HÌNH HỌC 10 NÂNG CAO TIẾP THEO

Hai vectd cung phiTdng, cung hirdng Vdi mdi vecto AB khac vecto-khdng, dudng thing AB dugc ggi la gid cua vecta AB.. N B D Hinh 4 Hai vecto AB vk CD ciing phuong, va hon the' cac mi

BO GIAO DUC VA OAO TAO

Ik

NANG CAO

I

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BO GIAO DUC VA DAO TAO

D O A N QUYNH (T6ng Chu bien) - V A N NHU CUONG (Chu bien)

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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 l<hac c6 the dijng di/dc

2 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

' ' ' ' i ' ' •' III 11

-01 - 2-010/CXB/733 - 1485/GD 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 g i ?

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 ?

Cdc em cd the trd ldi cdu hoi do khdng ? Vi sao ?

?1

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

la doan thing BA) Neu them dau *- vao ^

diim B thi ta cd vecto vdi di^m dSu la A vk

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

B Hinh 2

B b)

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 D C , 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

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 ?

' D

Ta bilt ring hai doan thing ggi la binjg nhau ne'u ^^^^^^^^^^"-^^

dd dai cua chung bang nhau Tren hinh 5 ta cd A ^ ^ ^^-^^ ^

hWi thoi ABCD Bdn canh cua hinh thoi la bdn ^""^^^^^ ^ ^ ^

doan thing bing nhau Bdi vay ta vilt ^^B

AB = AD = DC = BC 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 ?

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)

N l u 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

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 ) | A C | = |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 H A I 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

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

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

vecto ndo Id vecto a + ib +c)

c) Tii dd cd t h i rut ra ket ludn gi ?

Hinh 11

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

3 Cac quy tac c i n nhd

Tix dinh nghia tdng cua hai vecto ta suy ra hai quy tic sau day

QUY TAG BA DIEM (h 12) M

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 t i l p 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 Hinh 14

12

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

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

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

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

10 Cho hinh binh hanh ABCD vdi tam O Hay diln vao chd trdng ( ) dl dugc

ding thiic diing

12 Cho tam giac diu ABC ndi tilp dudng trdn tam O

a) Hay xac dinh cac dilm M, N, P sao cho

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)

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

• /

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

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

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-^ = CB-CD

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 = ^-CD

Tit do hay neu cdch chiing minh thii ba cCia bdi toan

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

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

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

—»

ndi ring : Vecta b bdng 2 nhdn vdi vecta

a (hodc bdng vecta a nhdn vdi 2), hodc

vecta b la tich ciia vecta a vdi sd'2

'a

/ / / /

k Ve hinh binh hdnh ABCD

a) Xdc dinh diem E sao cho AE = 2BC

U]cA

\ 2)

b) Xac djnh dilm F sao cho AF

18

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

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,

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, IAB

vd ^C qua vecto ^G va tCmg vecto 0 4 , ^ 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 Hinh 23

20

»"»'

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 ?

?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

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

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

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;

— OA +2,50B ;

4 '

lioA-loF

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

23 Ggi M va A^ ldn lugt la trung dilm cac doan thing AB va CD Chiing minh ring

2MAr = AC + BD = AD + BC

24 Cho tam giac ABC vk diim G Chiing minh rang

a) Nlu GA + GF + GC = 0 thi G la trgng tam tam giac ABC ;

b) Nlu cd dilm O sao cho OG = -{oA + 0B + Oc) thi G la trgng tdm

3

tam giac ABC

25 Ggi G Id trgng tdm tam gidc ABC Dat a = GA vk b =GB Hay bilu thi mdi vecto AB, GC, BC, CA qua cac vecto a vk b

26 Chiing minh ring nlu G va G' ldn lugt la trgng tdm tam giac ABC vk tam

giac A'F'C thi

3GG' = AA'+ BB'+ CC

Tix dd hay suy ra dilu kien cdn va du di hai tam gidc ABC vk A'B'C co

trgng tdm triing nhau

27 Cho luc giac ABCDEF Ggi F, Q, R, S, T, U ldn lugt la trung dilm cac canh

AB, BC, CD, DE, EF, FA Chiing minh ring hai tam gidc PRT vk QSU cd

trgng tam trimg nhau

28 Cho tii gidc ABCD Chiing minh ring

a) Cd mdt dilm G dUy nhdt sao cho GA + GF + GC -i- ^ = 6 Dilm G

nhu the' ggi la trgng tdm ciia bdn diem A, B, C, D Tuy nhien, ngudi ta vSn quen ggi G la trgng tdm ciia tic gidc ABCD

