1、 1 9.7 1 . 2 . 3 a b a , , b . 4 . 1. b a, 0 b a b b a . c a c b b a / , / , / . L A B a P L t a t OA OP L P L OP =(1 t) OB t OA . OP=x OB y OA x+y=1 2. b a, p b a, x y p = b y a x . P MAB x y MB y MA x MP O MB y MA x OM OP . 3. c ， b ， a p x y z p = b y a x c z . O A B C P x y z OP=x OB y OA + OC z
2、 x+y+z=1 P A B C . 4. cos a b a b a b b a b a b ， a cos a a a 2 . 0 b a b a 5. 2 1 2 3 1 2 3 ( , , ) ( , , ) a a a a b b b b 1 1 2 2 3 3 ( , , ) a b a b a b a b ; 1 1 2 2 3 3 ( , , ) a b a b a b a b 1 2 3 ( , , )( ) a a a a R 1 1 2 2 3 3 a b a b a b a b 1 1 2 2 / , , a b a b a b 33 () a b R 1 1 2 2
3、3 3 0 a b a b a b a b . 6. : cos | | | | ab ab ab 1 1 2 2 3 3 2 2 2 2 2 2 1 2 3 1 2 3 a b a b a b a a a b b b 7. 1 2 3 ( , , ) a a a a 2 2 2 1 2 3 | a a a a a a 8. 1 1 1 2 2 2 ( , , ) ( , , ) A x y z B x y z , 2 1 2 1 2 1 ( , , ) AB x x y y z z 2 | AB AB 2 2 2 2 1 2 1 2 1 ( ) ( ) ( ) x x y y z z 9.
4、a1 a2 a n a1 a2 a3 a n =0. 1. a b c A.a+b b a a B.a+b b a b C.a+b b a c D.a+b+c a+b c 2. ABCD A B C D B A D A BD A. B. C. D. 3. a= 2x 1 3 b= 1 2y 9 a b ( ) A.x=1 y=1 B. x= 6 1 y= 2 3C. x= 2 1 y= 2 1D.x= 6 1 y= 2 34. a= 1 1 0 b= 1 0 2 ka b 2a b k 5. ABCD AB=a 2c CD=5a+6b 8c AC BD E F EF=_. 3 6. A 1 1
5、 1 B 1 0 4 C 2 2 3 AB CA _. 1-3.CCB; 4. k= 5 7 . 5. EF=3a+3b 5c 5. : AD G EF EG GF =3a+3b 5c. 1 , 1 1 1 1 D C B A ABCD ,O 1 1 D B . : 1 C B 1 1 ODC . 2 E F G H K L AB BC CC1 C1D1 D1A1 A1A . : 1 CB 1 ODC . 1 : 1 , , , CB a CD b CC c 1 1 BCC B , 1 CB a c , O 1 1 D B , 1 1 1 1 1 1 ( ), 2 C O a b OD C D
6、 C O 1 () 2 ba 11 1 ( ) , 2 DO DD D O a b c , , y x 11 CB xC O ,( , ) yDO x y R , 11 ( ) ( ) 22 a c x a b y a b c 1 ( ) ( ) 2 2 x y a x y b yc c b a , , , 2 0 22 xy xy y , 1 1 y x . A B C C 1 D 1 A 1 B 1 D O L K H G F E O B 1 C 1 D 1 A 1 B A 4 11 , CB C O DO 1 1 , , OC OD C B , C B 1 1 ,OC OD , C B
7、1 1 ODC , 1 BC 1 ODC , C B 1 1 ODC . 2 11 ( ), ( ) 22 GH b c GF a c EF FL 2 S ABC SAB SAC ACB AC=2 BC= 13 SB= 29. 1 SC BC 2 SC AB . 3 E F G AB AC SB EFG ACG. G F E B C A S 1 1 1 AS = a, AB = b, AC = c 0, 0, ( ) 0 a c a b c c b , SC c a BC c b ( )( ) SC BC c b c a ( ) 0 c b c a c a b SC BC . 2 29 13
8、4, SC 2 3, 17 SA AB 5 , cos , | | | 2 2 17 ( ) 17 17 17 17 4 4 17 4 17 SC AB SC AB SC AB c a b c b SC AB arccos 17 17 . 3 11 ( ), , 22 EF c b EG a 1 ( ) 0, 2 AC EF a c b c 1 0 2 AC EG a c , AC EFG AC ACG ACG EFG 2 . 2 A AD AC AS x y z AC=2 BC= 13 SB= 29 C 0 2 0 B 13 2 0 S 0 0 2 3 SC = 0 2 2 3 BC = 1
9、3 0 0 . 1 SC BC =0 SC BC. 2 SC AB AB = 13 2 0 SC AB=4 | SC | AB|=4 17 cos = 17 17 . 3 13 ( ,1,0), (0,1,0), 2 EF _ y z x G F E B C A S 6 1 (0,2,0), 2 AC EF BC 13 ( ,0,0), (0,0, 3), 2 EG 0, 0 AC EF AC EG , AC EF AC EG AC , EFG AC ACG ACG EFG 1. . 2 . 3 AB= 2 2 1 AC= 4 5 3 ABC . ABC n= x y 1 n AB n AC
