1 1., BD 1 B 1 1 D 1, E F B 1 D 1. B = a, D = b, 1 = c. a, b, c : (1) 1 ; () BD 1 ; () F; D 1 F 1 (4) EF. : (1) B = D, D c b 1 E a B 1 1 = 1, B1 1 = B + B + 1, 1 = a + b + c. () BD 1 = BD + DD 1, BD = D B = b a, BD 1 = b a + c. () F = D + DD 1 + D 1 F, DD 1 = 1, DD 1 = 1. D 1 F = 1 D 1 1 = 1 B,
F = 1 a + b + c. (4) ( EF = F ) E = F ( B + BE) = F ( B + BE) = 1 F B + B = 1 a + b + c 1 1 a b = 1 a + b + c.. BD BD. = a, BD = b. B, D, D. :, D B = O + 1 OB = + 1 DB = 1 ( a b ), D = 1 B = ( a b ), D = 1 D = ( O + OD) = ( a + b ). O B M. B, M,N B. = a, B = b. M, N. :, M = 1 B, N = B, M = + M = + 1 B = + 1 ( 1 B ) = b + a, N = + N = + B = + ( 1 B ) = b + a. 4. L,M,N B B,,B. L, BM, N. N B
1 : L, BM, N,. L = 1 ( B + ), : L + BM + N = 0. 1 BM = ( B + B), 1 N = ( + B), N M N M O B L 4 B L 5 5. O B, : O + OB + O = 0. :, L,BM,N, O. O = L = L, OB = BM, O = N, 4, O + OB + O = ( L + BM + N) = 0. 6. O B, G B. O, OB, O OG. : GO = G + O, GO = GB + BO, GO = G + O. 5 G + GB + G = 0. GO = O + BO + O. OG = 1 ( O + OB + O). 7. BDEF, B + + D + E + F. : D = + F = E+ B, B+ + D+ E+ F = D.
4 F E O G D B B 6 7 8. BD, B = a + b, B = 4 a b, D = 5 a b ( a, b ). BD. : D = B+ B+ D = 8 a b = B, D// B. D = B, BD. 9.,B,,D, M,N B, D. : :, MN = 1 ( D + B). M = 1 ( + B), 1 N = D, MN = N M = 1 D 1 ( + 1 B) = ( D + B). B M N D O D M 9 10 10. M BD, O. : O + OB + O + OD = 4 OM. B
1 5 :, OM = 1 ( O + O), OM = 1 ( OB + OD), O + OB + O + OD = 4OM. 11., a, b? (1) a + b = a + b ; () a + b = a b ; () a b = a b ; (4) a b = a + b. : (1) : a // b. a + b = a + b : a b, a, b 0. () c = a + b, a = c b, : c b = c + b. b // c. : a // b,, a b, b = 0. () c = a b, a = b + c, : b + c = b + c. (1) : b // c. a // b. a b 0, a b, b = 0. (4) c = a b, a = b + c, : b + c = c b. () : c // b, b = 0,, c b. a // b, b = 0 a = 0. 1.,. (1) b a a b ; () a + b + c a + b + c. : (1),. 11(): a // b,, a b, b = 0. b b a a 1(1) () d = b + c. : a + b + c = a + d a + d = a + b + c a + b + c. (i) a, b, c, (ii) a, b, c 0 ( (i) ). 1. O 1 n. : O 1 + O + + O n = 0.
6 : n.. : O 1 + + O n = 0. n : B 1,,B n n 1 n : 1 B 1 B n 1 B n 1 n B n, n. O 1 + OB 1 + O + OB + + O n + OB n = 0. OB i O i π, : n p = O1 + + O n, q = OB1 + + OB n, q p π n. 0 < π n < π, q p, p + q = 0 p = q = 0. 14. O 1 n, P. : P 1 + P + + P n = n PO. : PO = P i + i O (i = 1,,,n), n PO = P1 + + P n + ( 1 O + + n O) = P 1 + + P n ( 1 ). 1. a, b, c = a + b d = a b? : k m : k c + m d = 0, k a + k b + m a m b = 0, (k + m) a + (k m) b = 0. a, b, a, b, k + m = 0 k = m = 0, k m = 0
7 c, d.. a, b,. : p = c 11 a + c1 b, q = c1 a + c b, r = c1 a + c b. x 1 p + x q + x r = (x1 c 11 + x c 1 + x c 1 ) a + (x 1 c 1 + x c + x c ) b. c 11 x 1 + c 1 x + c 1 x = 0 c 1 x 1 + c x + c x = 0, x 1 = k 1, x = k, x = k, k 1 p +k q +k r = (k1 c 11 +k c 1 +k c 1 ) a +(k 1 c 1 +k c +k c ) b = 0. p, q, r,.. k 1 a k b, k b k c, k c k1 a. : (k 1 a k b ) + (k b k c ) + (k c k1 a ) = 0,,. 4. a 1 = b 1 + b b, a = b b, a = b + b. a1, a, a b 1, b, b. : k 1 a1 + k a + k a = k 1 b1 + (k 1 + k + k ) b + ( k 1 k + k ) b. k 1 = 0 k 1 + k + k = 0 k 1 k + k = 0 k 1 = k = k = 0., k 1,k,k k 1,k 1 + k + k, k 1 k + k. a 1, a, a b 1, b, b. a 1, a, a b 1, b, b. 5. D B B, BD = k D. B, D.