b) Trgng tdm G la trung dilm cua mdi doan thing nd'i cac trung dilm hai canh dd'i eua tii giac, nd ciing la trung dilm cua doan thing ndi trung dilm hai dudng cheo eua tii giac

c) Trgng tdm G nim tren cae doan thing ndi mdt dinh cua tii giac va trgng tdm cua tam giac tag bdi ba dinh cdn lai

24

TRUC TOA DO VA Hfi TRUC TOA D O

6 ldp 7, chiing ta da lam quen vdi true va he true toa dd Dl-cac vudng gdc

Trong phdn nay, chung ta se ndi ki hon vl cac khai niem dd

1 True toa do

True toq do (cdn ggi la true, hay true so) la mdt dudng thdng trin do da xdc dinh mdt diem O vd mdt vecta i cd do ddi bdng 1 *

Hinh 27

Dilm O ggi la gd'c toq do, vecto / ggi la vecta dem vi cua true toa dd

True toa dd nhu vay dugc kf hieu la iO;l) Ta ldy dilm / sao cho 01 = 7,

tia 01 cdn dugc kf hieu Id Ox, tia ddi cua Ox la Ox' Khi dd true iO; i) cdn

ggi la true x'Ox hay true Ox (h 27)

Toa dp ciia vecto va cua dilm tren true

Cho vecto M nam tren true iO; i) Khi dd cd sd a xdc dinh di u = ai S6

a nhu thi ggi la toq dp cua vecta u dd'i vdi true iO; i)

Cho dilm M nim tren true iO; i) Khi dd cd sd m xdc dinh dl OM = mi

Sd m nhu thi ggi la toq do cua diem M ddi vdi true ( 0 ; i) (cung la toa dd

cua vecto OM)

h 1

Tren true Ox cho hai dilm A va B Ian liTdt cd toa dp Id a vd b Tim toa dp cQa vectd

AB vd vectd BA Tim toa dp trung dilm cOa doan thing AB

25

Do dai dai so cua vecto tren true

Nlu hai dilm A, F nim tren true Ox thi toa dd cua vecto AB dugc ki hieu

la AB vk ggi la do ddi dqi sdcixa vecto AB trdn true Ox

*Nhu vay

'AS = 1^1

Tix dinh nghia tren ta suy ra cac khing dinh sau day : Tren true sd,

1) Hai vecto 1^ va^ bing nhau khi va chi khi AF = CD

(hiln nhien);

2) He thiic H + ^ - ^ tuong duong vdi he thiic

AB + ^ = 'AC (he thiic Sa-lo)

That vay, 'A£+ 'BC = ~AC o~^l+ ~BCl = ~ACl

<:> (AF -i- 'BC'^ = 'AC7 ^JB + W = 'AC

2 l-le true toa do

Trtn hinh 28, ta cd mdt he true toa dd vudng

gdc Nd bao gdm hai true toa dd Ox vk Oy

vudng gdc vdi nhau

—»

Vecto don vi tren true Ox Id i, vecto don vi

tren true Oy la j

Diim O ggi la gdc toq do True Ox ggi la true

hodnh, true Oy ggi la true tung

He true toa dd vudng gdc nhu tren cdn ggi don gian la he true toq dd va thudfng dugc kf hidu la Oxy hay (G ; ^,7) •

CHOY

Khi trorig mat phang da cho (hay da chgn) mdt he true toa dd, ta

se ggi mat phing dd la mat phdng toq do

Hinh 28

26

3 Toa do cua vectd doi vdi he true toa do

^Quan sat hinh 29 Hay bilu thj

—•

moi vectd a, b, u, v qua hai

vectd r, 7 dirdi dang xl + y~]

vdi X, y Id hai sd thuc ndo dd

DINH NGHiA

>"

O

Hinh 29

Dd'i v&i hi true toq dd iO ; i,j), ni'u a = xi + yj thi cap sd

ix ; y) dugc ggi Id toq do cua vecta a, ki hieu Id a = ix ; y) hay aix ;y) So thic nhd't x ggi la hodnh do, sdthic hai y ggi la tung dd cua vecta a

?l| a) Tim toq dd cua cdc vecta a, b, u, v trin hinh 29

b) Ddi v&i hi true toq dd iO ;i,j), hdy chi ra toq do cua cdc vecta 0, i j l+'j 2]-1 -7-37 43i + 0,l4j

Nhdn x4t Tii dinh nghia toa dd cua vecto, ta thdy hai vecto bing nhau khi

vd chi khi chiing cd ciing toa dd, nghia la

f dix,

.»'

