Hinh Hoc 10 - C1

Khdi niem vectu di3 dwc sS dung nhi6u trong thqc tg, c h h g han de" bigu di6n n h h g d@ hung nhu lqc, van to"c, ... Nhilng dai luung nay d u ~ cdec t r m g b8i hai ye"u t 6 : cubng do va hu8ng ciia chOng. Hinh 1 m6 t i chuygn dong cfia hai mdy bay : mdy bay thB nhgt bay the0 hudng Bgc, v$n t6c 900 kmlgib, mhy bay thB hai bay the0 hudng D6ng Nam, van t6c 450 km/gi&.

Hinh 1.7

Hinh 1.1cho ta cci hai thang tin : hubng chuygn dong cfia m6y bay (I) 18 hlrbng b$c (til A t&i B), hubng chuygn dong ciia m6y bay (11) 121 h u h g dang nam (ti3 C tdi D), do d21i do* thdng AB ggp d6i do dai d o e th&ng CD the hien V@I t6c. mhy bay (I) gZip d6i v& t6c m6y bay (11).

Cdc hinh ve ,A

-

duuc goi la c6c vectu.

B

C

D

1. Khai niem vectd Cho d o g t h h g AB. N6u coi A 18 digm d&u,B la die"m cu6i thi ta n6i d o e t h h g AB c6 h u h g ta A de"n B. R6 rang vdi m8i doan thing, c6 the" x8c djnh dring hai h ~ d n g .

7

Dinh nghra. Vectu l i mot doan thing c6 h~dng,nghia I A trong hai didm mlit crja doan thdng dZ chi rd di6m nAo IA digm d$u, digm n i o l i di6m cu6i.

--

CHUY :

Ki hi@. Vecta c6 di&m d6u P, didm cu6i Q, d ~ u cki hi& la

V6i hai di6m phdn bi&t A vA B ta c6 hai van tBc kh6c

(doc 18 : vectd PQ). C6c vecta PQ, AB duuc the" hien trong hinh 1.2a).

v?i 6

nhau

.

i@,

Cho tam g16cABC. C6 bao nhieu vecta c6 di6m d i u vA di@mcudi la c6c dlnh c~iatam gi6c d6 ?

-L

/\*> P

b)

a) Hinh 1.2

--Vecta cbn duqc ki h i $ ~121 a,x,y , ... khi kh6ng c i n chi 1-6 die"m d i u va didm cu6i (xem h.1.2b).

2. Hai vectd cung phlldng, cung hddng Hay chi ra c6c vecta cbng phuong, khang cirng phuong trong H.1.3.

VBi m6i vectd

PQ] , d u h g t h h g PQ gpi la gi6

cha vectd d6.

Hai vecta d ~ q goi c la cling p h w g khi chring c6 gi8 song song hogc trfing nhau.

hang khi v5 chl khi va

Ad

cbng phuang.

Hinh 7.3

Trong hinh 1.3 hai vectu vh CD chng phuung vh c6 chng hudng tit tr6i sang phiii, trong khi d6 vh EF cirng phumg n h m g c6 hudng ngtfuc nhau. Ta n6i hai vectu vh CD 18 hai vectu cling h ~ h g cbn , AB vh EF la hai vecta ngwc hrldng.

3. De dbi cfia met vectd Do dhi c3a vectd Ta khang dinh nghia chat ch& khdi niem hai vecta cirng h ~ d n gma chl dbng hinh Anh trqc quan.

Cdn nhir : Hai vectu cbng hudng thi tr&c h&t phai cbng phitong.

18 khoHng c6ch tir di6m dhu d6n diem cu6i

c3a vectu 66, ki hieu lh Vdi vectu

PQ

lal.

ta c6 :

IFcl = p a . Vecta c6 do dhi bhng 1 goi la vectu dun vi.

4. Hai vectd bdng nhau Hai vectd vh duuc goi 121 bhng nhau n6u chdng c6 chng hudng v2I chng do dhi, ki hi$u = I;.

a

bang nhau trong hlnh binh hanh ABCD tam 0 (h.1.4).

C

D Hinh 7.4

-

5. Vectd kh6ng Cho dS'n biiy gid ta biS't m6i vectu c6 mat diem ddu vh met diem cudikhsc biet nhau, c6 h d n g til di6m ddu tdi diem eubi. Biiy gitf v6i m6i di6m A ba"t kki, ta quy udc c6 met vecta dec bi&t mh diQm ddu v& di6m cu6i d6u 118 A. Vecta nhy dude ki hi&u11 vh gqi lh vectd - khbng. Vectd

- kh8ng

Vectu - khdng Vectu c6 didm ddu v A diem cu6i trhng nhau goi lA vectu khbng, k i hi& la 6 :

-

Ta ciing quy Udc vectu - kh.bng chng p h m g , chng hlrbng vdi moi vectd vh do dai cba vecto - khbng bhng 0.

6. Cac dang bai tap cd b8n Dang 1. A'cic dtnh ?not vectu Phmng phdp. De" x6c djnh mat vecta ta chn bigt : - digm d5u vh digm cuo"i, ho$c do dhi v9 hu6ng ciia vecta 66.

