, z = k. =>& 4.1, j) a a b a c b a b b c c a c b c =. M9, a a b a c b a b b c c a c b c =, J ) Go-, " x = k 1, y = k, z = k, I p = k 1 a + k b + k c,. ) : ) p =, a, b, c &,*,. p = k 1 p a + k p b + k p c =, k.5wuj, a, b, c WU1,,1), 1,,), 1,,1), s a + b ) b c ) WU. : a + b = 4,, ), b c =, 4, 1), #$ a + b ) b ) c ) = 4 1, 4 1, 4 4 = 14, 1, 1).. ST a b ) a b. WXTV)R*+. : a b ) = a b = a b si a, b a b, VY*+b?cb si a, b =. : a / b.. a, b 7fdU, ss a b ) a + b ) = a b ), WXT8' M. 1

22 p p : a b ) a + b ) = a a + a b b a b b = a b ). ' M: $ a, b u*hfl86e78, J a b, a + b "Ol8675. y)xt: $5u*l86"Kl s c, ' c a, c b, <, 1) a = i j + k, b = 4 j 5 k ; ) a = i j + k, b = i + j k. : I c = a b. J c a, c b. xgp: 1) c = a b = i + 5 j + 4 k. ) c = a b = i + j + 5 k. )a*+u,h) 5. u = i + j k, v = 8 i 5 j + k. s v 1, v, ' v = v 1 + v, : I = u = 1,, 1), pr u 14 u v = v u ) u = 17,, 1). I 7 v = 17 7 i + j k ), J: v / u,? v 1 u =. 6. u "o, v " H. v1 = v v = 7 11 i + 8 j + k ), 1) ST: v >,H-" v = v 1 + v, <, v 1 u, v / u ; ) ~% v 1, v P). v1 u, v / u. : 1) v G7b-D: v = v 1 + v = v 1 + v, <. v 1 u, v 1 u, v / u, v / u, J v 1 v 1 = v v. q v 1 v 1 ) u =, v v ) u =, #$ v 1 v 1 u, v 1 v 1 / u, u, ^%: v 1 v 1 =, v v =, v 1 = v 1, v = v. ) I v = pr u v = v u ) u = v u ) u, u v1 = v v. J v / u, v 1 u = v u v u ) u u =, v 1 u.?: v = v 1 + v, N 1) wb-,h,!p)" { v = v u u u v1 = v v. ruu uurq q ru uuuuuq u q q q I P J I I I I I P P P P P P D 1 1 uuuuuuq qq , D D 1 /r, P DD 1 y Hf. x %f #d : 1) 1 P 1 ; ) P 1 ; ps ST Y ) 1 1. : [+.5UV [ 1 ; 1 1, 1 D 1, 1 ]. P " DD 1 y H. J: 1 P = 1 D 1 + D 1 P = 1 D 1 + k D 1 D = 1 D 1 + k 1 =, 1, k), 1) 1 P 1 =, 1, k),, 1) = 1,, ) = 1 1. D 1 = 1, 1, 1).

23 D 1 1. ) P = 1 1 P = 1,, 1 k), P 1 = 1,, 1 k),, 1) =, 1, ) = 1 D 1 = ) 1 1 = 1, 1, 1),, 1) = 1, 1, ) = D 1 = xgn a, b #.*l86: 1) a = i + 4 j + k, b = i + j + k ; ) a = i + j + k, b = i j. : 1) a b = i + j 5 k, a b =. #$ a, b.*l86". ) a b = i + j k, a b =, #$ a, b.*l86". 9. a, b, c "77U. ST: a + b + c = b?cb a b = b c = c a. : a + b + c =, JO) b : c / >P a b + c b = $h a c + b c =, a b = b c = c a. M9, p = a + b + c. NySV)>P p b = a b + c b = $h p c = a c + b c =. p, JN p, b $h p, b >P b c, a, b "7U, = a + b, = a +8 b, D = a b ). ST:,, D 4. : &S,, D 4, {ST: D = >. N D = + D = 5 a + 5 b = 5 a + b ) = 5, >P D =, :,, D f O, O, O -. O O + O O + O O =. ss: 1) O, O, O ; ) 4,,. : 1) &S: O, D, O {ST: 1ko >. I p = O O, J p O = p O =, Q! >P O O + O O + O O =, p O = O O) O O O) O =. p, ': p O = p O = p O = p, J O, O, O ; p =, J O O, gg O, O, O. #$ ) &S,, {ST: =.! = O O, = O O, = O O) O O) = O O + O O + O O =,!,, a = p, b = q, c = r, J a, b, c. ST9. : N< a = b = c =,, J a, b, c ; )J, N = >P a = b = c =, g. 1 8