4 Bieu thurc toa do cua cac phep toan vectd

Trong muc nay ta ndi vl bilu thiic toa dd cua cac phep toan vecto sau :

phep cdng, phep trii vecto vd phep nhdn vecto vdi sd

^Cho hai vecto a = i-3;2) va b = i4;5)

a) Hay bilu thj cac vectd a, b qua hai vectd / , j

b) Tim toa dp ciia cdc vectd c = a + b ; d = 4a ; U = 4a -b

3) Vecta b cdng phuang v&i vecta -a ^ 0 khi vd chi khi cd sd

k sao cho x' =kx, y' =ky

?2 Mdi cap vecta sau cd cUng phucmg khdng ?

a)d = iO;5)vdb = ( - 1 ; 7) ; b) M = (2003 ;0)vdv = il;0) ;

c)e= (4 ; - 8) va 7 = (-0,5 ; 1) ; d) fn = i42 ;3) vd n = (3 ; 42)

5 Toa do cua diem

Trong mat phing toa dd Oxy, mdi dilm M dugc xac dinh hoan toan bed

*• •

vecto OM Do vay, nlu bie't toa dd cua vecto OM thi dilm M se dugc xac

dinh Vi le dd ngudi ta dinh nghia

I Trong mat phdng toq do Oxy, toq do cda vecta OM dugc ggi la toq do cua diem M

Nhu vay, cap sd ix ; y) la toa dd cua dilm M khi va chi khi OM = ix ; y)

Khi dd ta vilt Mix ; y) hoac M = (x; y)

Sd X ggi la hodnh do cua dilm M, sd y ggi la tung do ciia dilm M

28

Nhdn xet (h 30) Ggi H, K ldn lugt la hinh ehilu cua

M tren Ox vk Oy Khi dd, nlu M = (jc ; j) thi

OM = ^ r + );7 = G ^ + G^ Suy ra

xi = ^ hay X = ^ ; y] = 'dk hay y = O^K

44i.4

Tren hinh 31

a) Toa dd cda mdi dilm O, A, B, C, D bang

bao nhieu ?

b) Hay tim dilm E cd toa dp (4 ; -4)

c) Tim toa dp cQa vectd AB

r V&i hai diem Mix^j; y^) vd Nix^; yp^) thi

MN = iXf^ -XM;yN -yM)'

?3| Hdy gidi thich vi sao cd ki't qud trin

CHUY

Dl thuan tien, ta thudng diing kf hieu ix^ ;y^f) di chi toa dd

cua dilm M

6 Toa do trung diem cua doan thang va toa do cua trong tam tam giac

KTrong mdt phlng toa dp Oxy, cho hai dilm Mixj^ ; y^^), iV(% ; y^^) Gpi P Id trung dilm cCia doan thing MN

a) Hay bilu thj vectd OP qua hai vectd OM va ON

b) TCr dd hay tim toa dp diem P theo toa dp cCia M va N

^Tim toa dd dilm M' ddi xulhg vdi dilm M(7 ; -3) qua dilm A(l; 1)

I Trong mdt phlng toa dp Oxy, cho tam giac ABC vdi trpng tdm G

a) Hay viet he thiic giiia cdc vectd OA, OB, OC vaOG

h) Jit do suy ra toa dp cCia G theo toa dp ciia A, B, C

b) Tim toq do ciia trgng tdm tam gidc ABC

Gidi a) Ta cd AF = (- 2; 4) va AC = (- 1; 3) Do — ^ - nen AB, 'AC

khdng ciing phuong, suy ra A, F, C khdng thing hang va chdng la ba dinh cua mdt tam gidc