8 : BD = D B, D = D. BD = kd, D 1 B = k kd, D = B + k. 1 + k 1 + k B D 5 B T 6 6. T B ( B T ), T B,. : BT = k B, T = (1 k) B., B : = k : (1 k), k = B. B + T = B + BT = B + kb = (1 k) B + k 1 = ( B + B ). B + 7 OB, B, D OB : 1, D O E. O = a, OB = b. (1) a, b O D; () OE : O. O E D 7 B : (1) B, B = B = ( a b ). OD = OB = 1 b, DB = OB = 1 b, D = DB + B = 5 a b. O = OB + B = a b.
9 () OE = ko = k a, OE = OD + md k a = b + ( m a 5 ) b. k = 4, OE : O = 4 : 5. 5 8. B, M B : 1, N :. M BN P, P B Q. B, P Q. : M = B, N =, BN = N B = 5 5 B, M = M = B. P = km, BP = mbn. P = B + BP ( k B ) = ( B + m ) B. 5 k =. P = + ( P = + ) B = 4 B + 1. 9 Q B P, Q = l P = B + sb. ) ( 4 l 9 B + 1 l = 9, Q = 4 B +. 7 7 7 = B + s( B). D Q N Q F P P E B M B 8 9 9. BD, P,Q B,D. P,Q BD E,F, BD. : B = a, D = b, BD = D B = b a,
10 : P = 1 B + Q = E = k P (k > 0), E = k a + k b, B = a + 1 b, 1 D + D = b + 1 a. F = m Q (m > 0), m F = m b + a. E = B + tbd = a + t( b a ) = (1 t) a + t b (t > 0). k a + k b = (1 t) a + t b, (k + t 1) a = ( t k ) b, a b, k + t 1 = 0 t k = 0 k = t = 1., F = B + sbd = (1 s) b + s b (s > 0), : m + s 1 = 0 s m = 0, : m = s =. : BF = BD, BE = 1 BD, E,F BD. 10. O, :,B,, M B k 1,k,k, OM = k 1 O + k OB + k O, k1 + k + k = 1.
11 : B, B,. M B M, B,, : M, B,, 0 m 1,m,m, O : m 1 M + m B + m = 0, m 1 ( OM O) + m ( OB O) + m ( O O) = 0, m 1 OM = (m1 + m + m ) O m OB m O. m 1 0, B,. : k 1 = 1 m 1 (m 1 + m + m ), k = m m 1, k = m m 1, OM = k 1O + kob + ko, k 1 + k + k = 1. 11. O, : M B ( ) k 1,k,k, OM = k 1O + kob + ko, k1 + k + k = 1. : M, B D. M = l D, 0 l 1. D B,.1, m 1,m, OD = m 1OB + mo, m1 + m = 1,m 1,m 0. OM = O + M = (1 l) O + l OD = (1 l) O + lm 1OB + lmo. k 1 = 1 l, k = lm 1, k = lm, OM = k 1 O + k OB + k O, k1 + k + k = 1, k 1,k,k 0.
1, k 1 1, 1 l = k 1 lm 1 = k lm = k l = 1 k 1, m 1 = k, 1 k 1 m = k, 1 k 1 m 1 + m = 1, m 1,m 0, 0 < l 1. OD = m 1OB + mo, D B. OM = (1 l) O + l OD M = l D, M D, B. 1. :,B, k 1,k,k, 0 = k 1 O + k OB + k O, k1 + k + k = 0. : B, l B + m = 0 (l,m ), l( OB O) + m( O O) = 0, (l + m) O + l OB + m O = 0. k 1 = (l + m), k = l, k = m,, : k 1 O + k OB + k O = 0, k1 + k + k = 0. 1. :,B,,D, 0 = k 1 O + k OB + k O + k4 OD, k1 + k + k + k 4 = 0. : B D B,, D, l,m,n : l B + m + n D = 0,
1 l( OB O) + m( O O) + n( OD O) = 0, (l + m + n) O + l OB + m O + n OD = 0. k 1 = (l + m + n), k = l, k = m, k 4 = n,, k 1 O + k OB + k O + k4 OD = 0, k1 + k + k + k 4 = 0. 14. : B B, B, F : FB = k 1 : k, BD : D = k : k 1, E : E = k : k,, k 1,k.k. B M, O OM = 1 (k O + k1ob + ko). k 1 + k + k E D M F B 11 : D E BD = k k 1 + k B, E = k k + k. D = k k B + BD = B + B = B + ( B) k 1 + k k 1 + k = k 1 k 1 + k B + k k 1 + k, D BE M, BE = E k B = B. k + k M = l D, BM = m BE.
14 : M = B + BM, ( ) k1 k l B + = ( B + m k 1 + k k 1 + k B, : k k + k ) B. lk = mk k 1 + k k + k lk 1 = 1 m k 1 + k l = k 1 + k k 1 + k + k m = k + k. k 1 + k + k M = k 1 + k k 1 + k + k D. D F M, M = k 1 + k k 1 + k + k D, M M, D,BE,F M. O, OM = O + M = O + k 1 + k k 1 + k + k = k 1 O + ( OB k 1 + k + k = 1 (k O + k1ob + ko). k 1 + k + k ( ) k k B + k 1 + k k 1 + k k O) + ( O O) k 1 + k + k 1. P, Q [O; e 1, e, e ] (,,1), ( 1, 1,). P,Q. :.
