1 CHNG III. QUAN H VUNG GC TRONG KHNG GIAN VN 1. VECT TRONG KHNG GIAN Cc kin thc cn nh v vect trong khng gian: -Quytccngvect:ChohaiimA,B.Khi 1 2 nM , M , ..., M btktalunc 1 1 2 1 n n nAB AM M M ... M M M B= + + + + ,-Quytctrvect:ChohaiimA,B.KhivimiimMbtktalunc AB MB MA = -Quy tc hnh bnh hnh: Cho hnh hnh ABCD. Khi ta lun cAC AB AD = +-Quytctrungim:ChoonthngABcIltrungim.Khitalunc ( )12MI MA MB = + .-Quytchnhhp:ChohnhhpABCD.A' B' C' D' .Khitalunc AC' AB AD AA' = + + . -imchiaonthng:imMgilchiaonthngABtheots1 k = nu . MA kMB =-Tnh cht ca im chia on thng:im M chia on thng AB theo t s1 k =khi v ch khi,1OA kOBOM Ok= bt k. -Trng tm ca t din:Trong mt t din cc on thng ni trung im cc cp cnh i din ng quy ti mt im. im gi l trng tm ca t din. -Phn tch mt vect theo hai vect khng cng phng: Cho hai vect, a b khng cng phng.Khivimivectx btktalunbiudinduynhtdidng . x ma nb = +-Ba vect khng ng phng: Trong khng gian cho ba vect, , a b cbt k khc vect khng. T mt im O bt k ta dng, , . OA a OB b OC c = = =Nu OA, OB, OB khng cng nm trong mt mt phng ta ni ba vect, , a b ckhng ng phng. Ngc li ta ni chng ng phng. -Tnhchtbavectngphng:Chohaivect, a bkhngcngphng.Bavect , , a b cng phng khi v ch khi tn ti duy nht cp s m, n sao cho. c ma nb = +-Cho tam gic ABC. Khi ta lun c ( )2 2 21. .2AB AC AB AC BC = + Dng 1. Phn tch mt vect theo ba vect khng ng phng -Phngphp:Sdngccquytcvecttrongkhnggiannhquytccng,quytc tr, quy tc hnh bnh hnh, quy tc hnh hp, quy tc trung im ... 1.Chohnhhp. ' ' ' ' ABCD A B C D c, , ' . AB a AD b AA c = = = GiMltrungimca on thng'. BCHy phn tch vectAMqua ba vect, , . a b c2.Cho hnh lng tr. ' ' '. ABC A B CtAA' , AB , AC . a b c = = =a) Biu th cc vect' , ' B C BCqua cc vect, , . a b c 2 b) Gi' Gl trng tm tam gic' ' '. A B CBiu th vevt' AGqua cc vect, , . a b c3.Cho t din ABCD. Hy xc nh hai im M, N sao cho: a)AM AB AC AD = + + b)AN AB AC AD = + Dng 2. Chng minh ng thc vect trong khng gian 4.Cho hnh chp S.ABCD. Chng minh rng: a) Nu y ABCD l hnh bnh hnh th. SA SC SB SD + = + b) Nu y ABCD l hnh ch nht th 2 2 2 2. SA SC SB SD + = +5.ChotdinABCDcM,NlnltltrungimcaABvCD,Gltrungimca 1, MN Gl trng tm. BCD AChng minh rng: a)AC BD AD BC + = + b) 1( )2MN AD BC = + c) 1( )2MN AC BD = + d)0 GA GB GC GD + + + =e) 1( ),4NG NA NB NC ND N = + + + g) 13 AB AC AD AG + + =Dng 3. Chng minh ba vect ng phng Phng php chng minh ba vect, , a b cng phng -Phng php 1: S dng nh ngha ba vect, , a b cng phng l ba vect c gi nm trn ba mt phng song song hoc trng nhau. -Phng php 2: S dng nh l chng minh rng. a mb nc = +6.ChotdinABCD.GiMNlnltltrungimcaABvCD.Chngminhrngba vect, , BC MN AD ng phng.7.Cho hnh hp. ' ' ' '. ABCD A B C DGi E, F ln lt l tm ca hai hnh bnh hnh' ' ABB Av' '. BCC BChng minh rng ba vect, EF, ' ' BD C Bng phng. 8.* Cho t din ABCD. Cc im M, N ln lt l trung im ca AB v CD. Ly cc im P, Q ln lt thuc cc ng thng AD v BC sao cho, , ( 1). PA kPD QB kQC k = = =Chng minh rng cc im M, N, P, Q cng thuc mt mt phng.9.*TrongkhnggianchohaihnhbnhhnhABCDv' ' '. AB C D Chngminhrngba vect', CC', ' BB DDng phng. p n: 1) 1 1AM a b c2 2= + +2a)B' C a b c, BC' a b c. = + = +3a) AM l ng cho ca hnh hp c ba cnh l AB, AC, AD 3b) Gi G l nh th t ca hnh bnh hnh ABGC. N l nh th t ca hnh bnh hnh ADGN 9)CC' BB' DD' = +