-

V i du. Cho vecta cho

Cd

=

v i mot di6m C. Hay dung digm D sao

AB .

Qua C kB d u h g t h h g d song song vdi durirng thhng AB ( d u h g t h h g nay sG trhng vdi d u h g t h h g AB ngu C thuijc d&ng t h h g AB).

Khi 66 c6 hai digm Dl vA D2 tr6n d sao cho CDl = CD2 = AB (h.1.5).

Hinh 1.5

Ne"u chli

9 de"n hudng thi di8m D ph6i d m g la di6m Dz

:

CLj2=AB.

N6'u

- = AB -= 6 . AB = 6 thi die"m D trhng vdi di6m C : CC

Dang 2. C h h g minh hai vectu bhng n.hau

- --

Ngu ttB gi6c ABCD la hinh binh hanh thl AB = DC, A D = BC .

III'

d

1

I1

De"c h h g minh hai vectu bhng nhau, c6 the" dhng c6c chch sau : - Hai vecta c6 didm dhu v9 digm cu6i trhng nhau ; - Hai vectu c6 chng 66 dai vh chng hudng ; - Hai vecta chng bkng vecta th13 ba. V i du. Cho tam gidc ABC nai tigp trong d&ng trbn t i m 0. Goi H I3 truc tam cda tam gi5c ABC. Tia A 0 cdt d&ng trbn (0)tai dikm D. ChQng minh =5:

GIAZ Ta c6 CH N BD vi chng vu6ng g6c vdi AB (h.1.6). Ta lqi c6 BH 11 DC vi chng v"6ng g6c vdi AC.

Do 66

tljr

gihc BHCD 18 hinh binh h8nh.

- V$y HB = CD .

Hinh 7.6

1. Cho hinh binh hinh ABCD. Hay chi ra c6c Veda kh6c vecta khang, c6 di6m d$u v i di6m cu6i l i mat trong b6n di6m A, B, C, D.

Trong so" c6c vecto tren, hay chl ra :

- cdc vectcr cirng p h ~ o n g; - c6c cap vecto cirng h~dng,nguuc hUbng ; - cdc cap vectd b3ng nhau. 2. Cho luc gi6c d&u ABCDEF c6 t2m 0.

Hay chl ra cdc vecta bang vecta I i 0 hoac c6c dlnh cGa lyc gidc. 3. Tir giAc ABCD l i hinh gi ngu

%=

c6 di6m d i u v i di6m cu6i

vi d

1Gl=1 E 1?

-

4. Ba didm A, B, C c6 vj tri the" n i o ne"u AB = BC ?

5. Cho hai hinh binh h i n h ABCD va ABEF. a) Hay dung c6c di&m MI N sao cho

--

b) Chl'lng minh CD = M N .

- -.

6. Cho tir gidc ABCD. Ggi M, N, PI Q I$n l w t I i trung didm c6c canh AB, BC, CD, DA. Chl'lng minh rang MN = QP

7. Hai tam gidc ABC v i AEF c6 cirng trung tuye"n AM. Chl'lng minh rsng

E=G.

1. ~ 6 n g cfia hai vectd

A

L

Tren h.1.7 ta hinh dung mot chi& thuykn di the0 hudng Nam tir' A tdi B r6i di the0 hubng D8ng Nam tir'B tdi C.

I \

Ta tang dat duuc k6't'quA nay ne"u cho thuy&n di t h h g tit A tdi C, vi vay, mat c6ch t g nhign tacoi A B + B C = A C .

---

Hinh 7.7

0,

I

.-.

, L$Y - - digm - - A' AIB'=a,

a vA 6. ~2'yrn6t digm A tuj. );, v( ddgc ggi la tdng c6a hai vectu a AB = a, BC = 6. Vecta va g. Ta k i hi@ tdng clja hai vectd a va 6 la a + 6. Va! Dinh nghia. Cho hai veCtd

+A,

ve

-

B I C ' = b . Hay

chirng minh AIC'= AC.

-

-0

AC = a

4

+ i;

(h.1.8)

C Hinh 7.8

2. Quy t6c hinh binh hanh -

.

-

.

A

Ne"u tll giAc ABCD la hinh binh hhnh thi A B + A D = AC (h.1.9). Dua vao dinh nghia tdng hai vecto, hay chirng minh quy tdc hinh binh hAnh.

Hinh 7.9

Nhu vay vdi hai vectu khSng chng p h ~ u n g vh i; ta c6 thi tinh t6ng +K bhng cBch dhng dinh nghia hogc dhng quy tdl hinh binh hanh.

52. T ~ N GVA HIEU CUA HAi VECTO

11

3. Tinh cha't ccd tdng cac vectd

Do tinh chgt ke"t hop ta s&

bd da"u ngoac trong bi6u thirc vectu

la

(i+GI + S va vigt

z.

i+G+

Ta dB biQt phep cgng cac so" c6 tinh chgt giao holn, kG't h ~ p . D6i vdi phep tong clc vectu cOng c6 nhsng tinh chst 66, cu the" 19 : Vdi ba vectu

i , g,

c tujr 9 ta c6

-----

V i du. Cho ng5 giAc d6u ABCDE nai tigp trong auang trbn

t a r n o . C h h g r n i n h rang O A + O B + O C + O D + O E = ~ . P h l l a g phcip. D6 chting minh vectd dhng hai cAch :

;bhng vectu 6

- Chirng minh di6m d6u v9 di6m cu6i ciia +

-.

ta c6 the"

trhng nhau, tir

-

66 suy ra v = 0 .