15 Q( 1, 1, ) D e O P(,, 1) O e B e1. 1. BD,,D, D, DB [;, BD] : = = ( 1) + 0 BD, ( 1,0); D D = 1 + 1 BD = 1 + 1 BD, ( 1, 1 ) ; ( 1 D, 1 ) ; DB (0, 1). D = 1 + 1 BD, DB = BD = 0 + ( 1) BD,. a, b, c (1,5,), (0,,4), (,, 1). a + c, a + b + 4 c. : a + c = (1,5,) + (,, 1) = (,10,4) + (,, 1) = (0,1,). a + b + 4 c = (1,5,) + (0,,4) + 4(,, 1) = (, 15, 6) + (0, 6,8) + ( 8,1, 4) = ( 11, 9, ). 4. B (1,,), (4,1,). (1) P, P B : ; () P, P B :.
16 : (1) P : PB = : P = PB..1, k = ( 14, P 5, 1 5, 1 ). 5 () BP = P, P (10,7,0). 5 (1, 1), B( 4,5), B = 5 B.. : (1) = 5 1 B, B + B = 5B. B = B, B 4 1. 4 x B = 1 + 1x 4 1 + 1 4 y B = 1 + 1y : ( 4,9). 4 1 + 1 4 () = 5 B, = 5B. x = x + 5x B 1 + 5 y = y : (6, 1). + 5y B 1 + 5 6. B (,0,) D(5,,0),,B. :,B,,D,,B (x,y,z ) (x B,y B,z B ), = D, D = DB. (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 B = x D x = 8 y B = y D y = 4 z B = z D z =.
17,B ( 1,,4) (8, 4, ) (). D B 6 D E F B 7 7.,B ( 6,5, 8), (4,0,7),,D,E,F,,D,E,F B. :,B,,D,E,F. = 1 B, D = DB, 4 E = EB, F = 4 FB. ( 4,4, 5), D (,, ), E (0,,1), F (,1,4). 8. BD., B (,1,5), (,,4), (1, 1,)., D. : M,,D (x,y,z ),(x D,y D,z D ). M,, 1 ( + x ) = 1 1 (1 + y ) = 1 1 (5 + z ) =, (5,, 1). M B,D, D (0,1,0). 9. (). : D,E,F B,,B. D BE G, D F G. G = k ( 1 D = k B + 1 ) = k B + k. [; ( k B, ], G, k ). BG = m ( 1 BE = m B + 1 ) B [ = m 1 B + 1 ( ] B)
18 m B, = m ( BG m, m ). G = B + BG,, ( k, k ) ( = (1,0) + m, m ), k = 1 m k = m k = m = ( 1. G, 1 )., G ( 1, 1 ). G = G. E D E D M F B 9 M F B 10 10.. : B D,BE F. D BE T. T = k D = k B + ( B + B ). [; B, ], B = ( a, = b. T k ) b a + b, k a a +. b BE = BT = m BE = 1 B + ( B B + B B), B m a + b a ( b a a + a ( b a ))
19 = m m a a + a + b. b a T = B + BT, ( k ) ( b a + b, k a a + m ) a = (1,0) + m, b a + b, a : : k b a + b = 1 m k a a + b = m a a + b a k = D F T, a + b a + b + b a. T = s s b D = a + s a a + b a + b. b ( s ) b T a + b, s a a +. b T = t t F = + ( t b B + B) = B b + a a t b, b T = + T, : ( s ) ( b a + b, s a a + t ) b = (0,1) + b b + a b, t, : : s b a + b = t b b + a b s a a + b = 1 t s = a + b a + b + a b.
0 s = k, T = T. 11 B, P P = k B + mt, k,m,t k + m = 1, k 1, m 1, 1 t 1. P B, k,m,t? : [; B, ]. (0,0), B(1,0), (0,1), P(k,mt). P B, k = 1, mt = 1. k + m = 1, m =, t = 1. 1., OB O 1 B 1 1, P 1, P = P 1, S 1, S = 1 S 1, Q,R O 1 1,B. : PQ RS. 1 P O 1 Q 1 O R 1 B B1 S B E H 1 : [O; O, O, OO1 ]. P = P 1, P = P 1 = 1, OP = O + P = O + 1 = O + OO 1, OQ = OO 1 + O 1 Q = OO 1 + 1 O, F G D, PQ = OQ OP = O + OO 1 + 1 O. OR = O + R = 1 B + OS = O + S = O, O + OO 1, RS = OS OR = O + OO 1 + 1 O.
1 PQ = RS, PQ// RS. 1. BD, B,D,D B,. :. E,F,G,H. E = k 1 EB = B, 1 + k F = k FD = 1 D, 1 + k EF = F E = 1 BD. 1 + k, H = k 1 B G = kd HG = BD. EF = HG, EFGH. 14. : ( ), : 1 ( ). ( : ) : [V ; V, V B, V ]. G B, G1 V B, 1 : V G = 1 ( V + ( 1 V B + V ) =, 1, 1 ), G 1 = 1 ( 1 B + + V ) = ( V B V + V V V ) = 1 ( V + ( V B + V ) = 1, 1, 1 ). V G : 1 M,, V M = V G = 4 V + G 1 = 4 ( 1 4, 1 4, 1 ), 4 ( 1 4, 1 4, 1 ) = V M. 4 M V G G 1., : G,G V B V. G BG M, V ( 1 M = 4, 1 4, 1 ), 4 M = M, : 1.