b) Ta cd X + Xg + Xr 2 + 0 + 1 ^A ^ ^B ^ ^C = 1 va yA^yB+ yc ^ 0 + 4-H3 ^ 7

3 3 3 3 3

vay toa do cua trgng tam tam gidc ABC la | 1; — |

Cau hoi va bai tap

29 Trong mat phing toa dd, mdi menh d l sau diing hay sai ?

a) Hai vecto a(26; 9) va 6(9; 26) bing nhau

b) Hai vecto bing nhau khi va chi khi chiing cd hoanh dd bing nhau va tung dd bing nhau

e) Hai vecto ddi nhau thi chiing cd hoanh dd d6i nhau

d) Vecto a cung phuong vdi vecto i nlu a cd hoanh dd bang 0

e) Vecto a cd hoanh dd bing 0 thi nd ciing phuong vdi vecto

7-30

30 Tim toa dd cua cac vecto sau trong mat phing toa dd

a) Tim toa dd cua vecto u = 2d -3b +c

h) Tim toa dd cua vecto x sao cho x + d = b -c

c) Tim cdc sd k, ldi c = ka + lb

32 Cho u, = —i-5j v = ki -4j

Tim cdc gid tri cua k di hai vecto u, v cung phuong

33 Trong cdc menh dl sau, mdnh dl nao diing ?

a) Toa dd ciia dilm A bing toa dd cua vecto OA, vdi O la gd'c toa dd

b) Hodnh dd cua mdt dilm bing 0 thi dilm dd nim tren true hoanh

c) Dilm A nim tren true tung thi A cd hoanh dd bing 0

d) F la trung dilm cua doan thing AB khi va chi khi hoanh dd dilm F bing

trung binh cdng cae hoanh dd cua hai dilm A, F

e) Tu" giac ABCD la hinh binh hanh khi va chi khi Xfi^+ XQ = x^+ xj^ vk

yh

'^yc=yB-^yD-34 Trong mat phing toa dd, cho ba dilm A(-3 ; 4), F(l ; 1), C(9 ; -5)

a) Chiing minh ba dilm A, F, C thing hdng

b) Tim toa dd dilm D sao cho A la trung dilm eua BD

c) Tim toa dd dilm E trtn true Ox sao cho A, F, F thing hang

35 Cho dilm Mix ; y) Tim toa dd cua cdc dilm

a) Mj ddi xiing vdi M qua true Ox;

b) M2 dd'i xiing vdi M qua true Oy ;

c) M3 ddi xiing vdi M qua gd'c toa dd O

36 Trong mat phang toa dd, cho ba dilm A(- 4 ; 1), F(2 ; 4), C(2 ; - 2)

a) Tim toa dd cua trgng tam tam giac ABC

b) Tim toa dd dilm D sao cho C la trgng tam tam giac ABD

c) Tim toa dd dilm F sao cho ABCE la hinh binh hanh

31

6 N TAP cHtraNG i

I - Tom tat nhCimg l(ien thurc can nhd

1 Vecto

- Vecto khac 0 la mdt doan thing cd hudng Vecto-khdng cd dilm ddu

va dilm cudi triing nhau Vecto-khdng ed dd dai bing 0, ed phuong va hudng tuy y

- Hai vecto bing nhau nlu chung cd cung hudng va cung dd dai

2 Tong va hi|u cac vecto

- Quy tdc ba diem : Vdi ba dilm M, A^, F bdt ki, ta cd

MN + NP = MP Quy tdc hinh binh hdnh : Nlu OABC la hinh binh hanh thi

OA + dc = OB

- Quy tdc vi hiiu vecta: Cho vecto MA'^ Vdi didm O bdt ki, ta cd

3 Tich cua mdt vecto vdfi mot so'

- Nlu b = kd id^O) thi |K| = \k\.\d\ vk

b ciing hudng vdi a khi ^ > 0,

b ngugc hudng vdi a khi ife < 0

- Dilm G la trgng tdm tam giac ABC khi va chi khi vdfi dilm O bdt ki, ta ed

OG = -iOA + ^ + OC)

4 Toa do cua vecto va cua diem

- D6i vdi he true (O ; 7, / ) hay Oxy

I) U = ia; b) <^ il = ai + bj ;

2) M = ix;y)oOM = ix;y)

- Nlu A = (JC; y),B = ix'; y') thi 'AB = ix'- x; y'- y)

- Niu M = ix; y) va V = ix'; y') thi

1) U + v = ix + x';y + y') ;

2) kU = ikx; ky)

ii - Cau hoi tir kiem tra

1 Hay ndi rd vecto khac doan thing nhu thi ndo

2 Nlu hai vecto AB vk CD bing nhau va ed gid khdng triing nhau thi bdn dinh

A, F, C, D cd la bdn dinh eua mdt hinh binh hanh hay khdng ?