- Chirng minh ivl= 0 hosc c h g phuong vdi hai vecto khlc vectd - kh6ng v9 khBng tang phudng, tir 66 suy ra = 6 .

Dutfng t h h g OC 19 met truc d6i xirng clia ngfi giAc d6u (h.l.lO).

cang phudng vdi

06. Hinh 1.10

Met khAc d u h g t h h g OA ciing 19 mat truc d6i xting cOa ngii giAc d6u n&nta c6 : -.

-

-

.

v = OA+OB+OC+OD+OE = od+(OB+OE)+(OC+OD)

ciing phudng vdi Vectd

G.

chng phumg vdi

vh

n&n ;= 6 .

4. Hi$u c6a hai vectd a) Vectu d6i cclia m6t uectu

Cho vectu G . Vectu c6 chng do dhi vB nwdc h ~ h vBi g a duuc goi 18 vectu doi. cria vectu ki hi& 18 (h.l.11).

a,

&13

-a

2 I

C6 phai mgi vectu d6u c6 vecto d6i kh6ng ? H%y chi ra c6c cap vecto d6'i nhau trong hinh binh hAnh ABCD tarn 0.

I

-2

-

Hinh 7 . 7 1

- -.

T a c 6 a + ( - a ) = 0. E)+c bigt, vectu ddi cda vectu

6 Z(1 vectu 6.

b) Hi& cria hai vectu

c.

Cho hai vectd i vh Ta goi hi& cria hai vecto hi& 18 a - b , 18 vectu i + (-G).

-. -

a-g=a+(-b).

a v8 g, ki

Ph6p 1a"y hieu cGa hai vectu goi lh phkp tril vectu. .-.

C) Cbch dqng hi& a

-

- b khi clto

-

a ud

-.-

-

La"y di6m M tujr 9, v5 MA = a, MB = 4b. Khi d6 BA (h.1.12).

-.

=a

-G

Hinh 7.72

That vay, ta c6

-i; = shi . Do 66

i-I;=;+(-C)=MA+BM=BM+MA=BA. -

-

-

-

A

5. Quy tbc ba digm Trong clc b8i toan v6 vectu ta thuang dhng quy t6c sau d6 phdn tich mat vectu th8nh t6ng hoac hi& clia hai vectd, goi la guy td'c ba di6m.

85-l Hay chirng minh c6c quy

I tdc ba digrn.

I

Cho ba digm M, N, P tujr9. Ta c6

-

92. T ~ N GVA HIEU C ~ H41 A VECTG'

13

6 . Cac dang toan cd b6r Dang 1. C h h g minh mot ddng thdc uecto. P h m n g phcip. D6 chirng minh mot ding thirc vecta ta cGng tie'n hhnh nhlr chdng minh c6c dAng thirc dai so" : bign d6i ve" nhy thrSlnh ve" kia hoec bie"n d6i c&hai ve' ciing bhng mot bi6u thdc, hoac ... Trong qu6 trinh bie'n d6i, ta c6 the" sd dung quy tbc ba die"m, quy t$c hinh binh hanh, c6c tinh chgt cua ph6p cong vh t13 vecta, bign d6i t ~ m gd ~ m gv6 mot dAng thirc dling. V i du. Cho bdn di6m A, B, C, D tuj, )i. ChQng minh rang

--AB+CD=AD+CB. GIAI

I

1

Ccich 1. Bie'n d6i ve" nay thhnh ve' kia

1

---

Crinb giac :

s + C D= AD+DB+CD = AD+(=+%)

Cdn chir )i tho tu c6c di6m d$u vA di6m cudi clia c6c vecto 6 hai v&. N6i chung

-

= AD+CB.

Z+CD*AD+Z.

\

Ccich 2. Bie"n d6i tumg ddmg -

-

A

-

AB+CD=AD+CB -

-

.

-

AB-AD=CB-CD

DB=DB

D&ngthirc cu6i dling, v$y dAng thirc c&nchrihg rninh la dling.

Dang 2. Xcic dinh met di6m nhd met ddng th2c vectu

- - -= G

--.

t a bie'n dhi ding thirc 66 v&dqng P M = v , trong 66 P la mot

N&u diCm M thod man di6u ki@n MA+MB-MC

PM

diem c6 dinh,

thl

M. Ne'u

M duqc dung n h th& ~ nao ? I

-

P h ~ u n gphcip. De" x6c dinh 6ie"m M nhfr mot dhng thiic vectd,

0,

18 mot vectd dB bie't. Khi 66 ta vB d l r ~ cdiPm

= P N thi M

=

N.

I

V i du 1. (Trung digm c6a doan thdng) ChQng minh rang di6m I I2 trung didm c6a doan thsng AB khi vA chi khi

z+z= G .

Ngu I la trung di6m cda AB thi

Z+IB=6.

