C. CAU HOI VA BAITAP
a)AfịA C; b) CẠCfi ;
c) AC.Cfị
GIAI
Ta cd BC = 2a , AC = a>/3 (h.2.9). a) AB.AC=IABỊ IACI cos 90° = o .
^
b) CẠCB ^|CA|.|CB|COS30° = aV3.2a — = 3â 2
c) AC.CB = |AC|.|CB|COS150° = a>/3.2ạ ' ^3^
V y
VAN d l 2
Chiing minh cac dang thiic ve vecto co lien quan den tich vo huong
1. Phuang phdp
• Sir dung tinh chit phan phd'i cua tfch vd hudng đ'i vdi phep cdng cae vectọ
• Dimg quy tic ba diim A8 + BC = AC hay quy tic hieu AB = 08-0Ạ
2. Cdc vi dii
Vi du 1. Cho tam giac AfiC. Chirng minh rang vdi diem Mtuy y ta cd /WẠfiC + MfịCA + MC.Afi = O .
GIAI
Tacd ~MASC = ldẠ{M6-~m) = JlAM6-lilAMB (I) MB.CA = MB.{MA-MC) = MB.MA-MB.MC (2) MC.AB = MC.{MB - MA) = MC.MB - MC.MA (3)
Cdng cac kit qua tir (1), (2), (3) ta dugc :
ldAM: + ~MB£A + ~M6AB = Q.
Vi du 2. Cho O la trung diem ciia doan thing Afi va M la mot diem tuy ỵ Chimg minh rang : /WẠMfi = OM^ - OẬ
GIAI
Tacd ldAJl^ = {Md + ^).(M6 + ^)
= 113 +lld.iOA +OB)+ 01.08 = 110 -at
6
(vi Ol + OB = 0 va dl.OB = -dl ) .
Vl du 3. Cho tam giac AfiC vdi ba trung tuyen la AD, BE, CF.
Chiimg minh rang fiC.AD + CẠfiE + AfịCF = Ọ
GiAl
Tacd Dođ
AD =-(AB + AC) (h.2.10).
2BC.AD = 8C.(AB + AC)
= B6.AB+B6.A6. (1)
Tuong tu 2C1.'BE = C1.'B6+CAm (2)
2AB.CF = AB.CB + AB.cl. (3)
Tit(l),(2),(3)tasuyra
2(B6.AD + cl.BE + AB£F) = 0 hay B6.AD + C1.M + AB.CF = 0.
VAN dE ?
Chiing minh su vuong goc cua hai vecto
1. Phuang phdp
Sir dung tfnh chit eiia tfch vd hudng : a ± fe •» ạfe = 0.
2, Cdc vi du
Vi du 1. Cho tam giac AfiC cd gdc A nhgn. Ve ben ngoai tam giac AfiC cac tam giac
vudng can dinh A la AfiD va ACẸ Ggi M la trung diem ciia fiC. ChCrng minh rang AM vudng gdc vdi DẸ
GIAI
Tacd
2JMJDE = (AB+^)(^-~^)
= AB.JE-ABAD+^.JE-'A6.AD = AB.'AE-A6.AD
= AB.AẸ COS(90° + A) - ACAD cos(90° + A) = 0
(viAB = AD,AE = AC).
vay AM J. DE suy ra AM vúdng gdc vdi DẸ
Vi du 2. Cho hinh chCr nhat AfiCD cd Afi = a va AD = a72.
Ggi K la trung diem ciia canh AD. Chiimg minh rang BK vudng gdc vdi AC.
GIAI
Ggi M la trung diim eiia canh BC.
Ta cd AB = a, AC = BD= ^2â+â = ậ Cin chiing minh MA6 = 0 (h.2.12).
Tacd M = ^ + 'BM = ^ + -^5 2 A6 = AB+AD.
vay M.A6 = (B1+-JD).(AB+JD)
= BẠAB+BẠA5+-AD.AB+-AB.A5 2 2 = -â + 0 + 0+-(a>/2)^ = 0 .
Do đ 'BK.JC = 0. Ta cd BK vudng gdc vdi AC.
VAN dE 4
Dieu thiic toa do cua Uch vo huong va cac iing dung : tinh do dai cua mot vecto, tinh khoang each giQa hai diem, tinh goc giQa hai vecto
1. Phuang phdp
• Cho hai vecto a = (aj ; ạ2) vk b = (fej; fe2). Ta cd ạ fe = a,fej + a2fe2.
~* | - » | j ^ ^
• Cho vecto u =(u^•, u^). Ta cd |M| = Ju^ + u^ . • Cho hai diim A = (x^; y^), B = (xg ; y^).
Tacd AB= \'XB\ = ^ix^-x^)^+(y^-y^)\ • Tfnh gdc giiia hai vecto a = (â; 03) va fe = (fej; 62):
cos i2S)=j^=^JfC^ ạfe, +âb^ .2