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N H O N G DIEU H O C SINH CAN CHU Y KHI SU DUNG SACH GIAO KHOA
1. Khi nghe thay co giao giang bai, luon luon c6 SGK tri/dc mat Tuy nhien
khong viet, ve them vao SGK, de nSm sau cac ban l2. Ve trinh bay, sach giao khoa c6 hai mang : mang chfnh va mang phu.

Mang chinh gom cac djnh nghTa, djnh If, tfnh chat,... va thi/dng di/dc dong
khung hoSc c6 dudng vien d mep trai. Mang nay duoc in ICii vao trong.
3. Khi gSp Cau hoi [ ? ] , can phai suy nghT, tra ldi nhanh va dung.
4. Khi g§p Hoat dong ^ , cac em phai dung but va giS'y nhap di thiic hi§n
nhufng yeu cau ma hoat dong doi hoi.

Ban quyen thudc Nha xuat ban Giao due Viet Nam Bo Giao due va Dao tao
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01 - 2010/CXB/733 - 1485/GD

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III

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Ma s6: NH002T0

CHUONG

VECTO

Vecta 1^ mOt kh^i ni^m todn hoc moi dd'i vdi cac em.
Hpc chuong ndy, cdc em phdi hieu duoc vecta la gl, the ndo
Id tdng, hi$u cCia hai vecta, tfch cua mot vecta voi mdt sd.
NhOng ki^n thuc ndy rSx quan trong, chung id co so de hpc
mdn Hinh hpc cua cd ba I6p 10,11 vd 12.


CAC D I N H N G H I A

1. Vectd la gi?
Trong Vat If, nhiJng dai lugfng nhu van t6'c, gia t6c, luc,... dugtc goi la
dai lugng co hudng. Di xac dinh cac dai luang do, ngoai cuomg d6 cua
chiing, ta con phai bi^t hudfng cua chung niia.
Vi du : Mot chiee tdu thuy chuyen dong thdng deu vdi tdc do 20 hdi li mdt
gid, hien nay dang d vi tri M. Hoi sau 3 gid niia nd sedddu ?
?1 Cdc em cd the trd ldi cdu hoi do khdng ? Vi sao ?
Hinh 1 la hai dd m6t viing bi^n tai m6t thofi di^m
nao do. Co hai tau thuy chuydn dSng thang d^u ma
van toe dugc bi^u thi bang mui ttn. Cac miii tdn
van tdc cho ta tha'y : Tau A chuydn ddng theo
hudfng Ddng, con tau B chuydn ddng theo hudng
Ddng - Bac. Tdc dd tau A bang mdt niia tdc dd
tau B (do miii t6n cua tau A dai bang mdt nira, mui
ten ciia tau B).

Nhu vay, cac dai lugng cd hudng thudng dugc bi^u thi bang nhiing miii t6n
dugc ggi la nhiing VECTO. Vecto la mdt doan thang nhung cd hudng. D^
bi^u thi cho hudng cua doan thang ta th6m mdt dau ' V vao mdt trong hai
diem mut cua doan thang dd.
Gia sit ta cd doan thang AB (ciing cd thi viet
B
B
la doan thing BA). Neu them dau *- vao ^
b).
diim B thi ta cd vecto vdi di^m dSu la A vk
Hinh 2
diiim cudi la B (h. 2a). Ne'u ta them da'u
"J' vao di^m A thi ta dugc vecto vdi'di^m dau la B vk diim cudi la A (h. 2b).
Nhu vay, vecto la mdt doan thltig da xac dinh mdt hudfng nao dd trong hai
hudng cd thi cd cua doan thing da cho. Hudng cua vecta la hudng di tit
diim ddu de'n di^m cud'i.


DINH NGHiA
Vecta Id mot doan thdng cd hudng, nghia la trong hai diem
miit cua doqn thdng, da chi rd diem ndo la diem ddu, diem
ndo Id diem cud'i.
KI hieu
N^u vecta cd di^m dSu la M vk diim cudi la A^ thi ta ki hieu vecto dd la
MN.
Nhi^u khi d^ thuan tien, ta ciing ki hieu mdt vecto xac dinh nao dd bang mot
chii in thudng, vdi miii ten d tren. Chang han vecto a, b, x, y, ....
Vecto-khong
Ta bie't rang mdi vecta cd mdt di^m ddu va mgt di^m cudi ; mdi vecta hoan
toan dugc xac dinh ne'u cho bie't dilm ddu va dilm cudi cua nd.

