Cuốn sách bao gồm các phần tổng hợp các kiến thức cùng chủ đề căn bản về hình học nhằm giúp các em học sinh nắm vững kiến thức trọng tâm. Bên cạnh đó, sách còn đưa một số bài tập cùng phương pháp giải nhằm rèn luyện kỹ năng thực hành của các em. Mời các bạn cùng tham khảo.
Bai tap 7: Mot hinh tru c6 ban Icinh day R va duong cao 2R Goi M , N Ian luot la diem bat ky tren day Tim gia tri Ion nhat cua doan M N HD-DS m qua2RV2 Bai tap 8: Mot hinh non c6 ban kinh day R va chieu cao bang 4R Tinh ban kinh day r va chieu cao h cua hinh tru noi tiep hinh non de dien tich toan phan cua hinh tru dat gia tri Ion nhat HDDS 4R Stp dat gia tri Ion nhat r = — R va h = CHU D E I X _ T06 t>0 KHONG Glf!N DANG TOAN TOA DO DIEM VA VECTd • Toa khong gian * > Ba vectadan vi i, j , k tren true Ox, Oy, Oz: i=(l;0;0), ]=(0;1;0), M k=(0;0;l) > M(x,y,z) hay M (x,y,z): y OM = X i + y j +z.k J > / >y a(x,y,z) hay a =(x,y,z): a = x.i + y j + z.k Phep toan vector Cho hai vecta: u = (x,y,z) va v = (x',y',z') thl: u ±v = (x±x';y ±y'; z ±z'); ku = (kx; ky, kz) -4? X- u.v =• xx' + yy'+ zz', cos(w,v) +y^ +z^ x.x+y.y+z.z ylx'+y'+z\^x"+y"+z" Cho hai diem A(xi, yi, zi) va B(x2, y2, 22) thl: AB = (X2 -xi; y2 -yi, Z2 -z,) 159 AB = ^{x,-xy+(y,-yy-+{z,-zy- Mchia AB theo tik ^ 1: M X| -kx^ ^ \-k ' y^ -ky-, 1-A: z, -kz, ' 1-A: j Chuy: 1) Gdc A cm tarn gidc ABC: cos A = cos(AB; AC) 2) Tog trung diem M cua doan AB, tdm G ciia tarn gidc ABC, tdm E ciia tu dien ABCD v&i toa doA(xi, y/, z/), B(x2, yi, 22), C(x3, ys, zs), D(x4,y4,Z4): X = X +x X = x^ +X2 + X = y = M y = y ^ G z= z, + z z z=• >'l +>^2 + X l +>'4 Z| + Z, + Z, + Z4 +z,+z z= Bai toan IrChobavecta a = (2;-5; 3), b = ( ; ; - l ) , c = ( ; ; ) a) Tim toa cua vecta e = a - b - c b) Tim toa cua vecto f = a ' - ^ b + c Gidi a) e = a - 4b -2c = ( - - ; - - -14;3 + 4-4) = (0;-27;3) b) f =4a - - b +3c =(8+ + ; - - - +21; + - +6) = (11; - ; — ) ^ 3' Bai toan 2: Tim toa cua vecta m cho biet: a) a + m = 4a va a = (0; -2; 1) b) a + 2m - b va a = (5; 4; -1), b = (2; -5; 3) Gidi a) a + m = a =^ m = a = (0; 6; 3) b) a + m = b = > m = - — a + — b = 2' 2' 2 Bai toan 3: Cho hai bo ba digm:A(l; 3; 1), B(0; 1; 2), C(0; 0; 1) va A ' ( l ; 1; 1;), B'(-4; 3; 1), C'(-9; 5; 1) Hoi bo ba dilm nao thing hang? Gidi Taco CA = ( ; ; ) , CB = (0; 1; 1) 160 Vi cac toa dp khong tuomg ung ti le nen khong c6 s6 k nao d^ CA = k CB, suy A, B, C khong thang hang Taco C A ' =(10; -4; 0), C B ' = (5;-2; 0) => C A ' = C B ' Do A', B' C thing hang Bai toan 4: Tinh tich v6 huong cua hai vecta moi truong hop sau: a) a = (3; 0; -8), b = (2; -7; 0) b) a = ( l ; - ; ) , b =(4; 3;-5) c) a = ( ; V ; V ) , b = ( ; V ; - V ) Gidi a) a b = 3.2 + 0.(-7) + (-8).0 = b) a b = 1.4 + (-5).3 + 2(-5) = -21 c) a b = + V2 V3 + V3(-V2) = Bai toan 5: Cho ba vccto: a =(1;-1;1), b = ( ; ; - l ) , c = (3; 2;-1) Tinh: a) ( a b ) c , a ( b c ) b) a l b + b ^ c a + c ^ a , a c + b ^ - c l Gidi a) Taco: a b = 1.4 + (-l).0+ l.