r
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'uOr roAN uOi
YIOT NAM
lft(*4t:!:?i:j.. a t i'
,"s..,:r,'r*,
,
:i-"+-";,Y"-'':
;ilbe roz
iir,- )i ;,
eAo
Cttfi nhiim; NGUyET{ Cixn roAx
I'ru s6 : 70 frhn Hung Eqo, Hi N6"i
i r'-
t,^-,
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i-iAt rHANG MoT K?
AA
l'ittr kg tdr.r soon i HOAKG CHIJI{G
DAy n6i: 5ZB2S
xfi cnuyEN vot cAc aAN rai yda fos,y
r&H 0UAFI TRSp€ffi flErA M&T eAgE{ [* **i$i
*
f
:
[.
*.
'r'f
t
C{c b4n thin m8n!
Ch^ung ta cti quen voi cac s6 lhr.ic (bao g6rn
td.t ci cic s6 hiru tj' vi cric sr1 vo l;i) vi thuong
nhin ehtng dudi g6c it( tt 16 n. IJAy gid ta h[y
c6 c{ch nhin mdi vE cAc s6 d6 Cac sd nhu 2, ll,
{ Z n, v.v... ilem dlt rruoc m6t eiri gi d6 dBu
q6 lf ngbia nhu mgt lQnh phii thgc hdntr tr,6n
cii tI6. Vi dU 2 denl dit tru6e S cd nghia la phrii
l{y 3 g 3, dtern ddt trudc c c6 nghia lI phii lfly
r * r, dem dft trrr<ic V (ky hi€u rrrdt rec to)
c6 nghia lir ph&i Iiy nr0t vecto cirng phtroug.
hudng vdi Ir-vir dti g$p hai lhr:7tliui, il,
NGUIEN ciNH TOAN
Duui g6c r10 ,t ICnh > niy thi phep c6ng vir
pbep nhrin nOn hi6u nhu thd nao? Vcri hai sci
thr,rc a, L"f ,g m,i[,vic lo I to.f,,
fu * b) V : aY -t bV, (ab) V - a(by) = bht,).
\-a)" :
1) tr{udn ti:uc irirnh ldnh a + b thi ta thi hlnh -
ridng rE tring lQnh r.r,r'a D (thco nr6t tht tqt nio
ct)ng duoc) rdi cgng cac kdt quir lai.
2) lludn thlre hirnh lAulr ab thi ta thr,rc hil
m$t trong hai l6nh a, b trtrdc rdi dlem l,:uir r.
lai thi hnnh ddi v6,i kdt qua clqt cTrlgc uau
dA thi h&ntr xong i6nh thir nhdt.
Edn {lny, ch$,c c6 ban dfl sdt rui}t rnurjn
nhtrng nhin duoi goc ilg a lgnh > nay thi
gi? Ta h5.y nhin hinh l. R6 rirng li ldy L
s6 thuc nao lAm l6nh ta etng kh6ng thr
kh6i duolg th&ng D, nghi,r li n6u ta ser r
ve,cto cung g6c 0 vi rudt ph6t trt nr6r (3)
V nIm trOn:ndt duong th$ng ,t") rhi d)
cac vecto aV ludn luOn bu,it: phir no 'Q'
dudng thfing D" Ch[ug tran vtri c:1,e sd thrni frI A Q.