24 1. } ): 1) a = i j k, b = 5 i + 4 j k, c = 11 i k ; ) a = i + j k, b = i 4 j k, c = i 5 j + k ; ) a = i + j + k, b = 4 i 5 j + k, c = 7 i 5 j + 8 k ; 4) a = j + k, b = i j, c = i + j k. : 1) a, 1 b, c ) = 5 4 = 88, #$ a, b, c U ) a, 1 b, c = = 5, #$ a, b, c U. ) a, 1 b, c ) = 4 51 = 4, #$ a, b, c U ) a, 1 b, c ) = 1 =, #$ a, b, c.. xgn a, b, c #.*: 1) a = i + j + k, b = i j + k, c = i + j + k ; ) a = i + k, b = i j + k, c = i + 5 j k ; ) a = i j k, b = i j + k, c = i + j ; 4) a = i, b = i + j ; c = i + j + k. : 1) V = a, 1 1 b, c ) = = 15. ) V = a, 1 b, c ) = 1 1 = 1 = ) V = a, b, c ) = 1 = 1 = ) V = a, 1 b, c ) = = 1.. dl): 1) 1,, 1),, 1, 5), 1,, 1), D, 1, ); ),, 1),,, 1), 1, 4, 5), D,, 4); ) 1,, ),, 5, 1),,, 7), D, 1, ); 4) 1,, 1),, 1, ), 1,, ), D,, 1). : &d,,, D l), {d,, D w4f,, D) )"o. ). #${h 4

25 1) = 1, 1, 6), =,, ), D = 1, 1, 4),,, D) = =, ) = 1,, ), = 4,, 4), D =,, ),,, D) = =, U. ) =,, ), = 1, 4, 1), D = 1, 1, 6),,, D) = =, U ) = 1, 1, 1), =,, ), D = 1,, ),,, D) = 4. d$,,, D "~l: = 45, 1 U. 1) 1,, 1),, 1, 4), 1,, ), D, 1, ); ), 1, 1), 5, 4, 4),,, 1), D4, 1, ); ) 1,, ), 1, 1, ),,, 1), D, 1, ); 4),, 1), 1,, ),,, 1), D1, 1, 1).. : 1) = 1, 1, ), =,, 4), D = 1, 1, ), V = 1 6,, D) = = 1 6. ) =, 5, ). =, 4, ), D =,, 1), V = 1 6,, D) = = 6 = 1. ) =, 1, ), = 1,, ), D =, 1, ), V = 1 6,, D) = = 1 6 = 5. 4) = 1,, 1), =, 4, ), D = 1, 1, ), V = 1 6,, D) = = 5. 5

26 p p 1 Duuuuu 1 H quuuuuq H H 4 puuuuuq 4 H H 4 H H 4 H H 1 4 H H 4 P H H 4 " S" P S" P T" P P P P P P P P P P 4 D S" S" T! S" 4 S" T! S" " , ; D D 1, = 4, D = 1 =. s 1) 1 1 D ; ). D 1 1 D. : D, D, DD 1 / i, j, k, [+.5WUj [D; i, j, k ]. J D 1 =,, ), D =, 4, ), D 1 =, 4, ). 1) 1 1 D d = D 1, D, D 1 ) D D 1 = , 4, 8) = 1 = 8. )!" D 1 / 1, #$ D 1 / 1 D.! D 1 y H 1 D " D 1 1 D. d = D, D, D 1 ) D D 1 = = 16 1 = a = a 1 e1 + b 1 e + c 1 e, b = a e1 + b e + c e, c = a e1 + b e + c e. ss a, a b, 1 b 1 c 1 c ) = a b c a b c e 1, e, e ). : a b = a 1 e1 +b 1 e +c 1 e ) a e1 +b e +c e ) = a 1 b e1 e +a 1 c e1 e +b 1 a e e 1 + b 1 c e e +c 1 a e e 1 +c 1 b e e = a 1 b b 1 a ) e 1 e +a 1 c c 1 a ) e 1 e +b 1 c c 1 b ) e e, a, b, c ) = a b ) c = [a 1 b b 1 a ) e 1 e +a 1 c c 1 a ) e 1 e +b 1 c c 1 b ) e e ] a e1 + b e +c e ) = [a 1 b b 1 a )c ] e 1, e, e ) [a 1 c c 1 a )b ] e 1, e, e )+[b 1 c c 1 b )a ] e 1, e, e ) = a 1 b 1 c 1 a b c a b c e 1, e, e ). {&4y)M4>xgP). 7. ss: a, b, c ) a b c. : a, b, c ) = a b ) c = a b c cos a b, c a b c = a b c si a, b a b c ST5>e6V): : N7a 7.7 : a b c ) + b c a ) + c a b ) =. a b c ) = a c ) b a b ) c,

