Phuang phdp:De tim cue tri trong khong gian chiing ta thuang su dyng hai <^ach lam:
^ach 1: Sii dung phuang phap hinh hpc Cach 2: Sii dung phuang phap dai só.
todn 1. Trong khong gian cho n diem A ^ A2,..., .
^) Tim M sao cho P = âMÂ + a j M A j +... + âMÂ a) Nho nhat khi ttj + ttj +... + a„ > 0
b) Lon nhát khi tti + ttj +... + < 0
itang phdp giai Toan Hinh hoc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu
) T i m M sao cho P = a j M A j + a 2 M A 2 +... + ar,MA„ nho nhat hoac Ion nhn'. at
ti
1
'Phixcmg phdp gidi:
Goi I la diem thoa man: ajLÂ + a 2 i A j +... + a ^ I A ^ = 0 diem I ton tai
n d u y nhat neu 0 . K h i do : i=l 1) P = a i ( M i + l A i f + a i ( M I + IA2 f +... + ( M I + IA„ f = (ttj + a2 +... + a „ ) I M ^ + £ a i I A f n Do k h o n g d o i n e n : 1=1
• N e u ttj + a2 +... + > 0 thi P nho nhat <=> M I nho nhat • Neu ttj + a2 +... + a „ < 0 thi P Ion nhat o M I nho nhat 2) P = |ai(m + iA7) + a 2( M + I A ^ ) + ... + a n( m + I A ; ; ) n . M I
i=l Do do P nho nhat hoac Ion nhat <» M I nho nhát hoac Idn nhát.
Chuy:
• Neu M thuoc d u a n g thang A (hoac mat phang (P)) t h i M I nho nhát khi va chi k h i M la hinh chiéu cua I len A (hoac (P)).
• Neu M thuoc mat cau (S) v a d u o n g thang d i qua I va tam ciia (S), cat (S) tai
hai diem A , B ( l A > I B ) thi M I nho nhát (Ion nhát) o M ^ B ( M = A ) .
^di todn 2. Trong khong gian Oxyz, cho cac diem A{xj^; y^; z^),
B ( x B ; y B ; i ^ B ) v a mat phang (P): ax + by + cz + d = 0. T i m d i e m M e ( P ) sao cho 1) M A + M B nho nhát.
2) | M A - M B | Ion nhát v o i d ( A , (P)) ?t d ( B , (P)).
'Phieong phdp:
• Xet v i t r i t u o n g doi ciia cac diem A , B so v o i mat phMng (P).
• Neu (axA + b y A + C Z A +d)(axB + b y B +CZB + d ) > 0 thi hai diem A , B cung phia v o i mat phSng (P).
• N e u ( a x A + b y ^ + c z A + d ) ( a x B + b y B + c z B + d ) < 0 thi hai diem A , B nam khac phia v o i mat p h i n g (P).
Cty TNHH MTV DWH Khang Vỉt
IJ M A + M B nho nhát.
Truang hap 1: H a i diem A , B 6 khac phia so v o i mat p h i n g (P).
Vi A , B 6 khac phia so v o i mat phang (P) nen M A + M B nho nhát bang ^ k h i va chi k h i M = (P) n A B .
, Truang hap 2: H a i diem A , B o ciing phia so v o i mat phang (P).
Goi A ' dói xiing v o i A qua mat phSng (P), k h i do A ' va B 6 khac phia
(P) va M A = M A ' nen M A + M B = M A ' + M B > A ' B .
Vay M A + M B nho nhát bang A ' B khi M = A ' B n (P). 2) | M A - M B | Ion nhát.
• Truang hap 1: H a i diem A , B 6 ciing phia so v o i mat phang (P).
V i A , B 6 cung phia so v o i mat phang (P) nen M A - M B Ian nhát bang A B khi va chi k h i M = (P) n A B .
• Truang hap 2: H a i diem A , B a khac phia so voi mat phang (P).
Goi A ' doi x u n g v o i A qua mat phSng (P), k h i do A ' va B 6 cung phia (P) va M A = M A ' nen | M A - M B | = | M A ' - M B | < A ' B .
Vay | M A - M B | Ion nhat bang A ' B k h i M = A ' B n (P).
to todn 3. Lap p h u o n g trinh mat phang (P) biet
1) (P) d i qua d u a n g thang A va khoang each t u A ^ A den (P) Ion nhat 2) (P) d i qua A va tao v o i mat phang (Q) mot goc nho nhát
3) (P) d i qua A va tao v a i d u o n g t h i n g d mot goc Ian nhát.
^hucmgphdp:
Cach 1: D i m g p h u o n g phap dai só
1) Gia su d u o n g thang A : x - x i _ y - y i _ z - z - va Ăxo;yo;Zo)
a b c
K h i do p h u o n g trinh (P) c6 dang: Ăx - X j ) + B ( y - y j ) + C(z - Z j ) = 0 bB + cC
Trong do A a + Bb + Cc = 0 => A - — ( a ^ O ) ( l ) K h i do d(A,(P)) = Ăxo - x j ) + B(yo - y i ) + Q Z Q - Z j ) (2)
|Thay (1) vao (2) va dat t = ta duac d(A,(P)) = Jui)
Trong do f(t)= '"^ + " * + P ^ khao sat ham f(t) ta tim dugc m a x f ( t ) . Tir do m ' t + n ' t + p'
i i y ra du(?c sv bieu dien cua A, B qua C roi cho C gia tri bat k i ta t i m dugc A, B.
Phuomg phdpgidi TodnHinh hoc theo chuyen de- Nguyen P/IM Khdnh, Nguyen Tat Thu
2) va 3) lam tuang t u Cach 2: Dung hinh hoc
1) Goi K, H Ian lugt la hinh chieu cua A len A va (P), khi do ta c6: d(A,(P)) = A H < A K , ma A K khong doị
Do do d(A,(P)) Ion nhat <=> H = K
Hay (P) la mat phang di qua K , nhan AK lam VTPT.
2) Neu A 1 (Q) =>((PMQ)) = 90° nen ta xet A va (Q) khong vuong goc vol nhau, • Goi B la mot diem nao do thuoc A , dung duong thang qua B va vuong voi (Q). Lay diem C co dinh tren duong thang dọ Ha CH 1(P), CK 1 d. G6c giua mat phang (P) va mat phang (Q) la BCH. Ta co sinBCH = — > ^ - ^
BC BC' BK
Ma — khong doi, nen BCH nho nhat khi H = K.
Mat phang (P) can tim !a mat phang chua A va vuong goc voi mat phk
(BCK). Suy ra np = - .-I - la VTPTcua (P),
3) Goi M la mot diem nao do thuoc A , dung duong thang d' qua M va song song vol d. Lay diem A co dinh tren duong th^ng dọ Ha A H 1(P), A K I d . Goc giiia mat phSng (P) va duong thSng d' la A M H . . g
1 a C O cos A M H = > . A M A M K M
Ma —— khong doi, nen A M H Ion nhat khi H = K. A M
Mat phang (P) can tim la mat phang chua A va vuong goc voi mat phang la VTPT cua (P).
-
ra np = " A ' " A Aid'
sao cho:
Vidui 3.5.1. Trong kh6ng gian Oxyz cho ba diem: A ( l ; l ; l ) , B(-l;2;0), C(3;-l;2).