18 vectd d6i cciia

E, ni5n

--t

z.

4

18 vectd d6i c6a Do 66 Nguuc lai, n6'u IA + IB = 6 thi IA = IB vh ba 6ie"m A, I, B t h h g hang. Viiy I I& trung digm clia doan AB. V i dl! 2. (Trong t2m cda tam gi6c) ChOng minh ring di6m --.

G IA trong t2m cda tam gi6c +

---.

ABC khi vA chi khi GA + G B + GC = i .

Trong tAm G clia tam gi6c ABC nhm tr6n trung tuye"n A1 vh GA = 2GI. La"y D lh di6m d6i xm vv6i G qua I (h.1.14), thi tir gi6c BGCD la hinh binh h&nh vh AG = GD.

Do

66

-GB+GC = G D

A

C

B

VA

---GA+GB+GC=GA+GD=~. 1.13 --N g e c lgi, giA sir GA + GB + GC 6. VS! hinh binh hanh BGCD c6 I 18 giao di6m cfia hai d&ng ch6o. Khi 66 -GB+GC = G D . Hinh

=

Suy ra G 18 trung digm clia doan AD. V$y ba die"m A, G, I t h h g hang vA GA = 2G1, nghia 18 G la trong tam cua tam gi6c ABC. Vi du 3. Cho tam gi6c ABC. H i y x i c dinh di6m M thoii m i n -

-

t

di&u kien MA - MB + MC = i .

C

B

Hinh 7.74

Vay M la dinh cGa hinh binh hanh ABCM.

.

- - --

8. ChQng minh rang ne"u AB = CD thi AC = BD . 9. Cho tQ gi6c ABCD. Hai di6m E, F IAn ldgt IA trung digm clja hai canh ddi AB v i CD, 0 l i trung di6m cira EF.

----

ChDng minh r i n g OA + OB + OC + OD = 6 10. Cho s6u digm M, N, P, Q, R, S ba"t kki. ChDng minh rang

11. Hai vectu gi de" :

a vi 6

la + 61 = la1+ li; b) la-61=lal-161

khdc vectu - khBng ph2i tho2 man dig" kien

?

a)

?

12. Cho tam gi6c ABC vuang tai A,

= 3 0 ° , BC = a. a) Ve v i tinh d6 dai vectu AD = AB + AC .

-AE=AB+AM.

b) Goi M l A trung digm cda BC. Ve v i tinh 66 dai vectu 4

-

C)ChQng minh ED = BM. 13. Cho luc gidc dbu ABCDEF tdm 0. -

-

-

+

6

a) ChQng minh O A + O B + O C + O D + O E + O F =

--

b) Goi MI N, P, Q I i n Idgt l i trung di6m clia AB, CD, AF, DE. ChQng minh M N = PQ . 14. Cho hinh binh h i n h ABCD. Dgng c6c digm M, N tho3 man :

-

d

c) ChQng minh M N = BA

---

15. C h h g minh rang hai tam gi6c ABC v i A'B'C' c6 tAm khi v i chi khi : A A ' + BB'+ CC' =

Cho vectd i # 6 . Ndu ta 16'y tong vdi chinh n6 thi ta d ~ q c vectu cDng hudng vdi vh c6 do dhi bhng 2 lhn do dhi clia Vectd nhy t a ki hieu 18 2 vh goi 18 tich cfia sb 2 vdi vectd i .

a.

a

Tdng quAt, ta c6 djnh nghia sau.

1. Dinh nghia

-

'Cho so" k

Mink hoa :

3

;t 0

v i vecta

i +g.

Tich ccia

s 6 k vdi

mot vecta, k i hi@ul i k i t cDng hudng vdi

3 3

hudng vdi

+

-2v

7

a n&

vecto.

i

Ii

a n 6 k > 0, ngmc

k < 0 v i c6 do d i i bang ikllal.

~ a ~ u ~ l r d c ~ ka 6== a6 .;

f-

-+g

-

V i du 1. Cho hinh binh h i n h ABCD t2m 0; I l i trung digm 4

-.

clja BC. Tinh c6c vecta 01, OB the0 c6c vecta a = AB va

6 =sd (h.1 .IS).

Ta c6 01 la d&ng trung binh trong tam gihc CAB nen

-u-2v:-

Hay vB

chc vecto 3; + 2;

vA

Hinh 7.15

2. Tinh chat

Dga the0 djnh nghia cfia ph6p nh3n mat so"vdi mat vecta, c6 the" chlihg rninh c6c tinh chgt sau. Vdi hai vectd

i

vh

bgt kki vB moi sd thgc h va k ta c6 :

V i dl! 2. ChOng minh rang

aNH$V XET :

-

a) Di6m I Ih trung di6m clja doan AB khi vh chi khi vdi

Ta v i du nay suy ra n6u A1 la trung tuykn cha tam gi6c 1= -(AB + AC) ABC thi

-

2

.

di6m M bgt kki ta c6 M A + M B = ~ ~ . b) Digm G I2 trong t3m clja tam gidc ABC khi v i chi khi _

.