Bay gid, vdi mdi dilm M bdt ki, ta quy udc cd mdt vecto ma dilm ddu la M
va dilm cudi ciing la M. Vecto dd dugc ki hieu la MM vk ggi la
veeta-khdng (cd gach ndi gitta hai tii).
II Vecto cd diem ddu vd diem cudi triing nhau goi la vectff-khong.
2. Hai vectd cung phiTdng, cung hirdng
Vdi mdi vecto AB (khac vecto-khdng), dudng thing AB dugc ggi la gid
cua vecta AB. Cdn dd'i vdi vecto-khdng AA thi mgi dudng thing di qua A
diu ggi la gia cua nd.
M

^'''

Hinh 3


a) Tren hinh 3, ta cd cac vecto AB, DC, EF, MN, QP.
Hay chii y de'n hai vecto AS va DC, chung cd gia song song vdi nhau. Hai
vecto AB va EF ciing cd gia song song. Cdn hai vecta DC va EF thi cd
gia triing nhau.
Trong cac trudng hgp dd, ta ndi rang : Cac vecta AB, DC, EF cd cUng
phucmg, hay don gian la ciing phucmg.
Hai vecto JWTV va QF cd gia cat nhau. Ta ndi hai vecta dd khdng cung
pfttrang. Vay ta cd dinh nghia

i

Hai vecta dugc ggi Id cUng phuang ni'u chung cd gid song
song hodc triing nhau.
Rd rang vecto-khdng ciing phuong vdi mgi vecto.

b) Bay gid hay chu y tdi cac cap vecto ciing phuang tren hinh 4.

N

B

D

Hinh 4

Hai vecto AB vk CD ciing phuong, va hon the' cac miii ten bilu thi AB va
CD cd cung hudng, cu thi la hudng tii trai sang phai.
Trong trudng hgp nay, ta ndi: Hai vecto AB vk CD cung hu&ng.
Hai vecto MN vk PQ cung phuang, tuy nhien ta thdy ring chiing khdng
ciing hudng vi vecto MN hudng len phia tren, cdn vecto PQ thi hirdng
xud'ng phfa dudi.
Trong trudng hgp nay, ta ndi: Hai vecto MN vk PQ ngugc hudng.
Nhu vay
Ne'u hai vecta ciing phuang thi hodc chiing ciing hudng,
hodc chiing ngugc hudng.
CHU Y
Ta quy udc ring vecto-khdng cung hudng vdi mgi vecta.


3. Hai vectd bSng nhau
Mdi vecto diu cd mdt dd ddi, do la khoang each giiia dilm ddu va dilm
cudi cua vecta dd. Db dai cua vecto a dugc kf hieu la \d\.
Nhu vdy, ddi vdi vecto AB, PQ,... taco
\AB\ = AB = BA, jpej = PQ = QP, ...
?2| Theo dinh nghTa dd ddi d trin thi veeta-khdng co do ddi bang bao nhieu ?

'

Ta bilt ring hai doan thing ggi la binjg nhau ne'u
dd dai cua chung bang nhau. Tren hinh 5 ta cd
hWi thoi ABCD. Bdn canh cua hinh thoi la bdn
doan thing bing nhau. Bdi vay ta vilt
AB = AD = DC = BC.

D

^^^^^^^^^^"-^^
A^ ^
^^-^^ ^
^""^^^^^ ^ ^ ^
^^B
Hinh 5

?3| Hai vecta AB vd AD tren hinh 5 cung cd dd ddi bdng nhau, nhung lieu
chiing ta cd nen ndi rdng chiing bdng nhau vd vii't AB = AD hay khdng ?
Vi sao vdy ?
Cdn ddi vdi hai vecta AB vd DC thi cd nhdn xet gi ve do ddi vd hudng
cda chiing ?
Mdt cdch tu nhien ta dinh nghia hai vecto bing nhau nhu sau
DjNHNGHiA
Hai vecta dugc ggi Id bdng nhau ni'u chiing cdng hudng vd
cdng dd ddi.
—»

—•

Ni'u hai vecta a vd b bdng nhau thi td vie't a = b.
CHDY

Hieo dinh ngMa tren thi cac vecto-khdng diu bing nhau :
AA = fifi = F ? = .... Bdi vay, ttr nay cac vecto-khdng dugc kf hieu
chung la 0.
/ V H a y ve mdt tam giac ABC vdi cac trung tuyen AD, BE, CF, roi chi ra cac bp ba
vecto khdc 6 vd doi mdt bang nhau (cac vecto ndy c6 diem dau va diem cudi difdc
j^y trong sdu dilm A, B, C, D, E, F).
Nlu G Id trong tdm tam gidc ABC thi cd t h i viet ^

= GD hay khdng ? Vi sao ?
7


Cho vectd a vd mdt diem O bd't ki. Hay xac djnh diem A sao cho OA = a. Cd bao
nhieu dilm A nhi/ vay ?