V V E G 1 G D B 14 B 15 15.,. :,. : [V ; V, V B, V ], M, ( 1 V M = 4, 1 4, 1 ). D,E B,V. 4 D = 1 B = 1 V B 1 V, V D = 1 V + 1 ( 1 V B =, 1 ),0. V E = 1 ( V = 0,0, 1 ), ED = V D ( 1 V E =, 1 ), 1. V E + 1 ED = ( 1 4, 1 4, 1 ) = V M, 4 M ED ED.. 4 1. : 1 1 : (1) = 5. () cos θ sinθ sinθ cos θ = 1.
4 1 1 1 1 () 0 1 0 = 1. (4) 4 =. 0 0 1 4 9 16 1 1 x y z (5) 1 1 5 =. (6) z x y = x + y + z xyz. y z x. : x y = 5, x y + z = 9, (1) () x 5y + z =, x + y = 1; x + y z = 6. 5 5 1 : (1) x = = 1 1 = 1, y = 1 = 1 1 = 1. 9 1 9 5 1 1 6 () x = = 4 1 49 = 6 7, y = 1 6 = 4 1 49 = 5 1 5 1 1 1 1 9 5 6 7, z = 1 6 = 189 1 49 = 7 7. 5 1 1. e 1, e, e. (1) : a = e 1 + e e, b = e 1 e 10 e, c = e 1 + e + 6 e ; () d = a b + c e 1, e, e ; () f, a + b c + f = 0. : (1) x 1,x,x x 1 a + x b + x c = 0,
4 x 1 + x x = 0 x 1 x + x = 0 x 1 10x + 6x = 0, 1 1 1 10 6 = 5 0,, a, b, c. () d = a b + c = ( e 1 + e e ) ( e 1 e 10 e ) + ( e 1 + e + 6 e ) = e 1 + 17 e + e, d (,17,). () f = 1 ( a b + c ) = 1 ( 6 e 1 +15 e +7 e ) = e 1 +5 e + 7 e. 4. a, b, c? c?,. (1) a (5,,1), b ( 1,4,), c ( 1, 1,5); () a (,,), b (6,6,4), c (1, 1,0); () a (1,, ), b (, 4,6), c (1,0,5). : x 1 a + x b + x c = 0. (1),. () 5x 1 x x = 0 x 1 + 4x x = 0 x 1 + x + 5x = 0, 5 1 1 4 1 = 11 0, 1 5 x 1 + 6x + x = 0 x 1 + 6x x = 0 x 1 + 4x = 0,
4 5 6 1 6 1 = 0, 4 0,. c a, b, x = 1, x 1 + 6x = 1 x 1 + 6x = 1 x 1 + 4x = 0,, c a, b. () x 1 x + x = 0 x 1 4x = 0 x 1 + 6x + 5x = 0, 1 1 4 0 6 5 = 0,,. c a, b, x = 1, x 1 x = 1 x 1 4x = 0 x 1 + 6x = 5,, c a, b. 5. a, b, c (1, 1,), (,k,1), (1,1 k,k). : k, a, b, c?, k, a, c? : 0, 1 1 k 1 = k k + = 0. 1 1 k k
6 k = 1. a, c, 1 1 = 1 k 1 = k, k =. k = a, c. 6. e 1, e, e. v a, b, c?,. (1) a = e 1 + e + 4 e, b = e 1 e + e, c = e 1 + e + e, v = 6 e1 + e + 15 e ; () a = e 1 e + e, b = e 1 + e + e, c = e 1 + 6 e + 5 e, v = e1 + e. : v = x 1 a + x b + x c,. (1) x 1 + x + x = 6 x 1 x + x = 4x 1 + x + x = 15, 1 1 1 1 1 = 0,. 4 1 x 1 = x x = 1 x = k v = ( k) a + ( x, k) b + k c, k. x 1 + x x = () x 1 + x + 6x = 1 x 1 + x + 5x = 0, 1 1 1 6 = 6, 5 : x 1 = 1 1 1 6 0 5 6 = 11 6, x = 1 1 1 6 0 5 6 = 16 9,
5 n 7 1 1 1 0 x = 6 v = 11 16 19 a + b c. 6 9 6 7. a, : = 19 6. M 1 (1,a,a ), M (1, 1,1), M (,1, ), M 4 ( 1,,). : 4.5, 4 1 1 1 1 1 1 a 1 a a = 7a 5a + = 0, 1 a a a a = 1 7. 5 n 1. n : n α, (1) 0α = 0; () ( 1)α = α; () k0 = 0 ( k); (4) kα = 0 k = 0 α = 0. : α = (a 1,,a n ), : (1) 0α = (0a 1,,0a n ) = (0,,0) = 0. () ( 1)α = (( 1)a 1,,( 1)a n ) = ( a 1,, a n ) = α. () k0 = (k0,,k0) = (0,,0) = 0. (4) kα = (ka 1,,ka n ) = (0,,0). α 0, a i 0, ka i = 0 k = 0.. :. : K, 1 K. n n = 1 } + {{ + 1 } K, n n K. K. n n 0, n K, 1 n n, 1 n K, m n ( m,n ), m = m n K, Q K. n
8. : a + b, a,b Q. : K, a 1 + b 1,a + b K, (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 ( ); a + b 0, 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 (). K,. 4. K, V K n. : k K, α,β V, (1) k(α β) = kα kβ; () } α + α + {{ + α } = nα; n () α + β = α + γ, β = γ. : α = (a 1,,a n ), β = (b 1,,b n ) (a i,b i K). k K, (1) k(α β) = k(a 1 b 1,,a n b n ) = (k(a 1 b 1 ),,k(a n b n )) = (ka 1 kb 1,,ka n kb n ) = k(a 1,,a n ) k(b 1,,b n ) = kα kβ. () α } + {{ + α } = (a 1 + a 1,a }{{} + + a,,a }{{} n + + a n ) = (na }{{} 1, n n n n,na n ) = nα. () γ = (c 1,,c n ), α + β = α+γ, : (a 1 + b 1,,a n + b n ) = (a 1 + c 1,,a n + c n ), a i + b i = a i + c i, b i = c i, β = γ. 1. : 6 (1) a = 5 i 6 j + k ; () b = 1 1 i k.