27 b c a ) = a b ) c b c ) a, c a b ) = b c ) a a c ) b, e P"# ST: pql,,, P &1, 8#,/Y p, b, c ) + a, p, c ) + a, b, p ) a, b, c ) =. :,,, P b?cb P, P, P. a = O, b = O, c = O, p = OP, a, b, c, p! P = a p, P = b p,!o P, P, P b?cb P, P, P ) =, b?cb b?cb b?cb b?cb 1. ST a p, b p, c p ) =, P = c p. a, b p, c p ) p, b p, c p ) =, a, b, c ) a, b, p ) a, p, c ) p, b, c ) =, p, b, c ) + a, p, c ) + a, b, p ) a, b, c ) =. 1) a b ) c d ) = a, b, d ) c a, b, c ) d ; ) a b ) c d ) = a, c, d ) b b, c, d ) a. : 1) a b ) c d ) = c d ) a b ) = [ c a b ) d d a b ) c ] = a, b, d ) c a, b, c ) d. ) a b ) c d ) = [ a c d )] b [ b c d )] a = a, c, d ) b b, c, d ) a. 11. ST lf a, b, c, d 8G b, c, d ) a + c, a, d ) b + a, b, d ) c + b, a, c ) d =. : N= 1 a"#>p 9: ;b<=>t 1. ST 6V): a, b, d ) c a, b, c ) d = a, c, d ) b b, c, d ) a, H? >SP. 1) a b, b c, c a ) = a, b, c ) ; ) b c ) a d ) + c a ) b d ) + a b ) c d ) = a, b, c ) d ; ) a d ) b c ) + b d ) c a ) + c d ) a b ) = ; 4) a d ) b c )+ b d ) c a )+ c d ) a b ) = a b + b c + c a ). : 1) a b, b c, c a ) = [ a b ) b c )] c a ) = [ a, b, c ) b b, c, b ) a ] c a ) = a, b, c ) c, a, b ) = a, b, c ). 7

28 ) b c ) a d ) = a d ) b c ) = b, c, a ) d + b, c, d ) a, c a ) b d ) = b d ) c a ) = c, a, b ) d + c, a, d ) b, a b ) c d ) = c, d, a ) b c, d, b ) a, #$ b c ) a d ) + c a ) b d ) + a b ) c d ) = a, b, c ) d. ) a d ) b c ) = a b a c b d + c d, b d ) c a ) = b c a b c d + a d, c d ) a b ) = a c b c a d + b d, vysv) 7aeJ: a d ) b c ) + b d ) c a ) + c d ) a b ) =. 4) a d ) b c ) = a b a c + b d c d, b d ) c a ) = b c + a b + c d a d, c d ) a b ) = a c + b c + a d b d, vysv) 7aeP: a d ) b c ) + b d ) c a ) c d ) a b ) = a b + b c + c a ). 1. ST< r i i = 1,,, 4), )*+: r 1 r ) r r 4 ) + r 1 r ) r 4 r ) + r 1 r 4 ) r r ) =. : => 8.7, r 1 r ) r r 4 ) = r 1 r ) r r 4 ) r 1 r 4 ) r r ), r 1 r ) r 4 r ) = r 1 r 4 ) r r ) r 1 r ) r r 4 ), r 1 r 4 ) r r ) = r 1 r ) r r 4 ) r 1 r ) r r 4 ), vysv) 7ae P"#. 14. ST a, b, c &1 b c, c a, a b. :!" b c, c a, a b b?cb a b, b c, c a ) =, qn 1 a 1) : a b, b c, c a ) = a, b, c ), #$ a. b. c b?cb a b, b c, c a fa", ),, ). ~ k. 7x 5y 5 = y9, s 456$. : $WU" x, y), J { x = x + x + x = x { x = x y = y + y + y = y, y = y. % x, y ) -. 7x 5y 5 =, QR P 1x 15y 5 =. 8

29 y N I I J I I I J I I I Q M m m m m + P O m m m m m m m + x. k;" l x ygh P. k D, $ P, P "8/V8456 MP : NP. s MN Q. J : $ O "K, $ " x E, [+.5WUj [O; i. j ]. < l ) ) l,,,, P, ), l < < l ). M l, + l ), N + l #$ MN Q WU": x Q = y Q = 4 l. Q ": y = 4 l l 4 < x < l ). 4 l < < l ), ) l. 9