Trong vat If, mdt luc thudng dugc bilu thi bdi mdt vecto. Dd dai cua vecto
bilu thi cho cudng dd ciia luc, hudfng cua vecto bilu thi cho hudng cua luc
tac dung. Dilm ddu cua vecta dat d vat chiu tac dung cua luc (vdt dd
thudng dugc xem nhu mdt dilm).
Tren hinh 6, hai ngudi di dgc
hai ben bd kenh va ciing keo
mdt khiic gd di ngugc ddng.
Khi dd cd cac luc sau day tac
dung vao khuc gd : hai luc
keo Fl va F2 cua hai ngudi,
luc F3 cua ddng nudc, luc
ddy Ac-si-met F4 cua nudc

len khiic gd va trgng luc F5
cua khiic gd.

Hinh 6

Uy-li-am Ha-min-ton (William Hamilton) Id nha
toan hpc ngirdi Ai-len. Ong da viet mot trong
nhiing cdng trinh toan hpc d i u tien ve vectd.
Ong la ngirdi xay dimg khai niem qua-tec-ni-dng,
mot dai li/gng gidng nhir vecto, cd nhieu iJng
dung trong Vat If.

Cau hoi va bai tap
1. Vecto khac vdi doan thing nhu thi nao ?
2. Cac khing dinh sau day cd diing khdng ?
a) Hai vecto ciing phuong vdi mdt vecto thii ba thi ciing phuofng.
8


b) Hai vecto cung phuang vdi mdt vecto thii ba khac 6 thi ciing phuang.
e) Hai vecto ciing hudng vdi mdt vecto thii ba thi ciing hudng.
d) Hai vecto ciing hudng vdi mdt vecto thii ba khac 0 thi cung hudng.
e) Hai vecto ngugc hudng vdi mdt vecto khac 6 thi cung hudng.
f) Dilu kien cdn va du di hai vecto bing nhau la chiing cd do dai bing nhau.
Trong hinh 7 dudi day, hay chi ra cac vecto cung phuong, cac vecto cung
hudng va cac vecta Tjing nhau.

rV
/

'

}^
/

^^ \

V

it
\

i

A

/

Hinh 7

4. Ggi C la trung dilm cua doan thing AB. Cac khing djnh sau day diing
hay sai ?
a) AC vk BC ciing hudng ;

b) AC va AS ciing hudng ;

c) AS vdfiC ngugc hudng ;

d)\'AB\ = \BC\;

e)|AC| = |fiC|;

f) | A5| = 2|5C|.

5. Cho luc giac diu ABCDEF. Hay ve cac vecto bing vecto AB vk cd
a) Cac dilm ddu Id B,F,C;
b) Cac dilm cudi la F, D, C.

T 6 N G CUA HAI VECTO

Chiing ta da bilt vecto la gi va thi nao la hai vecto bing nhau. Tuy cac
vecto khdng phai la nhiing con sd, nhung ta cting cd thi cdng hai vecto vdi
nhau dl dugc tdng ciia chiing, ciing cd thi trix di nhau dl dugc hieu eua
chiing. Hgc sinh cdn nim viing each xac dinh tdng va hieu ciia hai vecto
ciing nhu cac tfnh chdt cua phep cdng va phep trii vecto.


7

1. Djnh nghTa tong cua hai vectd
Hinh 8 md ta mdt vat dugc ddi sang vi
trf mdi sao cho cac dilm A, M, ... cua
vat dugc ddi din cac dilm A', M',... ma
AA' = MM' = .... Khi dd ta ndi ring :
•-••• ••>

vat duac "tinh tien" theo vecto AA'
?l| Tren hinh 9, chuyen ddng cua mdt vdt
dugc mo td nhu sau : Tic vi tri (I), nd dugc
tinh tie'n theo vecta AB di den vi tri.(II).

Sau do nd lai dugc tinh tien mot ldn nita
theo vecta BC dedi'n vi tri (III).
Vdt cd the dugc tinh tien chi mdt ldn de
tic vi tri (I) di'n vi tri (III) hay khdng ?
Ne'u cd, thi tinh tii'n theo vecta ndo ?