th} aio tim ra nr0t e{i l6ph d$ tu yd, ls
Itrinh. I
*+,
iiLr,,r,c vecto U tr€n hinh t"
'la mu6n thoit iih6i th€ gi6i chat hgp c&a
d.t&ng th8ng D d& di vuo virng -'riy trong thd
gi0i r$ng ldrn erla todn ni[t ph8.ng' Ldm sao
bAy gio ? M$t vecto' nhu 7 bu$c phli nLm trdn
du&ng th&n6 .* r'ie m$i vecio uhu-U c6 th6 c6
nhtrng vi tr1 triy ;f trong mlt ph&ng thi kh{c
nhau co b&n & ch6 nLo ? Ndu nhu ta dinb hu6ng
du&ng th&ng4 D thi ta c6 thE x6.c dinh V chl
bing Lqt sd thqrc thoi,j6 Ii hoauli dQ u cria
n6. Cdn ar6t v6cto nhu U thi kh6ng thi rAc dinh
ehi beng m$t sS thqc, mi phli blng hai.sd thr3c,
€6 lar hoann d6 a vir tung riQ u cria n6 (hinh 1)'
Dern lQnh c ttri hinh drii voi Y thi {li6u d6 citng
tuong tluong v6i vifc dem i$nh do tbi hnnh d6i
vdi hoi.nh iiO c'gn6, tuc li: dem s'j thuc a
nh&n v&i vectc, V thi hoinh d0 u cira V sE drroc
ahAc l6n v*i a, vtr nguqrc l4i dem hoinh il$r;
nh$n v&i a ihi ta duoc hoAnh d0 c'ia vecto aY'
fiiBu niy ggi cho ta 1i nghia tao nen uhirng l0nh
nrrvi e'r,ii vdi c6.e o*"io I.oug m5t phing b&n11
*:'icir dung nhirng lQnh cO il6i vtvi iroAnh t[6
vfi tung d[ cr"ru "iii ng. F]b cho huong suy ttghi
ni"r, cirig"c cr,i th6, ta hEy x6t thrl mQt tlutrng
llr,:p dfi.c iridt : 'f a nrudn t4o nOn m$t cai tdnh
eo"t6c dqrrg Ii. khi dem thi hirnh d6i v$i m$t
vecttr U thi quay nd di 604 thilnh mQt vecto
+
ktrie U; {hiuh ?i. Goi tqa dQ eria U lir n' g vir
lE a* *f," U' ltr o'r g'. Ndu g6c cfia trgc or vOi
u la cs thi
;r = OLIcosfr
g s {.iUsine.
c'=$U'c*s(w*6{}")
=: tlU'cosc{ cos60o 'OU' sina sin S0o
g'sOU'sitl{o(+60*)
- g{J'sipg,cos60o -!- OU'cosasin 60o.
Vi OU' = OU (phep iiuay), nbu Cudi ctlg ts c6: i
s'=-1 n- lT r,
22
' /!- r+ !s.
u=-, z
Yi du cg thb niy gqi eho ta f nghi tqo oln
rilriri:g IQnh mdi b&ng b6n l€uh ct a, b, c, d l6'c
dQrig IOn hoirnh d$ vi tung tIQ crlr cic vecto ::
t'= ar * bg
g'=cr*dg
(tlong vt clg tron a=112, b---litZ, "='liit,
cl= 71il.
Ta sip x{p b5n s6 thqc a, b, c, d thirnh m0t
bin g.
(::),' vi6t? = (i'r)n
tf *h m&i cho b&i na,,s (l l )otn"
dB chi ling
vecto Lr th{'nh veeto 7.
Ngudi ta ggi mCr. biog bdn s6 thgc nhu t$y
Iir n:0i ma traa. Nh'i vAv m6i sd thirc cho ta
m$t lQnh cir c6n m$t ma trin cho ta mQt IQnh
mdi. (l6c l$nh cil giam him chring ta lai trong
thd gioi ch{t hqp cia mOt dudng th&rrg con cdc
tenh m6i thi giii phong 'eho chring ta duqc vung
u'6y trong ci mit phing r$ug l6n' Kbi nhin m$t
*, t.{tr, "ho t" m6t lQnh m0i nhu tr€-n thi ta'
ft-,ai gtti nhd ring: Mu6n c6 hoinh iIQ cria vecto'
mdi (recto 7) tui dirng hai lQnh cfi & d6ng dlut
cria ma trin cdn mudn c6 tung d$ cfra vecto'