6 9 : (1) a 70 a 0 = a = 70 (5 i 6 j + k ). () b 1 b 0 = b = 1 ( i k ).. : (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 arccos 14 7. () a = 6, b =, a b = 7, cos a, b = 7 6 = 7 18, a, b = π arccos 7 18.. a e 0 : (1) a = (1, 1,), e = (1,1,1); () a = (,1,), e = (1,,0). : (1) e e 0 = e = (1,1,1), pr e 0 a = (Π e 0 a ) e 0 = ( a e 0 ) e 0 = e 0 = (1,1,1) = (1,1,1). () a e = 0, pr e 0 a = 0. 4. : (, 1,), B(0, 4,), (,,1). : B = (,,0), = ( 6,, 1), B = (,6, 1), B =, = B = 46 B, B. 5. : (,,1), B(7,6,9), (9,1, 5). : B = (4,8,8), = (6,, 6), B = 1( + 4) = 0, B. 6. a, b, c 60, a = 4, b =, c = 6. a + b + c. : a + b + c = ( a + b + c ) ( a + b + c ) = a + b + c + a b + a c + b c = 16 + 4 + 6 + (8 + 4 + 1)cos 60 = 100. a + b + c = 10.
0. 7. a =, b =, a, b = π 6. a + b a 5 b : ( a + b ) ( a 5 b ) = 6 a 10 b 11 a b = 54 40 11 = 14. 8., a, b, c (,5,7), (0,4,), ( 1,, 4). a + 4 b 5 c b + c. : p = a + 4 b 5 c = (14,1,5), q = b + c = ( 1,10,), p = 446, q = 105, p q = 0. cos p, q = 151 6180. 180915 9. : (1) a = (,, 6); () b = (,, 10). : (1) cos α = 6, cos β =, cos γ = 7 7 7. () cos α = 11 11, cos β = 11 11, cos γ = 10. 11 11 10 a = (1,,4), b = (1,1,1), c = b k a (k ). (1) k c a ; () a, c d. : (1), c a a c = 0 a b k a = 0. a b = 7, a = 1, k = 1. () d a, c d a = d c = 0, c = b k a, d a = d b = 0. d = (x,y,z), d a = x + y + 4z = 0, d b = x + y + z = 0. x = z,y = z, d = (k, k,k) (k ). 11 a, b, s,t, s a + t b = t a s b,
6 1 a, b. : s a + t b = t a s b (s a + t b ) = (t a s b ). s + t + st a b = t + s st a b. st a b = 0. st 0, a b = 0, π. 1., OB O 1 1 B 1 1, O = 8, O = 6, OO 1 = 1. P O, P = OP, M B, M = MB, N B 1 1. 1 P MN. : O, O, OO1 i, j, k, [O; i, j, k ]. O 1 = 8 i + k = (8,0,1); OP = 1 O = j = (0,,0); OM = 8 i + O = 8 i + 4 j = (8,4,0); ON = 1 O + 6 j + k = (4,6,1); 1 P = OP O1 = ( 8,, 1); MN = ON OM = ( 4,,1). cos 1 P, MN = 5 = 5 161. 69 1 69 D 1 1 1 O 1 O P B 1 N 1 1 D B1 M 1 B B 1 1.. : [; B, D, 1 ], 1. 1 = B + B + 1 = B + D + 1 = (1,1,1).