Hinh 8

Hinh 9

Nhu vay cd thi ndi : Tinh tiln theo vecto AC "bing" tinh tiln theo vecto
AB rdi tinh tiln theo vecto BC
Trong Toan hgc, nhiing dilu trinh bay tren day dugc ndi mdt each ngdn ggn :
Vecta AC la tdng cua hai vecta AB vd BC.
Ta di din dinh nghia (h. 10)
Cho hai vecta a vd b. Ldy mot diem A nab dd rdi xdc dinh
cdc diem B vd C sao cho AB = a BC = b Khi do vecta AC
dugc ggi Id tong eua hai vecta a vd b. Ki hiiu
AC =d + 'b.
Phep lay tdng cOa hai vecta dugc ggi la phep cdng vecto.
B
—•

b
^

Hinh 10

10

C


Hay ve mdt tam giac ABC, rdi xdc djnh cac vecto tong sau day

a) A5 + CB ;
b) AC + BC.
Hay ve hinh binh hdnh ABCD vdi tam O {O Id giao dilm hai di/dng cheo). Hay vilt
vectd AB dudi dang tong cOa hai vecto ma cac dilm mut cCia chung dirpc lay
trong ndm dilm A, B, C, D, O.

2. Cdc tinh chat cua phep cong vectd
^Chijng ta biet rang phep cong hai sd cd tfnh chat giao hoan. Ddi vdi phep cdng hai
vecto, tfnh chat do cd diing hay khdng ? Hay kiem chufng bang hinh ve.

^Hay ve cac vecto OA = d, AB = b, BC = c nhirtren
' hinh 11. Tren hinh ve dd
a) Hay chi ra vecto ndo Id vecto a + b, va do dd,
vecto ndo Id vqcta (a + b) + c .
b) Hdy chi ra vecto ndo Id vecto b +c vk do do

6
Hinh 11

vecto ndo Id vecto a + ib +c).
c) Tii dd cd t h i rut ra ket ludn gi ?

Tix cdc hoat ddng tren, chiing ta suy ra cac tfnh chdt sau day cua phep edng
vecto (ciing gidng nhu cac tfnh chdt cua phIp cdng cac sd')
1) Tinh chat giao hodn : a + b = b +a ;

2) Tinh chdt ket hgp :

(a + b) + c = a + ib + c);

3) Tinh chdt cua veeta-khdng : a + 0 = d.
CHUY
—•

—»

Do tfnh cha't 2, cac vecto id + b) + c vk a + ib +c) bing nhau,
bdi vay, tur nay chiing dugc vilt mdt each don gian la a + b + c,
vk ggi la tSng cua ba vecta a, b, c.
11


3. Cac quy tac cin nhd
Tix dinh nghia tdng cua hai vecto ta suy ra hai quy tic sau day
QUY TAG BA DIEM (h. 12)

M

V&i ba diem bdt ki M, N, P,
ta ed
MN + NP = MP
Hinh 12

QUY TAG HINH BINH HANH (h.l3)
s

y

Ni'u OABC Id hinh binh hdnh
thi ta ed
OA + OC = 0B.
^
^
^
?2

Hinh 13

a) Hdy gidi thich tai sao ta ed quy tdc hinh binh hdnh.
b) Hdy gidi thich tai sao ta cd \d + b\ < \d\ + \b\.
Bai toan 1. Chiing minh rdng vdi bdn diem bd't kiA, B, C, D, ta cd
AC + W = AD + BC.
Gidi. Dung quy tie ba dilm ta cd thi vilt AC = AD + DC. Bin vky
AC + ^

= AD + DC + 5D = AD + 5D + DC (do tinh chdt giao hoan)
= AD + BC (quy tie ba dilm dd'i vdi B, D, C).

^Dung quy tac ba dilm, ta cung cd the viet AC = AB + BC. Hay tilp tue d l cd mdt
cdch chufng minh khdc ciia Bdi todn 1.

Bai toan 2. Cho tam gidc diu ABC cd canh
bdng a. Tinh dd ddi cua vecta tdng AB + AC
Gidi. Ta ldy dilm D sao cho ABDC la hinh binh
hanh (h. 14). Theo quy tic hinh binh hanh ta cd
'AB + AC = AD

12

Hinh 14


vay

\AB

+ AC\ = \AD\ = AD.