*Oi ,fri dnng hai lQnh ctr & ddng thu hai eira
"r" ,ta". rrJng ci hai trulng hqp' lQnh ctr &
;;;,* [ og-*,i tinh d6i 'noi honnh d0 cfia
vectcv cfi (vectcr7) cdn lQnh ct 6^ b6u phli lir
db thi hinb d6i vfi tung d0 cria vecio cfl'Trong
cac rna trfn nhu tr€n ta hdy chl y d6n cdc ma
/a 0 \
trfln risng t; ;J. MQt ma tran c6 dsrg niJ
cho ta nr$t lQnh mi khi thi trirnh d6i v0i m6J
;;;,;" cd roa d0 (r, v) thi se co-50t vecto rr'
c6 toa d0 (ac, ag), tuc l?r vecto aV'" V{y lQnh
ir,ri=*t o b&i ma trin r.r6i ri'€n cri tf C dgng honn
;;;;;"; t nrrtr't6'c dr.rng cira l€nl cii cho b6l
sii thurr a. Nhn vqy cati ltnh ct zhirng qua chi
Ii DoJt truong liqP diic biQt ctra c{c lQnh m6'i"
ir ilul 2
t'
I
t
I
I
1","
i:
N
i-
I
I
fr€n co s& do tluug & goc dg < i6nn r ta c6 tht
vidt:
, lQnh
hay v[n ti.t :
(:")((:,"' )7) =
vi l€ d6, ta c6 thB vi6
(:')(: i) =(:;
vi goi ma tr6.n (l€nh)
ma tran (lenh) & vd tru
0 dAy, hh6c v<ri tru&u
niry ndi chuug khdng gi
rem, ndu ta thi hhnh I
sE rti tl6n kdt qui :
tr& nOn c6 nghidm: r:
lo 0\
(o , /:l€nh a
(;:):,
I oa' * bc' ab' * bd") *'r
( .r'+ dc' cb' + dd' )v
t:
' * bc' ab' * bd'\
'*dc'cb'*dd') 0)
& v6 sau li tich cia hai
6c, l$y theo thri tU 116.
g hqp cAc s6 thgc, tich
ao hoin. B4n ilgc cri thfr
e,h (: 3),."u. thi ta
I
i
Ta ctng sE xet tSng vA tich cic l6nh m6i
gi6ng y nhu khi x6t t6ng vi tich cac lQnh ct
n6i & hai di6m 1) vi 2) & tr6n.
Gii sfr ta c6 hai ma trgn (: ,) ,, (:', i',).
Chtng cho ta hai l€nh mdi md t_a rlem thi
hanh ri6ug rE tldi vdi mgt vecto V cri toa 6Q
{r, g). Ta dtugc k€t qui Ii hai ,r..to 'iiirr, yr)
vi lr't(r't, U'r) c6 toa d0 nhu sau:
1 *r: ux i ba { "'t : u'r -l b'g
I gr: cr L ctg i c'r: c'r* d'g
Jbng 7i* i?cfia hai vscto ito lir mQt vecto'
V' c6 toa rIQ li :
x'= il-1- r'l = fu * a')r + $ * b')!l
. A'= y * g't = (c * c')* * @+ d')g
V{y kdt qui dqt iluoc ctng gi6ng y nhu khi
tr {.hi hAnh d6i r,6i vecto V chi mgt lQnh
la*a' b+D'\
tt
\c*c' d+d'l
Tron co s& d6, te c6 tb6 vi6t:
(: : ) . (:, 1')= (iii :i'r)
vA goi ma trAu 3 r'6 eau li t6ng c,ia hai ma
tr{n & v6 dnu, vi ctng ggi lQnh & v6 sau la
t6ug cfia hai l6nh & vd allu. Ndu ta thi hinh
/u'b'\-
lcnh [ "' "-' I trudc thi ta ituoc vecto v'i A
' \c d )
A,- tr6n vi ti6p theo rlo ta lai thi hinh l€nh
rr / n h \
,-. f " ", I d6i voi V'r thl ta rluqc vecto I''' c6
* \ r. ,I/
toa d0 ld :
r" : ar't * bg't * u(u'r + b'il * b(c'r *d" u)
= fua' * bc\t; * kb' *bd')g
g *, c_n 'L i dg't = c(a'r + b' gi + d(c'r I rt'g)
: Ga' *dc')r { (cb' { dd.' )g.
Vay kdt qul tlat il.uqc ctng giSng y nhu khi
rs thi hinh tl6i vdi Y chi mOt l0nh
I aa']bc' ob'*bd'\ .
