C. CAU HOI VA BAITAP
A M= —B Tim sd /c trong cac dang thCfc sau:
5
a)AM = kAB ; h)MA = kMB ; c) MA = /cAS.
GIAI
(Xem h. 1.19)
AM B
' • • >
a) AM = kAB => 1^1 = = - . Vi AM va AB ciing hudng nen k=-.
AB 5 5 b) MA = kJiB => \k\ = = - . Vi MA va MB nguoc hudng nen k = —-.
' ' MB 4 6 . e 4
c) M4 = itAB => \k\ = = - . Vi MA va AB nguoc hudng nen k = --.
' ' AB 5 5 Vi du 3. a) ChCfng minh vecto đi cQa vecto 5a la (-5)ạ
b) Tim vecto đi cCia cac vecto 2a + 3b, a - 2b.
GiAi
a) -(5a) = (-l).(5a) = ((-l)5).a = (-5).a
b) -(2a + 36) = (-l).(2a + 3^) = (-l).(2a) + (-l).(3fe) = (-2).a + (-3).6 = - 2 a - 3 6 .
-(a - 2b) = (-l).(a - 2b) = (-l).a + 2.b = -a + 2b.
VAN dE 2
Phan tich (bieu thi) mot vecto theo hai vecto khong cung phuong
1. Phuang phdp
a) Dl phan tich vecto x = OC theo hai vecto khdng cung phuong a = OA , b = OB ta lam nhu sau :
• Ve hinh binh hanh OÁCB'
cd hai dinh O, C va hai canh
OÁ va OB' Ian luot nim tren hai gia ciia OA, OB (h.1.20). Tacd x = OÁ + OB'.
• Xac dinh sd h di OÁ = hOẠ Xac dinh sd kdiOB'^ kOB. Khi do x = ha + kb.
b) Cd thi sir dung linh boat cac cdng thiic sau :
• AB = OB-OA, vdi ba diim O, A, B bat ki ;
• AC = AB + AD nlu tii giac ABCD la Mnh buih hanh. 2. Cdc vidu
Vi du 1. Cho tam giac ABC c6 trong tam G. Cho cac diem D, E, F Ian Icfot la trung diem ciia cac canh BC, CA, AB va / la giao diem cOa AD va EF. Dat tJ = AE, v = AF. Hay phan tich cac vecto Al, ^ , ^ , DC theo hai vecto u, V.
GIAI
Vi tii giac AEDF la hinh binh hanh nen AD = AE + AF = u + v va 'AI = -AD (h.1.21). 2 — 1 - - 1 - 1 - vay A/ =—(M + V) =—M + -V. 2 2 2 —• 2—• 2 -* - 2 - 2 - ' AG = — AD =—(u + v) = —u +—v 3 3 3 3 DE = FA = -AF, vay D £ = (-1).V + 0.M. 'DC='FE=~AE-'AF, vay DC = M - V.
Vi du 2. Cho tam giac ABC. Bilm M tren canh BC sao cho MB = 2MC. Hay phan tich vecto AM theo hai vecto u - AB, v = 7^.
GlAl 2^ Tacd AM = 'AB+^ = AB+-'BC 3 = AB + -(AC-AB) 3 = -AB + -AC. 3 3 vay AM = -M + - v (h.1.22). •^ 3 3
• Ta cd thi giai bai toan bing each dung dinh If Ta-let nhu sau :
Ke ME II AC va MF II AB, ta cd AM = AF + AF. Theo dinh If Ta-let AE = -AB,AF=-AC. Dođ AE = -AB = -u,AF = -'AC = -v.
3 3 3 3 3 3
vay AM^-u + -v.
3 3
VAN dE J
Chiing minh ba diem thang hang, hai du6ng thang song song
1. Phuang phdp
Dua vao cac khing dinh sau :
• Ba diim phan biet A, B, C thing hang <=> AB va AC cung phuong «• 'AB = kAC.
• Nlu AB = kCD va hai dudng thing AB va CD phan biet thi AB II CD. 2. Cdc vi du
Vi du 1. Cho tam giac ABC cd trung tuyen AM. Ggi / la trung diem ciia AM
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va K la diem tren canh AC sao cho AK = -AC. Chufng minh ba diim fi, /,