Vi ABC la tam giac diu nen ABDC la hinh thoi va dd dai AD bing hai ldn
dudng cao AH cua tam giac ABC, do dd AD = 2 x

= a-43.
2

Tdm lai, | A 5 + ACJ = a-43.
Bai toan 3
a) Ggi M la trung diem doan thdng AB. ChUdng minh rdng MA + MB = 0.
b) Ggi G Id trgng tdm tam gidc ABC. Chifng minh rdng GA + GB + GC = 0.
Gidi
a) Theo quy tac ba diem, ta cd MA + AM = MM = 0. Mat khac, vi M la
trung dilm ciia AB ntn AM = MB. Vay
MA + MF = 0.
b) (h. 15) Trgng tam G nim tren trung tuyin
CMvkGC = 2GM. Dl tim tdng GA + GB ta
dung hinh binh hanh AGBC Mudn vay, ta chi
cdn ldy dilm C sao cho M la trung dilm GC.
Khi dd GA + GB = GC' = CG Bdi vay

Hinh 15

GA + GB + GC = CG + GC = CC = d.
|?3l Trong ldi gidi ciia Bdi todn 3, ta dd diing ddng thicc GC' = CG. Hdy gidi
thich tai sag cd ddng thicc do.
GHI NHd
Ni'u M Id trung diem doan thdng AB thi MA + MB = 0;
Ni'u G Id trgng tdm tam gidc ABC thi GA + GB + GC = 0.

cr-

CHUY
Quy tic hinh binh hanh thudng dugc dp dung trong Vat If dl xac
dinh hgp luc cua hai luc cung tac dung len mdt vat.
13


Trdn hinh 16, cd hai luc Fj va F2 ciing
tac dung vao mdt vat tai dilm O. Khi dd
cd thi xem vat chiu tac diing cua luc
F = Fl + F2, la hgp luc cua hai luc Fj
va Fj . Luc F dugc xac dinh theo quy tie
hinh binh hanh.

Cau lioi va bai tap
6. Chiing minh ring neu AF = CD thi AC = SD
7. Tii giac ABCD la hinh gi nlu AS = DC va JAFI = | F C | ?
8. Cho bdn diem bdt ki M, A^, F, Q. Chiing minh cac ding thiic sau
a) PQ + NP + MN = MQ ;

b) NP + MN = QP + MQ ;
c) MN + PQ = MQ + PN.
—»

9. Cac he thiic sau day diing hay sai (vdi mgi a vk

b)l

a) \d + b\ = \d\ + \b\ ;
h)\d + b\-< \d\ + \b\.
10. Cho hinh binh hanh ABCD vdi tam O. Hay diln vao chd trdng (...) dl dugc
ding thiic diing
a)~AB + ~^ =

h) AB + CD =

c) AF + OA =

d)dA + dC =

e)'dA + 'dB + 'dC + 'dD =
11. Cho hinh binh hanh ABCD vdi tam O. Mdi khing dinh sau day diing hay sai ?
a) IAF + ADI = IFDI ;
c) OA + OB = 0C + ^

h)'AB + ~BD = ^
;

d)'BD+

AC = JD+

;
'BC .

12. Cho tam giac diu ABC ndi tilp dudng trdn tam O.
a) Hay xac dinh cac dilm M, N, P sao cho
OM = OA + OB ; ON = OB + OC ; OP = OC + OA.
h) Chiing minh ring OA + OB + OC = 0.
14


13. Cho hai luc F^ va F2 cung cd dilm dat tai O (h.l7). Tim cudng dd luc
tdng hgp cua chung trong cac trudng hgp sau
a) Fj va F2 diu cd cudng dd la lOON, gdc hgp bdi ^ va ^ bing 120°
(h. 17a);
b) Cudng dd cua ^ la 40N, eua ^ la 30N va gdc giiia ^ va ^ bing 90°
(h. 17b).
O

lOON

JPJ"
30N

O

40N

a)

b)
Hinh 17

HlfiU CUA HAI VECTO

1. Vectd doi cua mot vectd
Ni'u tdng cua hai vecta a vab la veeta-khdng, thi ta ndi a la
—»

—»

vecta ddi cua b, hodc b Id vecta dd'i ciia a.
?t| Cho doan thdng AB. Vecta dd'i ciia vecta AB Id vecta ndo ? Phdi chdng
mgi vecta cho trudc diu cd vecta dd'i ?
Vecta dd'i cua vecta a dugc ki hiiu la -a.
Nhu vay

a + i-a) = i-a) + 5 = 6.

Ta cd nhdn xlt sau ddy
Vecta dd'i cua vecta a la vecta ngugc hudng vdi vecta a vd
cd cdng do ddi vdi vecta a.
Ddc biit, vecta dd'i ciia vecta 0 Id vecta 0.
15
• /


B

y i du. Gia sit ABCD la hinh binh hanh (h.l8).
Khi dd hai vecta AB vk CD cd cung dd dai
nhung ngugc hudng. Bdi vay
AB =-CD
Tuong tu, ta cd
'BC

vkCD =-AB.

D'*'
Hinh 18

= -'DA va DA = -'BC

^Gpi O la tam cCia hinh binh hdnh ABCD. Hay chi ra cac cdp vecto ddi nhau md c6
dilm dau la O va dilm cud'i Id dinh cda hinh binh hanh dd.