( ca'*ac' cb' I dd' )' tuc li :
(: l) (:;)=(:;I"i:"ii"I) Q,
Trong th6 gioi cic l0nh mOi niy xudt hi6n r{t
irhiEu di6u kf la kh6ng c6 trong thd giOi cic
l$nh ct. Sau diy lir rnQt vii hi6n tuqug kf la d6 I
1) ra goi i rn rQnh (: -;).
^rdsng
cdng
Dilrg thirc i2 = -t c6 vE bi hiBm niy fhuI ra
l3i r6t d6 hi6u. Ban itoc dE lh{n thsy rXug lQnh
( I -l ) .Hare qua li lQnh a suay soo nsuEe
chi6u kim rl6ng h6r, vi tl&ng thuc i2 = -1 c6
nghia li: quay 90o nguoc chiEu krm dbng hb
li€n ti€p hai 'lh.u thi tue li quay 1800. Vi bay
gio mOt phuong trinh b{c hai khOng c6 nghigm,
viduacz+r+l:0
thuc (l) ho[c (2) & tren, ta sE c6:
,'= (: -:)(l -;)= ( -; _l)= -,
o\ /o -/srz\
-1i2i*\?,t,, o)
-dirz \
-il2 I
-r+/Ti vi J nghia
r{t a6
2
hiEu :
c6.c nghi€m d6 ctng
| .lB t-tiz
2 2 \0
-- I -ttz
-l \?elz
v8 r /e {-ilz o\,/ o /
- +'+'= (; -,2)- \-v;,, t'i)
,_,12 lttz ),
: [-i*,, -,2 t
tl6u ta.vi6t phu<rng trinh ducri .lqug rr,2 f r = -1
thi ta thdy ngay r5.ng hai ngtriQm t-rOn chiLrh
ll hai'lQnh cri tinh chf,.t sau diy: dem m6i lonh
iI6 thi hiuh li€u ti6p hai l6.n (ecz) d6i voi bf,t
cir v6cto nio r6i lly kdt qui c,)ng vrvi vecto
c6 tluqc khi chi thi hinh l6nh dd c6 rn6t lln
thdi (o) thi k6t qu& cu6i cring Ii ituEc veeLo itdi
cria vectq hic tIEu.
2) ragei cn ronh ( I l). "" ..
l)='"
Nhu vAy, ta thf,v ring trong th6 gi&i niy 1
kh6ng phii chi c6 hai c[n bA.c hai l.i f t vi -1
rni cdu c6 nhtug ciln b{c hai khric, vt dB *e
vi-e.
Do d6, -ii tty thta cia o ttEu bing a. Ngu&i
ta n<5i rfi.ng a co ttnh ch{t nlty d&ugr,,
4) ia gei s ra renh (: :)" ra c6
,'= (l ;)(l :)= (; 3)=,
i-i.,-
,'=(l ;)(l;)=(l
Nhu v{y ta th{y rBng 0 cd nhtng ctrn
kh{c khdng, vi dg 6 vl -8. Ngudi ts
€ c6 tinh chdt <lty linhr.
Nh&ng di6u tr6n ctng c6 vl bi hilm ahung
ctug r5.t d6 hiEu. B4u dgc tr; miuh c6 'thE th[y
ugay ri.ng: e li l6nh sau dALy ( bi€n d6i vecto ili
cho bing ph6p cl6i xuug qua phlu giic thu
nhdl. D ; a ln lQnh sau tliy < chi6u vu6ng g6c vGeto
i15 cho xu6ng true Or:: cbn I li l€nb gl thl rin
gi-i' bi m{t tlB ban rigc tu tim l[y. Rd ring ti
m6t clr&n trdi nrdi cli nr& ra trtrdc, m6.t etc bq.n.
Bii hgc ta cln rirt ra & aIAy li: n0n c6 gfiug
tinr nhtng c6ch nhin mfi ildi v0i cii ct. Tti
c6.ch nhin moi hy vong sE c6 cii rndi ra iloi.
Giita cii ct vi. cii nrdi ni3r c6 mQt ml.u thuin
(dirng & g6c dQ ct mi uhin: r'i dg s6 thuc vl
ma trln lI kh6c nhau) vi c6 rn{t th6ng nhdt
(tlung d g6c rn&i mi nhit'r: l€nh cho b&i m0,
s6 thr;c etng lir lQnh cho b6'i rngt ma tr{.n).