(a) B 1 = B + BB1 = B + 1 = (1,0,1), cos 1, B 1 = 6 =., 1 1 1, D 1, B 1, D 1, 6. (b) BD = D B = ( 1,0,1), cos 1, BD = 0 = 0, 1, BD = π.. 1 B 1 D, D 1, B 1, B 1, D 1 π. 14. ( a b ) c = a ( b c )?. : c, a. 0, a c, k 0, a = k c. a = k c, = k( c b ) c = (k c )( b c ) =. 0, a, b, c ; 0, a b = b c = 0, b a, c. 15. a x = b x. : ( a b ) x = 0, : (a) a b 0, x a b ; (b) a b = 0, x. 16. a, b, c, a x = 0, b x = 0, c x = 0. x = 0.. : a, b, c,, x, : x = k 1 a +k b +k c. x = k 1 ( a x )+k ( b x )+k ( c x ) = 0, : x = 0. 17. a, b, c a a a b a c b a b b b c = 0. c a c b c c : a, b, c k 1,k,k : k 1 a + k b + k c = 0. k 1 a + k ( a b ) + k ( a c ) = 0 k 1 ( b a ) + k b + k ( b c ) = 0 k 1 ( c a ) + k ( c b ) + k c = 0,
6 x a + y( a b ) + z( a c ) = 0 x( b a ) + y b + z( b c ) = 0 x( c a ) + y( c b ) + z c = 0 ( ) x = k 1, y = k, z = k. 4.1, a a b a c b a b b c = 0. c a c b c, a a b a c b a b b c c a c b c = 0, ( ), x = k 1, y = k, z = k, p = k 1 a + k b + k c,. ( ) : p = k 1 p a + k p b + k p c = 0, p = 0, a, b, c,. 18. B, B H. B, H. B H 18 : H = B + k B H B = 0, B B + k B = 0. k = B B, B H = ( B B) B B. B
4 B = B, H = 1 ( B) [( ( B)) B ( B ( B)) ]. 19 BD, B = a, B = b, D = c, D = d., a b = b c = c d = d a, BD?? : a + b + c + d = 0, a + b = ( c + d ) a + d = ( b + c ) : a + b + a b = c + d + c d a + d + a d = b + c + b c a + b = c + d, a + d = b + c. a = c, b = d. B = D, B = D, BD, a + c = 0, b + d = 0. a b = b c = a b 0 = a b = b c = c d = d a. BD. 0. a,b,c,d,. a,b,c,d. :, : = B + B = D + D, BD = B + D = B + D. BD = ( B+ B) ( B+ D) = B + B B+ B D+ B D; BD = ( B+ B) ( B+ D) = B + B D B B+ D B; BD = ( D+ D) ( B+ D) = D + D B+ D D B D; BD = ( D+ D) ( B+ D) = D B D+ B D D D. 4, : 4 BD = D + B B D + B D + D B.
6 5 B + B + D + D = 0 B + D = D B. ( B + D) = ( D B) D + B B D = ( B D + D B). 4 BD = ( D + B B D ) = (b + d a c ). a + c = b + d,. D Q O B 0 B P 1 1 B. B = a, = b, B = c,, r. PQ BP Q : (1), (). PQ, a,b,c r. : PQ, Q = P, Q = P = r. BP = B + P, Q = + Q, BP Q = ( B + P) ( + Q) = B + ( B) P r = B + B P r.
6 (1) BP Q B P, P = k B, k > 0; () BP Q B P, P = m B, m > 0. P = r, k = m = r a, PQ = P. : (1) r PQ = B, a cbcos B + ar r = b + c a + ar r ; () r PQ = B, a cbcos B ar r = b + c a ar r. S = { x x + a x 1}, a. : x, y S 0 t 1, t x + (1 t) y S. : x + a ( ) x 1 x a a + 1 + 4. a r = 1 + 4, x S a x, r (). 0 t 1 x, y S, t x + (1 t) a y + a r = 1 + 4. t x + (1 t) ( a ) ( y + = t x a ) + + (1 t) y a + t a x + + (1 t) a y + tr + (1 t)r = r. t x + (1 t) y S. 1 4, H 1,H,H,H 4 4, 4 1, 4 1, 1. : H 1,H,H,H 4. (: ) : O. r, O i = r. 4, OH 1 = O + O + O 4, H 1 4 = ( OH 1 O )( O 4 O ) = ( O 4 + O )( O 4 O ) = 0.
6 7, H 1 4 = 4 H 1 = 0, H 1 4.. OH i = ( O 1 + O + O + O 4 ) O i, (i = 1,,,4) OH 0 = 4 O i,, OH i OH 0 = O i = r. i=1 H 1,H,H,H 4 H 0 r. 4 0 < a < 1, 0 < b < 1. : a + b + (1 a) + b + a + (1 b) + (1 a) + (1 b). : [O; i, j ]. (0,0), B(1,0), (1,1), D(0,1), 1. P(a,b), P BD, P = a + b, PB = (1 a) + b, P = (1 a) + (1 b), PD = a + (1 b). P + P =, PB + PD BD =,. 5 P 1,P,,P 6 O 1 6. : OP 1, OP,, OP 6 OP i OP j (1 i j 6) OP i + OP j. : a ij OP i OP j ( π), a ij = P i OP j π. a 0 = min{a ij 1 i j 6}, 0 < a 0 π. OP 1 OP,, OP 6, a 1 > π, a > π,..., a 16 > π, π = a 1 + + a 61 > 6 π = π,. a 1 = P 1 OP π. OP 1 + OP = + cos a 1 + 1 =. OP 1 + OP. 6 a,b,c : a + b + c = 0, a + b + c = 1. r i = (x i,y i,z i ) (i = 1,,6), {x i,y i,z i } = {a,b,c}. r i r j, ri r j 1. : p = (1,1,1). r i (1 i 6), r i p = 0. r 1,, r 6 p,. r i = 1 (1 i 6), r1,, r 6 6. 5,, r i r j, r i + r j. r i + r j + ( r i r j ), r i r j 1.
8 7 1., a, b, c (1,0,1), (1,,0), ( 1,,1), ( a + b ) ( b c ). : a + b = (4,,), b c = (, 4, 1), ( ) ( a + b ) ( b c ) = 4 1, 4 1, 4 4 = (14,10, 1).. ( a b ) a b.. : ( a b ) = a b = a b sin a, b a b, sin a, b = 0. : a // b.. a, b, ( a b ) ( a + b ) = ( a b ),. : ( a b ) ( a + b ) = a a + a b b a b b = ( a b ). : a, b, a b, a + b. :. 4. c, c a, c b,, (1) a = i j + k, b = 4 j 5 k ; () a = i j + k, b = i + j k. : c = a b. c a, c b. : (1) c = a b = i + 5 j + 4 k. () c = a b = i + j + 5 k. ( ) 5. a, b : (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 = 0. a, b 0. () a b = i + j k, a b =, a, b.