2. Hieu cua hai vectd

' -

DINH NGHiA
Hieu ciia hai vecta a vd b, ki hiiu a -b. Id tong eua vecta a
va vecta dd'i cua vecta b, ticc la
—»

d-b

—*

= d + i-b).

Phep ldy hiiu cua hai vecta ggi Id phep trie vecta.
Sau day la each dung hieu a - b nlu da
cho vecto a vk vecto b (h. 19). Ldy mdt
dilm O tuy y rdi ve OA = a vk OB = b.

Kiiid6BA=d-b.
Hinh 19

'T2\ Hdy gidi thich vi sao ta lai ed BA = a -b

(h. 19).

Quy t^c ve hieu vecto
Quy tie sau ddy cho phep ta bilu thi mdt vecto bdt ki thanh hieu cua hai
vecto cd chung dilm ddu.
Ni'u MN Id mdt vecta dd cho thi vdi diem 0 bdt ki, ta ludn cd
MN = 0N - OM.

\

>

Bai toan. Cho bdn diem bdt ki A, B,C,D. Hdy ddng quy tdc ve hiiu vecta
de chicng minh rdng
'AB+ 'CD = 73+ 'CB.
16

Gidi. Ldy mdt dilm O tuy y, theo quy tic vl hieu vecto, ta cd
AB + CD = OB-dA

+

OD-dC

AD + CB = dD-dA

+

dB-OC

So sdnh hai dang thiic tren ta suy ra AF + CD = AD + CF .
^

2 (Giai bai toan tren b^ng nhumg each khac)
\a) Ding thiic can chufng minh ttrong di/ong vdi ding thiic AB-^
TCr do hay neu ra cdch chiing minh thuf hai ciia bdi toan.
b) Ding thiid c^n chCfrig minh cung tuang dirong vdi ding thiic AB-CB
Tit do hay neu cdch chiing minh thii ba cCia bdi toan.

=

CB-CD

=

^-CD

c) Hiln nhien ta cd AB + BC + CD + DA = 6. Hay neu each chiing minh thuf ti/.

Cau hoi va bai tap
14. Tra ldi cac cau hdi sau day
a) Vecto ddi eua vecto -a Ik vecto nao ?
b) Vecto ddi cua vecta 0 la vecto nao ?
c) Vecto ddi cua vecto a + b Ik vecto nao ?
15. Chiing minh cac minh dl sau day
a) Nlu a + b = c thi a = c - b, b = c -a ;
b) a- ib + c) = d-b-c

;

c) a -ib -c) = a - b + c.
16. Cho hinh binh hanh ABCD vdi tam O. Mdi khang dinh sau day diing hay sai ?
a)dA-OB

= AB;

b ) C O - a S = FA;

c)AB-AD

= AC

e)CD-Cd

= BD-'Bd.

;

d)AB-AD

= 'BD ;

17. Cho hai dilm A, F phan biet.
a) Tim tap hgp cac dilm O sao cho OA = OB ;
b) Tim tap hgp cac dilm O sao cho OA = -OB.
18. Cho hinh binh hanh ABCD. Chiing minh ring 'DA-DB + DC = 0.
17
2 - HHIONC-A


19. Chiing minh ring AF = CD khi va chi khi trung dilm cua hai doan thing
AD va BC triing nhau.
20. Cho sau dilm A,B,C, D, E, F. Chiing minh ring
AD + 'BE + CF

= AE + W + CD = AF + 'BD + CE.

TICH CUA MOT VECTO V 6 I M 6 T sd
Ta da bilt the nao la tdng ciia hai vecto. Bay gid nlu ta ldy vecto a cdng
vdi chfnh nd thi ta ed thi ndi kit qua la hai ldn vecto a, vilt la 2 3 , va goi
la tich cua sd 2 vdi vecto a, hay la tfch eua a vdi 2.
Trong muc nay ta se ndi din tfch eua mdt vecto vdi mdt sd thuc bd't ki.
1. Djnh nghTa tich cua mot vectd vdi mot so
Xet cac vecto tren hinh 20. Ta hay chii y
den hai vecto a vkb.

Hai vecto dd cd

Cling hudng, va dd dai vecto b bing hai
ldn dd dai

vecto a , tiic la |S| = 2\d\.