(crin nfra)
bic hri
n6i ring t'
+-
*
B) ra goi o ri ro.h ( I :) r, '.
oz = 11 o\ 11
\0 0l \0
0\_/1
o/-\o 0\l=4"
0l
fi
a
;
[*
seo cho tdt c& ctic sii
dl:44-ol
di, = az- at
ds-os--at
d2*rrg- 12
a4-44 -02
dt;: a4 - (rl
Bni 1/99. TimI, sdt:anhi6n q (aa ( as{a4
1) dt, dz, ds li cie sd nguy€u t6 16. Khi d6 d4
vir ds Ia crte a6 chEn ldn hon 2, L.h& ntng niy
hh6ng thb xiy ra vi d4, ds li cic nguy€n t6.
2) dt, dz, d3, Ii ci.c s6 nguytn t6 chdn (chi
c6 [hB bing 2). Khi dd do:6, khi. nEng ruy
ciing khOng th6 xiy ra yi do li s6 nguy0u t6.
3) 1'rong 3 sd dr, dz, ds chi c6 mdt s6 chErr,
hai s6 kia 16. Khi d6 do ll mOt s6 chfi.n l6n
hon 2, trdi w0i do li s6 nguyOn t6.
4) Trong 3 s6 d1, d:, da c6 n:Qt s6 l8 vL hai
s,I chEn (bing 2):
-- dr 16 thi d? = ds = 2. Khi tI6 ds = {, trii
v&i ds tir s6 nguy0n t6.
- de 16 thi dr = d2:2. Khi itd dn=4, trai
vli da li sd nguy0n t6.
- ilz lE thi d1 =.1s : 2. I(hi Q1iid+ = ds = dz * 2
vi d6 = dZ* 4. Irnat khic, troug'B s6 tl|, dy*2,
d2 + 4 lu6n lu0n c6 m$t s6 chia h€t cho 3 vir
dEu lir cAc s6 nguy€n 16. Kha nlng ntry chi xiLy
ra khi dZ = 3, dL: d6 = 5 vi dO.= 7. I't aI6 d6
dirrg tim iluqc:
a2=al*d3== at*.2,
a4: il? + dl'= at'1' 7.
.J
1t
illu Id cac sd ngugtn td, trottg do co tlti cd
nlr&ng sd ngugdn td tring nhau.
Lbi gi&i (c&a cdc trqrr fl6 :Di'c Lau, lrfi Dfi.c
Minh, l6p 9N H0ng Quang, tlii llung vir nhiBu
b1n kh6c).
'l'u tliBu ki6u cfia bAi to{n ta c6:
da = dt -f de, ds = dg, * dg, tla: d1 + d2 |. ds
Xet c{c kb6 ntng:
L
I
t,
li"
1,"
,
i
i"
i
lo
I
i
I
I
i.
[$
I
tt'
t,
I
I
I
I
ti
i
I
I
Yly cric sd at, a2, as, e4 th.6a m6n di€u ki6n
crla bii lodn phli c6 dang:
ol=n, a..a=n*2, o3 =n*5, a4=n|7,
troDg dd n ld mQt s6 tir nhi6n nio tI6.
T. T.
Bni 2/99. Cdc di?m ddi r*ng o6.i trqrc tdm
efia m6t tant gia. qua cac canh ctta tam gidcdg
tqo thi,nh ntt)t tam gid.c ntbi, Ctic cqnh cfia hai
tam gidc ndg cdt nhou tao thdnh m6t luc giac.
Chfing minhrdng cdc dwb'ng 17fi,,g ndi cdc d?nh
i16i di|n cfta luc gidc nd11 cdt nhuu tqi mgt dibm.
Ldi giai (cria ban Fhan Vd.n Vrin, lop 9N Hbng
Quarlg, Hli IIung).