7 9 B D P B 1 1 D 1 1 6 6., BD 1 B 1 1 D 1, P DD 1. 1 B 1. : (1) 1 P 1 ; () P 1 ; () 1 1. : [ 1 ; 1 B 1, 1 D 1, 1 ]. P DD 1. : 1 P = 1 D 1 + D 1 P = 1 D 1 + k D 1 D = 1 D 1 + k 1 = (0,1,k), 1 = (1,1,1). (1) 1 P () P = 1 1 = (0,1,k) (0,0,1) = (1,0,0) = 1 B 1. 1 P = (1,0,1 k), P 1 = (1,0,1 k) (0,0,1) = (0, 1,0) = 1 D 1 = D 1 1. () 1 1 = (1,1,1) (0,0,1) = (1, 1,0) = 1 B 1 1 D 1 = 1. 7. u = i + j k, v = 8 i 5 j + k. v 1, v, v = v1 + v, v 1 u, v // u. : u u 0 = u = 1 (,, 1), pr 14 u 0 v = ( v u 0 ) u 0 = 17 (,, 1). 7 v = 17 7 ( i + j k ), v1 = v v = 7 ( 11 i + 8 j + k ), : v // u, v 1 u = 0. 8. u, v. (1) : v v = v 1 + v,, v 1 u, v // u ; () v 1, v. : (1) v : v = v 1 + v = v 1 + v,. v 1 u, v 1 u, v // u, v // u, v 1 v 1 = v v. ( v 1 v 1 ) u = 0, ( v v ) u = 0, v 1 v 1 u, v 1 v 1 // u, u 0, : v 1 v 1 = 0, v v = 0, v 1 = v 1, v = v. () v = pr u 0 v = ( v u 0 ) u 0 = ( v u ) u, v1 = v v u. v // u, v1 u = v u ( v u ) u = 0, v u 1 u. : v = v 1 + v, (1)
40, v u v = u u v1 = v v. 9. a, b, c. : a + b + c = 0 a b = b c = c a. : a + b + c = 0, b c a b + c b = 0 a c + b c = 0, a b = b c = c a., p = a + b + c. p b = a b + c b = 0 p c = a c + b c = 0. p 0, p, b p, b b c,. 10. a, b, B = a + b, B = a + 8 b, D = ( a b ). :,B,D. :,B,D, : B BD = 0. BD = B + D = 5 a + 5 b = 5( a + b ) = 5B, B BD = 0, :,B,D. 11. O, OB, O OB O + O O + O OB = 0. :,B,. :,B, : B = 0. B = OB O, = O O, B = ( OB O) ( O O) = OB O+ O OB+ O O = 0,,B,. 1. a = p n, b = q n, c = r n, a, b, c.. : n a = n b = n c = 0, n 0, a, b, c ;, n = 0 a = b = c = 0,.
8 41 1. : 8 (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, b, 1 c ) = 5 4 = 88 0, a, b, c. 11 0 1 () ( a, b, 1 c = 1 4 = 5 0, a, b, c. 1 5 () ( a, b, 1 c ) = 4 51 = 40 0, a, b, c. 7 5 8 (4) ( a, b, 0 1 c ) = 0 = 0, a, b, c. 1. 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, b, 1 1 c ) = 1 1 = 15. 1 1 () V = ( a, b, 0 1 c ) = 1 1 = 1 = 1. 1 5 1 () V = ( a, b, 1 1 1 c ) = 1 = 1 = 1. 1 1 0
4 (4) V = ( a, b, 1 0 0 c ) = 1 1 0 = 1. 1 1 1. : (1) (1,, 1), B(0,1,5), ( 1,,1), D(,1,); () (,,1), B(,0, 1), ( 1, 4,5), D(,,4); () (1,, ), B(,5, 1), (0,,7), D(,1,); (4) (1,0,1), B(0, 1,), (1,, ), D(,0, 1). :,B,,D, B,, D. ( B,, D). (1) B = ( 1, 1,6), = (,0,), ( B,, D) = D = (1, 1,4), 1 1 6 0 = 0,. 1 1 4 () B = ( 1,, ), = ( 4,,4), D = (0,0,), ( B,, D) = 1 4 4 = 0 0,. 0 0 () B = (,,), = ( 1, 4,10), D = (1, 1,6), ( B,, D) = 1 4 10 1 1 6 = 0 0,. (4) B = ( 1, 1,1), = (0,, ), D = (1,0, ), ( B,, D) = 1 1 1 0 = 45 0,. 1 0 4., B,, D : (1) ( 1,0,1), B(,1,4), (1,, ), D(, 1,); () (, 1,1), B(5,4,4), (,, 1), D(4,1,); () (1,0,), B(1, 1,0), (,, 1), D(,1,0);
8 4 (4) (,, 1), B(1,, ), (,,1), D(1,1,1). : (1) B = ( 1,1,), = (,, 4), D = ( 1, 1,), V = 1 1 1 1 ( B,, D) = 6 6 4 = 1 6. 1 1 () B = (,5,). = (0,4, ), D = (,,1), V = 1 1 ( B,, D) = 6 6 5 0 4 1 = 0 6 () B = (0, 1, ), = (1,, ), D = (,1, ), V = 1 1 ( B,, D) = 6 6 = 10. 0 1 1 = 10 6 = 5. 1 (4) B = ( 1,0, 1), = (0, 4,), D = ( 1, 1,), V = 1 1 ( B,, D) = 6 6 1 0 1 0 4 = 5. 1 1 D 1 1 1 B 1 D 5 B 5., BD 1 B 1 1 D 1, B = 4, D = 1 =. (1) 1 1 BD ; () D 1 1 BD. : D, D, DD 1 i, j, k, [D; i, j, k ]. D 1 = (,0,), DB = (,4,0), D 1 = (0,4,).