(-l) = D o d : ( a b ) c = c =(9;6;-3) Taco b c =4.3 + 0.2 + (-l)(-l)= 13 Dodo a ( b c ) = 13a =(13;-13; 13) b) Tac6 a^ = b ^ = 17, c^ = 14nen a l b + b l c + c l a - b + 17c + ul = (77; 20;-6) vaa.c = 0=>4a.c + b -5c 53 27t = 2, Bai toan 6: Cho = 5, goc giUa hai vecta u va v bang — Dat p = ku + 17v v a q = u - v Tim k de vecta p vuong goc vol vecta q Gidi Fa CO = = 5, cos(u, v) = cos 2^ = '2 Do p q o p q = 0\,y > , z > va x + y + z = 22 Chung minh bat dang thuc yjx-1 + 2-^y-2 + 3yjz-3 < 14 Cf/di Trong khong gian Oxyz, xet vecta M = (l;2;3);v = (Vx^;7>^^-^;Vz^) M ( x ; y; 0) va diem c6 dinh A ( l ; -3; 3), B(2; - ; -5) a khac phia vai mp(Oxy) Ta c6: M.V = | M | | v | C O S ( M ; V ) hay i nen ^ + 2^y-2+3y[7^ | M.V | ' - + ^ ^ < V M V M = DANG TOAN TICK CO Hl/OfNG r/c/i CO hu&ng: Tick CO huang ciia hai vecta a = (x,y,z) va b ^ (xy'z) va CO toa do: y [a,h] m X Z vy Dinh thuc X z' x' ^' y y X' la mot vecta, ky hieu = iyz'-zy'; zx'-xz'; xy'-yx') n = PP mq-np q q Kit qua: - Vecta [a, h\ chung vai a, b 164 - Do dai ciia vecta [ a , A ] ; \ a, h ] | = | sin( a, b) -2 vecta a, h cungphuang: [ a, h ] = -3 vecta a, h, c dongphdng: { a, b ].c = -3 vecta a,h,c khongddngphdng: [ a,b ].c Dim tich vathitich Dien tich tarngidc ABC: S = - [AB, AC] The tich tit dien A BCD: V = [AB ,AC ].AD The tich hinh hop ABCD.A 'B'C'D':V= [AB, AD^AA The tich hinh Idng tru ABCA "B'C: V = ^ [AB, AC] AA Chiiy: 1) Ba vecta dan vf i, j , k tren true Ox, Oy, Oz: i =(1:0:0), "j =(0:1:0), k =(0:0:1) 2) Khodng each giira hai diemA(xi, yi zi) vd B(x2, y2, ^2): AB = 7U, - X , ) - +{y,-y,)- +(Z2-z,f 3) G6c giua hai vecta: u = (x,y,z) va v = cos(w,v) = (x'y'z): x.x+y.y+z.z + y~ + z\^jx'^+y'^+z' Bai toan 1: Tim tich c6 huong cua cac cap vecta: a)fl=(l;3;-2)va6=(2;-l;4) a) Tich CO huong cua cap vecta la: f -2 -2 [a,b] = 4 V- b ) a = ( l ; - ; ) v a f t =(5;1;4) Giai 1J \ 4/ = (10;-8;-7) 165 b) Tich CO huong cua cap vecta la: ^ [a,b]= ^ ^ ^ (-2 ; ; 1-2^ J =(-ll;ll;ll) ^ Bai toan 2: Tim tich c6 huong cua cac cap vecto don vi: a) / = (1 ;0;0) va } = (0; ;0) b) } = (0; ;0) va ^ = (0;0; 1) Gidi a) Tich CO huong cua cap vecto la: / V 0 0 1 \ 1/ = (0;0;1)= k b) Tich CO huong cua cap vecto la: 0 0 1 0' 0 = (1;0;0)= / Bai toan 3: Xet su dong phang cua ba vecto a) a - ( - ; l ; - ) , b = ( ; 1; 1), c = (-2; 2; 1) b) a = ( ; ; ) , b = ( ; - l ; ) , c = ( l ; ; l ) Gidi -3 a) Ta CO [ a , b ] = -2^1 1 = (3;l;-4) D o d [ a , b ] c =-6 + - = - ^ Vay vecto khong dong phang b ) T a c [ a , b ] = ( ; ; - ) ^ [ i , b].c = Vay vecto dong phang Baitoan4:Cho a = ( ; - ; 1), b =(0; 1;2), c = (4; 2; 3) va d = (2; 7; 7) a) Chung minh cac vecto a , b , c khong dong phang b) Hay bieu thi vecto d theo cac vecto a , b , c Gidi a)Tac6[a, b ] = (-3;-2; l ) = > [ a , b ] c =-13:i^0 Vay vecto