Trong trudng hgp dd ta vilt b = 2d vk

'a /

/

/

4

/

u

/

X
'c

<.
^ ^

—»

ndi ring : Vecta b bdng 2 nhdn vdi vecta
Hinh 20
a (hodc bdng vecta a nhdn vdi 2), hodc
vecta b la tich ciia vecta a vdi sd'2.
Lai chii y din hai vecto c va Khi dd ta vilt c = i-2)d vk ndi ring : Vecta c bdng -2 nhdn vdi vecta d
(hodc bdng vecta d nhdn vdi -2), hodc vecta c Id tich cua vecta d vdi -2.
k Ve hinh binh hdnh ABCD.
a) Xdc dinh diem E sao cho AE = 2BC.
b) Xac djnh dilm F sao cho AF

U]cA
\ 2)

18

3 • HH10NC-B


DINH NGHiA
Tich cua vecta a vdi sd thuc k la mdt vecta, kl hiiu Id ka,
dugc xdc dinh nhu sau
1) Ni'u k>0 thi vecta ka cUng hudng v&i vecta a ;
Ni'u k<0 thi vecta ka ngugc hudng vdi vecta a ;
2) DO ddi vecta ka bdng \k\.\d\.
Phip ldy tich cua mdt vecta vdi mdt sd ggi la phep nhdn vecta
vdi sd (hodc phip nhdn so vdi vecta).
Nhdn xet. Tii dinh nghia ta thdy ngay ld = d, (-l)a la vecto dd'i cua a,
tiic la (-l)a = -a.
Vl du. Tren hinh 21, ta cd tam giac ABC ydi MvkN ldn lugt la trung dilm

hai canh AB vk AC. Khi dd ta cd
a)'BC = 2'MN ; MN =

-BC;
2
f

b) BC =i-2)NM

; MN =

c) AF ^ 2MF ; AA^ =

1
CB
^ 2)

--ICA.
^ 2.

Hinh 21

2. Cdc tinh chat cua phep nhan vectd vdi so
Dua vao dinh nghia phep nhan vecto vdi sd' ta ed thi chirng minh eac tfnh
chdt sau ddy
Vdi hai vecta bdt ki a, b vd mgi sd'thuc k, I, ta cd
1) kild) = ikl)d ;
2) ik + l)d = kd + la ;
3) kia + b) = ka + kb ; kia - b) = ka - kb ;
4) ka =0 khi vd chi khi k = 0 hodc a = 0.

19


^

2 {DikiSm chdng tinh chat 3 vdi k = 3)
^a) Ve tam giac ABC vdi gia thiet ^

= d va^

= b.

b) Xac djnh diem A' sao cho A'B = 3d vd dilm C sao cho BC' = 3b.
c) Cd nhdn xet gi ve hai vecto AC vd A'C">
d) Hay k i t thiic viec chufng minh tfnh cha't 3 bang each diing quy tac ba dilm:

CHUY
1) Do tfnh chdt 1, ta cd i-k)d = i-l.k)d = (-1)(H) = - ( H ) . Bdi vdy
ca hai vecto i-k)d vk -ika) diu cd thi vilt don gian la -ka.
2) Vecto — 3 cd thi vilt la — . Ching han — a cd thi vilt la —.
n
n
' 3
3
Bai toan 1. Chicng minh rdng diem I la trung diem cua doan thdng AB khi
vd chi khi vdi diem M bdt ki, ta cd MA + MB = 2MI.
Gidi. ih. 22) Vdi dilm M bdt ki, ta cd
'MA = ~MI + 1A,

~MB = ~MI + 1B .

M

Nhu vay
MA -I- MF = 2M/ + 1A + 1B

Hinh 22

Ta bilt rang / la trung dilm cua AB khi va chi khi M -i- ^ = 6. Tii dd suy ra
dilu phai chiing minh.
Bai toan 2. Cho tam gidc ABC v&i trgng tdm G. Chicng minh rdng v&i diim
M bd't ki, ta cd
MA + MB + MC = 3MG.
3 (BSgiai Bai toan 2) (h. 23)
' a) Tirong tir Bdi todn 1, hay bilu thj cac vecto MA,
vd ^C qua vecto ^G va tCmg vecto 0 4 , ^ .

IAB

GC.

b) Tfnh tong MA -^ JiB + MC. Vdi chu y rang G Id trpng
tam tam giac ABC, hay suy ra dieu phai chiing minh.

20

Hinh 23


»"»'

3. Dieu kien de
hai vectd cung phi/dng

Ta da bilt ring nlu b = ka thi hai vecto a vkb ciing phuong. Dilu ngugc
lai cd dung hay khdng ?