44 (1) 1 1 BD d = ( D 1, DB, D 1 ) DB D 1 = 0 4 0 0 4 (8, 4,8) = 1 = 8. () D 1 //B 1, D 1 // 1 BD. D 1 1 BD D 1 1 BD. 0 0 4 0 ( D, DB, D1 ) 0 4 d = DB = D 1 1 = 16 1 = 4. 6. a = a 1 e1 + b 1 e + c 1 e, b = a e1 + b e + c e, c = a e1 + b e + c e. ( a, b, a 1 b 1 c 1 c ) = a b c ( e 1, e, e ). a b c : 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, a 1 b 1 c 1 e ) = a b c ( e 1, e, e ). a b c ( ). 7. : ( 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 sin a, b a b c. 8. : a ( b c ) + b ( c a ) + c ( a b ) = 0.
8 45 : 7.7 : a ( b c ) = ( a c ) b ( a b ) c,. b ( c a ) = ( a b ) c ( b c ) a, c ( a b ) = ( b c ) a ( a c ) b, 9. :, B,, P, a, b, c, p ( p, b, c ) + ( a, p, c ) + ( a, b, p ) ( a, b, c ) = 0. :,B,,P P, PB, P. a = O, P = a p, b = OB, c = O, PB = b p, p = OP, P = c p. P, PB, P ( P, PB, P) = 0, ( a p, b p, c p ) = 0, ( a, b p, c p ) ( p, b p, c p ) = 0, ( a, b, c ) ( a, b, p ) ( a, p, c ) ( p, b, c ) = 0, ( p, b, c ) + ( a, p, c ) + ( a, b, p ) ( a, b, c ) = 0. 10. (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.
46 () ( a b ) ( c d ) = [ a ( c d )] b [ b ( c d )] a = ( a, c, d ) b ( b, c, d ) a. 11. a, b, c, d ( b, c, d ) a + ( c, a, d ) b + ( a, b, d ) c + ( b, a, c ) d = 0. : 10 ( a, b, d ) c ( a, b, c ) d = ( a, c, d ) b ( b, c, d ) a,. 1. : (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 ) = 0; (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 ). () ( 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, : ( a d ) ( b c ) + ( b d ) ( c a ) + ( c d ) ( a b ) = 0. (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,
8 47. : ( a d ) ( b c ) + ( b d ) ( c a )( c d ) ( a b ) = ( a b + b c + c a ). 1. r i (i = 1,,,4), : ( r 1 r )( r r 4 ) + ( r 1 r )( r 4 r ) + ( r 1 r 4 )( r r ) = 0. : 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 ),. 14. a, b, c b c, c a, a b : b c, c a, a b ( a b, b c, c a ) = 0, 1 (1) : ( a b, b c, c a ) = ( a, b, c ), a. b. c a b, b c, c a. 15 a,b,c, a + b + c < 1. p = (1 a,0,0), q = (0,1 b,0), 1 r = (0,0,1 c). : V, V = ( p q ) 1 a 0 0 r = 0 1 b 0 = (1 a)(1 b)(1 c). 0 0 1 c (1 a)(1 b) = 1 (a + b) + ab 1 (a + b) > 0, (1 c)(1 b)(1 a) = (1 b)(1 a) c(1 b)(1 a) 1 a b c > 1. 16 a = (x 1,y 1,z 1 ), b = (x,y,z ). : x 1 x + y 1 y + z 1 z x 1 + y1 + z1 x + y + z. : ( a b ) ( a b ) = a b ( a b ).
48 a b = ( y 1 z 1 y z, x 1 z 1 x z, x 1 y 1 x y ) ( a b ) = (y 1 z y z 1 ) + (x 1 z x z 1 ) + (x 1 y x y 1 ), a b = (x 1 + y 1 + z 1)(x + y + z ), ( a b ) = (x 1 x + y 1 y + z 1 z ). (x 1 x + y 1 y + z 1 z ) = (x 1 + y 1 + z 1)(x + y + z ) [(y 1 z y z 1 ) + (x 1 z x z 1 ) + (x 1 y x y 1 ) ], x 1 x + y 1 y + z 1 z x 1 + y 1 + z 1 x + y + z. x 1 x = y 1 y = z 1 z. 9 1. B B(,0), (,0). 7x 5y 5 = 0,. : (x,y), x = x + x B + x = x x = x y = y + y B + y = y, y = y. (x,y ) 7x 5y 5 = 0, 1x 15y 5 = 0. M y Q N P O B x
9 49. l B P. B, P, PB MP BNP. MN Q. : B O, B x, [O; i. j ]. M ( l ) ( ) l,0, B,0, P(t,0), t l ( ), t + l, N MN Q : x Q = t y Q = 4 l. Q : y = 4 l ( l < t < l ). t + l ( ) l, t. ( l < t < l ) ( l 4 < x < l 4 ).