10、n AB=0 n AC=0 1 2 2 1 0 , 2 4 5 3 0 1, xy x xy y n= 2 1 1 1 n0 | n | n 3 1 3 2 3 2 . . n 1 . . . 1 , . 2 . 9.7 1. ABCD A 1 B 1 C 1 D 1 M AC BD 1 1 B A =a 1 1 D A =b A A 1=c M B 1 A. 2 1 a+ 2 1 b+c B. 2 1 a+ 2 1 b+c A A B B C C D D 1 1 1 1 M 7 C. 2 1 a 2 1 b+c D. 2 1 a 2 1 b+c 2. P x y z P x P 1 x y
11、z P yOz P 2 x y z P y P 3 x y z P P 4 x y z A.3 B.2 C.1 D.0 3 A B C D AB+ BC + CD + DA =0 |a| |b|=|a+b| a b a b a b O A B C OP =x OA +y OB +z OC x y z R P A B C A.1 B.2 C.3 D.4 4. A 1 2 1 B 1 3 4 D 1 1 1 2 AP PB | PD | _. 5. C 2a+1 a+1 2 P 2 0 0 A 1 3 2 B 8 1 4 a ; 6 A BCD M N ABC ACD BD=4 MN . : 1-
12、3, ACC; 4. 3 77 ; 5. a=16 http:/ http:/ 6. 3 4 7 A B C A 1 B 1 C 1 l 1 l 2 M N P Q AA 1 BA 1 BB 1 CC 1 . M N P Q . 11 2 , 2 MN AB NP A B A B C A 1 B 1 C 1 1 1 1 1 , AC xAB AC y A B j Q P N M L 2 B 1 C 1 L 1 A 1 C B A 8 1 1 1 1 MQ MA AC CQ MQ MA AC C Q 1 1 1 1 2 2 2 MQ AC AC xAB y A B xMN yNP , MQ
13、MN NP M N P Q . 8. OABC , , AOC BOC AOB , OC OB OA N M , BC OA, ,G MN . : BC OG : , ON : ) ( 4 1 ) ( 2 1 2 1 2 1 ) ( 2 1 OC OB OA OC OB OA ON OM OG ) ( ) ( 4 1 OB OC OC OB OA BC OG ) ( 4 1 2 2 OB OB OA OC OC OA , cos AOC OC OA OC OA , cos AOB OB OA OB OA AOC AOB OA OB OC , , BC OG BC OG , 09 1 ABCD
14、A 1 B 1 C 1 D 1 BD 1 ACB 1 E 1 BD 1 ACB 1 2 BE= 2 1 ED 1 . 1 1 , DA DC DD x,y,z A(1,0,0), B(1,1,0), C(0,1,0), D 1 (0,0,1), B 1 (1,1,1) A A D D B B C C 1 1 1 1 E M 9 11 1 1 1 1 1 1 1 1 1 1 (0,1,1), ( 1,1,0), ( 1, 1,1) 0, 0, , AB AC BD BD AB BD AB BD AB BD AB BD ACB 2 11 , ( , ,0), 22 AC BD M M 1 1 1
15、1 1 1 ( , ,1), ( , ,0) 2 2 2 2 2 MB BM BD 11 ( , , ), ( , , ) 22 BE BD ME MB BE BM ME 1 1 1 1 , ,0) ( , , ) ( , , ) 2 2 2 2 2 2 1 1 2 3 11 11 32 BE BD BE ED 10 ABC A 1 B 1 C 1 (1) AB 1 BC 1 AB 1 A 1 (2) AB=2 AA 1 =4 BC 1 A 1 C (1) AB =a AC =b 1 AA =c 1 AB =a+c 1 BC =ba+c C A 1 =bc 1 AB 1 BC (a+c) (b
16、a+c)=0 c 2 a 2 +ab=0 AB = AC =x 1 AA =h h 2 x 2 + 2 1 x 2 =0 x 2 =2h 2 1 AB C A 1 =(a+c) (bc)=abc 2 = 2 1 x 2 h 2 =h 2 h 2 =0 (2) 1 BC = C A 1 = 20 1 BC C A 1 =(ba+c) (bc)=b 2 c 2 ab= 14 BC 1 A 1 C cos =|cos|= C A BC C A BC 1 1 1 1 = 7 10 BC 1 A 1 C 7 10 ABC A 1 B 1 C 1 ABC CA=CB=1 BCA=90 AA 1 =2 M
17、N A 1 B 1 A 1 A . 1 BN A A B B C C 1 1 1 x y z M N 10 2 cos 11 BA CB 3 A 1 B C 1 M. 1 B 0 1 0 N 1 0 1 BN = 2 2 2 ) 0 1 ( ) 1 0 ( ) 0 1 ( = 3 . 2 A 1 1 0 2 B 0 1 0 C 0 0 0 B 1 0 1 2 1 BA = 1 1 2 1 CB = 0 1 2 1 BA 1 CB =3 1 BA = 6 1 CB = 5 . cos 1 BA 1 CB = | | | 1 1 1 1 CB BA CB BA = 10 30 . 3 C 1 0 0 2 M 2 1 2 1 2 B A 1 = 1 1 2 M C 1 = 2 1 2 1 0 B A 1 M C 1 =0 A 1 B C 1 M. 11