3 Nlu ed nhilu vecto thi xdc dinh tdng cua chiing nhu thi nao ?

4 Hieu hai vecto dugc dinh nghia qua khdi niem tdng hai vecto nhu thi nao ?

5 Cho hai dilm A, F phan biet Vdi mdt dilm O bdt ki, mdi ding thiic sau ddy diing hay sai ?

a)'AB = dA-OB ; h)OA-0B = BA;

c) OA + OB = -BA ; d)dA + 'Bd = -AB

6 Cd thi dung phep nhan vecto vdi mdt sd dl dinh nghia vecto dd'i cua mdt vecto hay khdng ?

7 Cho hai vecto a vk b khdng ciing phuofng Trong eac vecto c, d, U, v, x,

y sau day, hay chi ra cac vecto cung hudng va cdc vecto ngugc hudng

33

1 0 1

c = —a + —b ; d = -d + -b ; u =3d + 4b ;

2 3 3 _ _ - , l l - » _ - >

V =3d - b ; x = — a — b ; y = - 9 a + 3b

4 3

Hai vecto c vk d co cimg phuong hay khdng ? Tai sao ?

8 Cho tam giac ABC vdi trung tuyin AM- vk trgng tam G- Mdi khang dinh sau

day diing hay sai ?

1

a)AM = 2AG; b)AG = - A M ; c) MG =-GA ;

d)AG = -(AB + AC); e)GB = AG + 'BG

9 Cho bilt toa do hai dilm A va F Lam the' nao d l

a) Urn toa dd eua vecto AB ?

b) Tim toa dd trung dilm eua doan thing AB ?

10 Cho bilt toa do ba dinh cua mdt tam giac Lam-thi nao d l tim toa dd cua

trgng tam tam giac dd ?

Ill - Bai tap

1 Cho tam gidc ABC Hay xac dinh cdc vecto

AB + BC ; CB + BA; AB + CA; BA + CB;

'BA + CA ; CB-CA ; AB-CB ; BC-AB

2 Cho ba dilm O, A, F khdng thing hang Tim dilu kien cdn va du dl vectd

OA + OB cd gia la dudng phan giac cua gdc AOB

3 Ggi O la tam cua hinh binh hanh ABCD Chiing minh ring vdi dilm M bdt

ki, ta cd

MO = -iMA + MB + MC + MD)

4 Cho tam giac ABC

a) Tim cae dilm M vkN sao cho

MA- MB + MC = 0 vk 2NA + NB + NC = 0

34

b) Vdi cac dilm M,Nb cau a), tim cdc s6p vk q sao cho

MA^ = pAB + qAC

5 Cho doan thing AB vk diim I sao cho 2JA + 3 ^ = 0

a) Tim sd k sao cho AI = kAB

b) Chiing minh ring vdi mgi dilm M, ta cd

'MI = -'MA + -'MB

5 5

6 Trong mat phang toa dd Oxy, cho ba dilm A(-l ; 3), F(4 ; 2), C(3 ; 5)

a) Chiing minh ring ba dilm A, F, C khdng thing hang

b) Tim toa dd dilm D sao cho 73 = -3FC

c) Tim toa dd dilm E sao cho O la trgng tam tam giac AFF

IV - Bai tap trac nghiem

1 Cho tam gidc ABC Ggi A', F', C ldn lugt la trung dilm cua ede canh BC,

¥

CA, AB Vecto A'B' ciing hudng vdi vecto nao trong cac vecto sau day ?

(A) AF ; (B) AC ;

(C) BA ; (D) CB

2 Cho ba dilm M, N, P thing hang, trong dd dilm A^ nim giiia hai dilm M

va F Khi dd cae cap vecto nao sau day cung hudng ?

(A) MA^ va PN ; (B) MA^ va MP ;

(C) MP vk PN ; (D) A^M va A^F

3 Cho hinh chii nhat ABCD Trong ede ding thiic dudi day, dang thiic nao diing ?

(A) AF=CD ; (B) FC=DA