+ MB+ MC= 3%.

vdi di6m M b2't kki ta c6 MA

M~+MB=~MI+(IA+IB) Hinh 7.16 -

Theo tinh chgt trung di6m cSa doan t h b g trong $2, I 18 trung di6m cSa AB khi v8 chi khi IA + IB = 6.Ttf 66 suy ra di&uc i n c h h g minh. M 4

b) Tlrcmg tq cAu a) ta c6 (h.1.17) :

-MA MG,+GA - -MB=MG+GBOM =

MC=MG+GC Suy ra N @ u' M tdng vdi G thi ta c6 m@nh d& : G la trpng tarn clia tam gi6c ABC khi vA chi

--khi G A + C B + G C = ~ .

C

6 A

_

.

-

-

MA+MB+MC=3MG+(GA+GB+GC) Hinh 1.77 Ta bigt G lla trong t6m cGa tam giAc ABC khi vh chi khi

GA+GB+GC=~. T a d 6 suy ra di&uc i n c h h g minh.

3. ~ i e kien u de"hai vectd ccng phddng

a

'm NHAN XET: Ba d i h phan biet A, B, C thing hang khi va chi khi c6 sa'kdk z = k z .

Vecta &ng p h ~ u n gudi vecto k saocho kg.

a=

6# 6

khi ud chi khi e6 mot so"

Chdng minh

a

Ne"u = k6 thi vA 6 cirng phddng. cirng phucmg. Nwuc lqi, giii sa vla

a

W 6 t 6 n h 161 t 0. ~a IQ k

Khi 66 ta c6

= kg.

la1

= - nhu

lil

a

VA

6

chng h ~ h g

4. PhBn tich met vectd the0 hai vectd kh6ng .cung

Cho i vA vectu tujr 9.

Ginh giae : N h digm C n i m trGn d&ng thing OA hoac nam tren d&ng thang OB thi h va k se la c6c so" nhd thh nao ?

18 hai vectu khang ccng p h ~ n gvh

c

18 met

-

N& c6 hai so' h vh k sao cho -d = h i + kg thi ta n6i vecta c . phcin ttch d ~ u c(hogc bi6u thi d q c ) qua hai vecta khbng cung phmg vb 6 . Ta c h h g t6 ring tan tai va duy nhgt cap so" h, k nhu vay.

La"y didm 0 tuy

Ji,

_.

-

8

-.-

-

-8

vE OA = a, OB = b, OC = c (h.1.18).

+ b

7 f

___)

c*o

3

:.

B

B'

Hinh 7.78

KB CAI I/ OB (A' thubc d&ng thing OA) vA CB' /I OA (B' thuac duang thing OB). Khi 66 c = OC = OA ' + OB ' . 4

Cho i va 6 khang cirng ph~ong. ChOng tb rang hi+kG = 6 khi va chi khi h = k = 0.

-

F

-

- chng phwng n&n c6 so" Tlldng tu, OB' v& c h g phlldng OB'=~=.

Vi OA ' vB

-

h dk OA ' = hOd

.

n&n c6 so" k de"

~066;=hi+kg.

5. Cac dang toan c d bdn

Phmng pkdp. Dd c h h g minh ba dikm A, B, C t h h g hang ta chimg minh = kAd hogc Bd = k s d .

AB

V i du. Cho b6n digm A, B, C, M thoi man he thUc

=+2E-3Z=6. ChBng minh ba di6m A, B, C thdng hing.

Ta bi6n d6i he that trong d&b8i v&dang

AB = kAd . Ta c6

V$y ba di6m A, B, C thhng hang. Dang 2. Phcin tich mot vecta the0 hai vectu kh6ng cling phumg phumg phbp. D6 phdn tich vecto

a

theo hai vectd kh6ng

c h g p h ~ u n g vB 6 ta vG hinh binh hanh OACB sao cho c h g p h ~ u n gv6i a, OB chng phuung v6i 6 . Vi = ha,

--

O B = k i ; nsn O C = h i + k i ; . Ta ciing c6 th6 v$n dung linh hoat vi$c phdn tich vecta the0 quy t&cba digm, quy t i c hinh binh hanh, ... V i du. Cho tam giAc ABC c6 trong t2m G. a) Hay ph3n tich

theo hai vecto

ii

z.

b) Goi E, F I i hai digm xAc dinh bdi c6c di&u ki?n :

Hay phan tich

Til di6u kien Vi@c phan tlch mot vectu the0 hai vectd khang cirng ph~fongla co sd d6 dt.ta ra dinh nghia.to9 do trong 54.

EA = hay

Ed = 2

-2(EA + AB)

=2

the0

vi

a,

E suy ra

G .

Til di&ukien 3 z + 2FC = 6 suy ra

hay 5 2 + 2 z = G .

16. Cho tlll gidc ABCD. a) H2y dung di6m G tho4 m2n di6u kien

--GA+GB+GC+GD=Z.

b) ChQng minh rang vdi moi di6m M ta c6

--

17. Cho tam gidc ABC. Hai di6m M va N thay d6i sao cho

G=~M~+~MB-MC.

b) ChOng minh rang d&ng thsng MN lu6n di qua mot di6m c6 dinh. 18. Cho tam gi6c ABC. Goi D, E, F I$n hut la trung digm cda BC, CA, AB.