^ ^

~a
b
—>

'c

X

^ <
t

Hinh 24
I

—•

?1| %im hinh 24. Hdy tim cdc so k, m, n, p, q sao cho b = ka ; c = ma ;
b = nc ; X = pU ; y = qU.
Mdt each tdng quat ta cd
/

— •

— - —

[

Vecta b ciing phuang v&i vecta a (a J^O) khi vd chi khi
ed sd'k sao cho b = ka .
?2| Trong phdt bieu d trin, tai sao phdi cd diiu kiin a ^ 0 ?
Dieu yA^n dl ba diem thang hang
Dieu kiin cdn vd dii de ba diem phdn biit A, B, C thdng
hdng la cd sd'k sao cho AB = kAC.
Chvcng minh. Ba dilm A,B,C

thing hang khi va chi khi hai vecto AB vk

AC cung phuong. Bdi vay theo tren ta phai cd AF = kAC.
Bai toan 3. Cho tam gidc ABC ed true tdm H, trgng tdm G vd tdm du&ng
trdn ngoai tiip O.
a) Ggi I la trung diim cua BC. Chdng minh AH = 201.
b) Chicng minh OH = OA + OB + OC.
C) Chvmg minh ba diem 0,G,H thdng hdng.
21


Gidi ih. 25)
a) De thd'y AH = 201 nlu tam giac ABC vudng.
Neu tam giac ABC khdng vudng, ggi D la dilm dd'i xiing cua A qua O.
Khidd
A.

BH II DC (vi ciing vudng gdc vdfi AC),
BD II CH (vi ciing vudng gdc vdfi AB).
Suy ra BDCH la hinh binh hanh, do do
I la trung dilm eua HD. Tix dd
AH = 201.
b) Ta ed
OB + OC = 201 = AH
Hinh 25

nen
OA + OB + OC = 0A +AH = OH.

c) Ta da bilt OA + OB + OC = 30G . Vdy OH = 30G.
Suy ra ba dilm 0,G,H thing hang.
Dudng thing di qua ba dilm ndy ggi la dudng thdng O-le ciia tam giac ABC.
»•»'

4. Bieu thj mot vectd qua hai vectd Ithong cung phifdng
Cho hai vecto a vk b . Nlu vecto c cd thi vilt dudi dang c = ma + nb.
v6i mvk n la hai sd thuc nao dd, thi ta ndi ring : Vecta c bieu thi dugc
qua hai vecta a vd b .
Mdt cau hdi dat ra la : Ni'u dd cho hai vecta khdng cUng phuang a vd b
thi phdi chdng mgi vecta diu cd the bieu thi dugc qua hai vecta do ?
Ta cd dinh If sau day
DINH U
Cho hai vecta khdng ciing phuang a vd b Khi dd mgi
vecta X deu cd the bieu thi dugc mdt cdch duy nhdt qua
hai vecta a vd b, nghia Id cd duy nhdt cap sd'm vd n sao
—*

cho X = md + nb.

22


ChUcng minh
Tix mdt dilm O nao dd, ta ve cac vecto
'OA = a, 'OB = b, 'ox = X (^. 26).
Nlu dilm X nim tren dudng thing OA
thi ta cd sd m sao cho OX = mOA.
vay ta cd
-4

X = ma + Ob (luc nay n = 0).
Tuong tu, nlu dilm X nam tren dudng
thing OF thi ta cd
X = Oa + nb (We nay m = 0).
Nlu dilm Xkhdng nim tren OA vk OB thi ta cd thi ldy dilm A' tren OA va
dilm F' tren OB sao cho OA'XB' la hinh binh hanh. Khi dd ta cd
OX = 0A' + OB', vk dd dd cd cac sd m, n sao cho QX = mOA + nOB, hay
—•

X ^ ma + nb.
Bay gid nlu cdn cd hai sd' m' vk n' sad cho x = ma + nb = m'a + n'b. thi
im - m')a = in' - n)b
n'-n-*
"
Khi dd, nlu m^m' thi a =
^—b . tiic la hai vecto a vk b cung phuong,
m-m'

trai vdi gia thilt, vay m = m'. Chiing minh tuong tu ta ciing co n = n'.

Cau Ii6j va bai tdp
21. Cho tam giac vudng cdn OAB vdi OA = OB = a. Hay dung eac vecto sau
ddy va tfnh dd dai cua chiing
OA + OB;

OA -OB ;

30A

+40B;

lioA-loF.
— OA +2,50B ;
4
'
4
7
22. Cho tam giac OAB. Ggi M, N ldn lirgt la trung dilm hai canh OA vk OB.
Hay tim cdc s6mvkn thfch hgp trong mdi ding thiic sau day
OM =mOA +nOB ;

MN =mOA

JN =mOA

'MB =mOA +nOB.

+nOB;

+nOB;

23