Goi H IiL trqc t6.m cfia tam gi6c AEC; I,J,K
lhn lugt la cdc dibrrr ddi xung crio FI qua cdrc
canlr BC, CA, AB,
1. Truoc h6't ta chr'tng minh 1,,1, K n$.nr lr€n
iludng tron ngoai tidp tarn ginc ABC. Th{t vfy,
Do i/ r'd J ddi xung n'.r au qua AC n€rr
€it = c.-ri r, rai c6 cil : Cii (961 c6 canh
trrong r'rng vudng gdc), do v$y : CAJ: eB./, nghia
lri { rliFrn A, B, C, J cr)ng nfim tr€n tr$t du.dng
Ilr'rn h'i-r, ./ nim trdn ilulng tron ngor.i tidp ABC.
li[ng lip iuqn ttrong lq, I vir I( ci,ng niru tr6n
clr:ong trdn ngoai ti6p ABC.
2. Goi M li. giao di6m ciLa KJ ,t:a AC, N Ii
giao itiinr cia /(1 r'i ]3C. 'Ia s6 c:hfing minh B
alibrn ,lJ, M, {jhjng ir:ing. Th:\r vny itreo rrcn
tr co Alf =jJK (cr\ng chin cung C,r). rie t n<rp
vii AII{: fIfN (ao g r.i ,I dtii u.ng Ihau qua
i;C)rvATtk =,u,Iu (Oo fi_jaJ ddi xirng nh*u
r[ul '.''.4 ) la ,1 ucrc f r/N = ,l F/,]iI di€u nlv chung
t6 3 di6m M, H, H th&Dg hnng.
- ii[ng cach l.rip luSu tucrng !u, ia chung mir:L
duqrc cfic-d,rong thiug n6i hai cnp dinh Oil aiqrr
cdn Iqi c&a luc giic (phin chung cfia hai tam
giitc AtlC vn IJI|) d6u di qua truc rAm II cria
tam giic ABC.
Nhdn rit. r.\hibu ban ald eri ldi gili dirng vi
theo cirng mdt luoc tI6 chring minh nhu tr0n
(truoc h6t chi'rng LA A,B,C, 1,"I,-Il, cring nim tr.6n
m$t du<rng trol, rbi sau d6 ch{rng minh cac
ilucrng th&ng u6i cdc tiinh tldi di€n crla luc siic
cf.t nhau tal tl l.
Ch&ng hqn, dE chung minh M, I/,N th&ng hAng
c6 th6 la? lui._nrrlarr sau:
Do A-E-I: ,4CIi _(gbc c6 can!.f uo,giy,g urrOng
$cI ':,f :1il=-d). 1'* .16 ui-a,. Nhurrg
AIK : .IJIN (dn 1 vd II d6i xung nhau qua BC)
n6n ,,1I"I: ffl.\r, n*hia lir ilN ll f,I. 'I'trrrng tu
ta r:irns cd ]tM ll tl. 't'* d6 suy ra LI , H, N
thEug hirg.
T.T.
Bdi 3/99. Cjrc tla thttc bitc tttti
Ii r.) - 4.xt * br * c.
Bi6r ring l/io)li 1, l/(-1)l>{ r, l/(r) l< t,
hdy chung rr,inh rlng
if(rrll < 5/+ vdi moi I r I -( t.
Lb'i gi&i. 'frrrdc h6t ta lr5.y lAp tla thirl li{c
hai g(r) : Ar(r- 1) + B(r2 - 1) + Cr(- + i).
'I'a hiiy chgu c{c he s6 A, B, C sao cho
/(- 1) - Q(-7):2A,
/(0) :9(0) :-8,
/(t) -9(l) : 2C,
1l
hay,{ : -- ft - :), B--l(0), L': --:- if(l).
,,
Ydi cflch cl-ron A, E, C nhu v{y, thi tla thuc
h@):f@) - s(r) c6 bAc ( z var c6 (it nhat) 3
nghiirn khdc uhau ln o, + 1, thinh th& ta *rii
cd lr(c) - 0 hay f (r): g(r), tuc lir
yoo= J-]GI) r(c-l ) -f$) @z -r) * f() r(r*1).
2
Vi vly
I /(r) I * -|,, f(- r) l.lrtr-r)l+lrtor I.lrz-tl+
1
+ : t/tr)l.lc(r+t)l
2
-2 " 2
Ntnh"t