-

4

-

ChOng minh rang AD+BE+CF = 6 . 19. Cho tQ gidc ABCD va c6c di6m M, N, P, Q I$n l l l ~ l tlri trung digm cda AB, BC, CD, DA.

ChQng minh rang hai tam gidc ANP vA CMQ c6 chng trong tdm. 20. Cho tam gidc ABC. Ggi C v i H IAn l w t lri trong tam v i twc t i m cllla tam gidc d6,O la t i m va AA' la d&ng kinh cda dcrdng trbn ngoai tigp tam gidc. Chlllng minh :

Ttf d6 chlllng minh ba di6m 0, G, H thsng hang. 21. Cho tam gi6c ABC. Tren BC Idy digm D sao cho Goi E I i digm tho4 m i n di&u kien 4EA+2E+3Ed=6. a) Phdn tich

Ed theo

vi

Ed.

b) ChQng minh A, E, D thdng hring.

2sd = -BC.

5

54.

HE TRUC TOA DO

21 22. Cho tam gidc d&uABC tdm 0, M I2 mat di6m ba"t kki trong tam gidc. K6 MD, ME, MF I$n luut w8ng g6c vdi cdc canh BC, CA, AB. _. 3a) ChUng minh M D +ME + MF = -MO . 2

-

- +- c6 gi6 fri khdng d6i. IMD ME + M F ~

b) Tim tap hap trong tdm tam gidc DEF khi M chuygn dong sao

tho

23. Cho tam gidc ABC. Cdc digm M, N thod man :

ChUng minh dudng thang M N di qua trong tdm G ctja tam gi6c ABC. 24. Cho tam gidc ABC. Tim tzp hop cdc di6m M tho3 man :

25. Cho tam gidc ABC va dllirng thing d. Tim digm M tr@nd sao dat gid tri nh6 nhdt. 26. Cho tam gidc ABC noi tigp trong d ~ d n gtrbn tdm 0 d d n g kinh AD. Goi G va H Idn ll@t l i trong tdm va twc tam clja tam gidc d6. ChUng minh :

27. Cho tam gidc ABC. Goi D va I l i cdc di6m xdc dinh bbi c6c dang thrlrc vectd :

a) Phdn tich

Ad the0

vA

z.

b) ChUng minh ba digm A, I, D thang hang. c) Goi M l i trung di6m clja AB. Hay xdc dinh digm N tr@nAC sao cho ba duang thang AD, BC v5 M N dbng quy.

54.

HF TRUC TOA DQ

1. Truc toa do a) Truc toa da (cbn goi t?it 18 truc) 18 mat d ~ h thdng g tren dlr dii x6c djnh mot di6m 0 goi 18 di&m g6c v8 mot vecta d m vj + e . Ki hi@ truc 66 18 (0; ) (h.1.20).

I

CHUY:

x'

Thay vi ve mfii ten 6 digm cusli cda vecta dun vi

g,

ta thuang vi! mUi ton 6 cu6i tia Ox.

I

0

i

'

M

L

b

2r

X

Hinh 7.20 -, sao cho 0 1 = e . Goi tia d

LS'y digm I

c higu la truc (0; ) cbn d ~ q ki

01 18 Ox,tia 66'i 18 Ox'thi

true XIOX.

b) Toa do ciia digm d6i vdi mat truc Cho M 18 mot diem tug jl tr&ntruc (0; ). Khi d6 cb duy nh6t mot so" k sao cho OM = k; . Ta goi s6' k db la toq dB cria di6m M d6i vdi truc toa do 65 cho.

2% NH&N XET : Ngu

-

C)

ccbng hudng vdi

AB = AB, cbn n@'u

thi nwc

4

h d n g vdi

e

thi

= -AB.

Cho hai di6m A, B tr&ntruc (0; ;).

AB = d;.

Khi d6 c6 duy nh6t m@ s6 d sao cho 18 do dbi dai sb ciia vectd kB d6i vdi d=

a.

true

Ta gqi so" d 66 dB cho v8 ki hi&

N6'u - hai di6m A v8 B c6 toa 68 l6n l ~ d t18 a vB b thi AB = b - a .

2. HQ truc toa dQ

VG hai truc to? 60 vu6ng g6c (0;T ) vB (0; 3 ) (h.1.21).

;,3)

Ta goi he gem hai truc to? do 66 18 h$ true toq d6 (0; hoac h$ truc toa do Oxy.

Hinh 1.21.

Di6m 0 goi la gd'c toa dB. Truc Ox dwc goi la truc hoanh, truc Oy goi 18 true tung. Mat ph&ngma tr6n 66 dB cho mat he truc toa do Oxy duqc goi t$t 18 m@tphdng Oxy.

3.T o a dQ cGa vectd ddi v6i met he truc toa dQ a) Cho vecta ;trong mat ph?ing to? do Oxy. N&u phAn tich -, vectd ;the0 hai vectd i vB ta d m ;= xi + y j

3

-.

,

54. HE TRUC TOA 00

2 i3

thi ta n6i cap so"(x ; y) 18 toq do cria vectu ;d6'i vdi he toa dQ Oxy vB vi6t ;= (x ; y) hoac (x ; y). So" thir nhgt x goi la hoanh do, so"tho hai y goi 18 tung do ciia vectu ;.

SL NHAN x h: Tb dinh nghia toa dG clja vecta ta thgy : ngu

= (x ; y),

2 = (x' ; y') thi u=ule

b) CAch tim to? 66 ciia vecta

~5 vectu

;

Y'

OA = G .

Goi Al , A2 1An luut la hinh chi& ciia A li3n Ox va Oy (h.1.22).

y = y'.

A

-. OA, = y j . 4

OAl = xi ,

Ta c6

"

vectd i,j

-#

'

f

f

, 2i-3j,-1+j

.

Hay chQng minh cang thQc ben.

+

-.

= xi+ yj .

C

A,

X

-. -

Hinh 1.22

-

Do d6 u = x i + y j . c) Toa do c6a mot di6m

'1

Nhd vay cap so" (x ; y) 18 toa do c3a di6m M khi vh chi khi = (X ; y). Khi 66 ta vigt M(x ; y) hogc M = (X ; y). So"x goi la hoanh do, so" y goi la tung do c6a M.

0

.y 7

Hinh 7.23

d ) Li&nhe gi2a toa do cda di&m va toa do c3a vectu Cho hai di6m A(XA; yA)v8 B(xB ; yB).Ta c6

-

AB = (xB-xA ;yB - y A )

4. TOU do Hay chirng minh c6c c6ng

-

--

Trong mat ph&ng Oxy cho di6m M tujr 9. Toa do cfia vectu d6i v6i h$ tog do Oxy duuc goi la toa do c3a digm M d6i v6i he truc 66 (h.1.23).

0,

7

0

= OA1 + OA,

V4y Q------7 tog dij cfia CAC

-

+

-+-+

-

cuc vectd u + v , u - v , k G

-

Ta c6 cdc c8ng thilc sau : Cho

-

-

u = (ul ; u2) ,. V = (vl; v2). Khi 66 -, u + v = (u1 + v1; Ug + v2) -.

.-.

u - v = (UI

- v1; u2 - v2)

I

X

C

x

a = (3 ; -4) , b = (2 ; 5). Sim toa do c6a 4

V i d". Cho hai vecto 4

-

4

c6cvectu a + b , a-6.5;. Tac6

a+g =(3+2;-4+5)=(5;1)

- -

a-b =(3-2;-4-2)=(1;-6) 5

= (5.3 ; 5.(-4) = (15 ; -20)

Lm 9 : Vdi khAi ni$m ha do, vectu d ~ u ccoi 18 mot c$ 16p t o h tren chc vectd trb t h h h ph6p toAn tr6n chc s6. MMai dZmg thI3c vectu tuung dudng v6i hai d h g th13c trGn cdc s6.

5. Cac dang toan cd b6n Dang 1. Tim t o a do cria c6.c di6m va. toa d6 cria cac vectu P h u w phbp. Trong mgt phfing Oxy, dd xdc dinh toa do c6a mat di6m M ta lam nhu sau :

KG MM1 I Ox ; MM2 I Oy. Khi 66 x y=

Ohia

-

= OM1 la hohnh 66 vh

18 tung dij c6a didm M.

Ne"u M 18 di6m dgu hoac di6m cu6i ccba mat vectu i nho 66 vA bi6t to? do c6a thi dirng c8ng thI3c lien he giSa to? 66 cba vectu vh c6a digm dd tim toa do c6a M.

D6 tim toa do cia mot vecta ta c6 th6 s3 dung dinh nghia hoec tim toa dij didm d6u vh didm cu6i cclia vectd 66 r6i sir dung c6ng thI3c lien he giila toa dij c6a vectd va toa do c6a di6m. Vi d". Trong mat ph$ng Oxy cho luc gihc d6u ABCDEF canh a, c6 hai dlnh A v i D n5m tr6n truc Ox vA c6 tam Ii g6c toa do. Tim toa do cdc dlnh clja luc gidc 66 v i toa do -

d

d

tic vecta AD, BF, BE.

Hai dinh A vA D n h tr6n Ox vh A 0 = OD = a n6n A = (-a ; O), D = (a ; 0) (h.1.24).

Tam gidc OCD 18 tam gihc 13th nkn chiin d&ng cao H k& ta C cda n6 1h trung di6m cclia OD v8 CH = . DO 86 2

54.HE TRUC TOA Do

25

Hinh 7.24

Ta c6

AD = 2ae

n&n

AD

= (2a ; O), chn

@ = -a&;

n&n

dQ cfia 3 the0 to? dQ ciia B vh E ta duqc BE = (xE- xB ; y~ - yB)= (a ; -a& = (0;- a &). Tinh to3

).

Dung 2. Tim m6i ZiZn he gigs toa dQ cua cbc di8m va toa do cua cac vectu dga trdn mot ddng tit2c uectu P h m g phcip. MBi vectu ddqc hohn tohn xAc dinh b6i mot cap so" lh toa do ciia n6. Hai vectu bhng nhau khi va chi khi cAc toa do tuung h g ciia chcng bhng nhau. M6i ding thirc vectu cho ta hai 68ng thoc giaa cAc so" hoac he hai phuung trinh b8c nhgt hai i n . V i du 1. Cho doan thang AB c6 A(xA ; yA) VA B(xB ; ye). Khi 66 toa 60 trung di6m I(xl ;yl) c3a doan thang AB IA

--

Ta c6 I 1h trung di6m ciia AB khi v9 chi khi IA + IB = 6 . Vi

G .

= (xA -

-

; YA - yI) ~h IB = (xB LA + IB = (xA+ x~ - 2xI ; y~ + YB - 2~1). XI

XI

; YB

- yI). DO 66

4

-

.

-

Vi LA + IB = 0 n6n ta c6 xA + xB- 2xI = 0 vh yA + y~ - 2yr = 0. Suy ra

-

0, ,

Hay chirng minh cang thirc d i nPu 6 vF du 3.

Vf du 2. Cho tam giic ABC c6 ba dinh A = (xA ;yA), B = (xe ; ys), C = (xC ; yc). ChUng minh rang toa do trong tsm G cGa tam gi%cABC dlwc tinh the0 c6ng thllrc : XG =

X A +XB +XC

3

YG =

YA+YB+YC 3

--

Gdi j, : Dhng cbng that GA + GB + GC = 6 v&i G 18 trong tiim tam gi6c ABC. V i dl! 3. Cho tam giiic ABC c6 ba dlnh A = (2 ; 3), B = (6 ; 71, C = (1 ; 2). Tim tog do trung digm I cda canh AB v i trong t2m G cda tam gi6c ABC.

Viiy I = (4 ; 5). Theo cbng thilc trong tiim tam gihc, ta cb :

28. Cho ba di6m A, B, C b$t kki tren true ( 0 ;

- - ChUng minh r i n g AB+ BC = AC .

c).

29. Cho b6n digm A, B, C, D b$t kki tr6n tr"c ( 0 ;

).

Goi E, F, G, H I$n i ~ u la t trung digm clja AB, BC, CD, DA. ChUng minh :

30. Cho tam gidc d&uABC canh a c6 B trcng vdi g6c toa do, C nhm tren ttyc Ox.

H%ytim toa do c6c dinh, trung digm canh AC v i trong tdm G cda tam gidc ABC.

-.

-

4

31. Cho u = ( 2 ; -5), v = ( 3 ; 4 ) , w = (-5; 7).

a) Tim tog do clja vecta

+ 33 - 5 ; . 4

b)Tim toadovecta

-

-

*

sao cho U + ~ V - ~ W + X = ~ . *

+

= (7; 2) theo hai vecta u v i v .

C) Phdn tich vecta

d) Tim x b i k rang

6

= (6;

X)

cDng p h ~ a n gvdi

;.

-

94.HE TRUC TOA DQ

27

32. Cho c6c digm D(2 ; 3) , E(5 ; -1) , F(-3 ; 4) I i n lUut la trung digm clja chc canh AB, BC, CA clja tam gi6c ABC. Hay tinh toa d$ c6c dinh c6a tam gi6c. 33. Trong mat phing Oxy cho ba digm A(2; 5) , B(1 ; 2 ) va C(4 ; 1). =fi.

a) Tim tog da di6m M sao cho z + 3 =

b) Tim toa d$ digm D sao cho tii gi6c ABCD la hinh binh hinh. C) Tim di6m E tren d ~ d n gthing song song vdi Oy v i cdt Ox tai di6m c6 hoanh d$ bang 3 sao cho ba di6m A, B, E thing hang.

-

-.

34. Cho ba vecta a = (2 ; 1) , b = (3 ; -4) a) Tim tog do clja VeCtd 2 i b)Timtoad$clja C) Hay phan tich

-

+4G -5c.

saocho the0

-

, c = (-7 ; 2).

vi

+ 2 ; =5G -

z.

6. 3

35. Cho ba di6m A(-2 ; -1) , B(3 ; - 1 2

, LIL

; 1). Hay

ba di6m A, B, C thing hang. Hai vectd hay n g ~ u ch ~ d n g?

vi

AC

C I I I J I I minh ~

chng h ~ d n g

36. Cho ba digm A(-1 ; 4) , B(-3 ; -2), C(2 ; 3). a) HZy tim toa d$ di6m D sao cho tQ gi6c ABCD la hinh binh hAnh. b) Tim digm E trGn Oy sao cho ba digm A, C, E thang hting. 37. Trong mat phdng toa do tim tgp hup c6c di6m M(x ; y) tho3 man : a)x=O;

-

138. Cho u=-1-5T,;=m~+10T.

3

Tim m

d6

;v i ;cang ph~ang.Khi d6 ;v i ;cang hudng

hay nguuc hudng ? 39. Cho Iuc gidc ABCDEF. C6c di6m P, Q, R, S, T, V IAn Iuut l i trung digm cdc canh AB, BC, CD, DE, EF, FA. Chiing minh rang hai tam gidc PRT v i QSV c6 chng trgng tam.

Hay gidi bAi todn tren bang p h ~ n ph6p g vecta v i phudng phdp toa dij.