khong dong phSng b) Gia su d = ma + nb + pc n =3 - m + n + 2p = m = -2 m + 4p = m + 2n + 3p = p =l Vay d - - a + b + c 166 Bai toan 5: Trong Ichong gian Oxyz clio ba diem A ( l ; 2; 4), B(2; - ; 0), C(-2; 3; -1) Goi (x; y; z) la cac toa cua diem M nam tren mat phang ( A B C ) T i m sir lien he giua x, y, z Giai AB = ( l ; - ; - ) , A C (-3; l ; - ) , A M = (x - 1; y - 2; z - 4) Ta C O M nam trcn mat phang (ABC) [ A B , A C ] A M = 19(x - 1) + 17(y - 2) - 8(z - 4) = « 19x + 17y - 8z - 21 = Vay quan he la 19x + 17y - 8z - 21 = Bai toan 6: Chung minh cac tinh chat sau day cua tich c6 huong: a)la,i]=0 b ) [ i , b] = -[b, a] Gidi Goi vecta a = ( x i , y i , zi) va b = (x2, y2, zi) c a) [ a , a ] = = (0; 0; ) = Yi b) [ a , b ] = yi Yi X2 Y2 ^2 = (yiZ2 = -(Y2Z1 Y2 V YI \ X| / - y Z i ; Z | X 2- Z2Xi; X i y - Y1Z2; X2yi) Z2X1 - Z X ; X y i - xiy2) \ Z2 Z2 ^2 z, Z, ^2 y2 X| Y, -[b,a] / Ket qua [ a , a ] = - [ a , a ] = ^ [ a , a ] = Bai toan 7: Chung minh cac tinh chat sau day ciia tich c6 huang: a)[ka, b] = k [ i , b] = [ i , k b ] b) [a.b] _ -(a.b)l Gidi a) Goi vecta a = ( x i , y i , zi) va b = ( X , y , Z ) k[a, b ] = Yi k V Y2 z, Z, ,k Z, Z2 f ky, kz, V Y2 z^ X, ,k X j kz, kx Z2 ^2 X| YI X2 Y2 J kx, ky X2 Y2 \ = [ka,b] Tuang tu: k [ a , b ] = [ a , k b ] 167 b) VP = ^-(a.b)^=^ b ^.cos^a b P(l -cos^a) = •^sin^a = [ a b ] P = VT Bai toan 8: Chung minh cac tinh chat sau day cua tich c6 huong: a)[c, a + b] = [c, a] + [c, b] b) a [ b , c ] = [ i , b ] c Giiii Goi vecta a = ( x i , y i , Z | ) ; b = ( x , y2, zj) va c = (X3, ys, Z3) a) Ta CO [ c, a + b ] ^2 y.-t y., z,+Z2 y,+y2 + V yi y, Z, + Z , Z3 z z, X, z, yi X3 X, + y3 X, X, + x Z3 Z, X3 X, X3 X1+X2 X3 y? V yj X3 Yj y,+y2 y3 + Xi yi X2 X3 Zj + X, y2 y3 X3 X, z y2 X2 = [c,a] + [c,b] b) a [ b c ] = X X|| z, z y2 x + z, +yi Z] yi Z3 y3 X3 Z3 X3 Z, +y3 y2 Z2 X| Z, X2 y2 y, X3 Xi +Z3 Xj yi X2 =[a.b]c y2 Bai toan 9: Trong khong gian cho ba vecta a , b , c timg doi khong cung phuang Chung minh rang dieu kien can va du de vecta tong: a + b + c = 1a[a,b] = [ b , c ] = [c,a] Gidi Tira + b + c = = > a = - ( b + c ) = > [ a , - b - c ] = D o d [ a , - b - c ] = [ a , - b j - [ a , c ] = =>[c, a ] = [ a , b ] Tuong tir ta cung CO [ b , c ] = [ a , b ] Vay: [ a , b ] = [ b , c ] = [ c , a ] Ngugc l a i , t u [ a , b ] = [ b , c ] [ b , a + c] = Matkhac, [ b , b ] = = > [ b , a + b + c ] = => b cung phuang v a i vecta a + b + c Chung minh tuong t u ta ciing c6 vecto a ciing phuong v o i vecta a + b + c Nhung a va b khong cung phuang, vay: a + b + c = 168 Cau Tir he phuong trinh da cho, ta c6: S= X 5' -4P>0 0 + 12m^ + 21m + = f(m) +y=m P = xy = m - Vai < m < 2, he CO nghiem x, y > khi: • Khi T = Khao sat ham so f(m) vai m e Ta duac minT = ' ^1 m = — va maxT = 100 m = 2 Cau 7a Goi A(a, b) thi B(4-a; 2-b) V i A, B e (E) nen: 10 (4-a)^ ^ (2-b)^ Vay A z wn 4ViT , + 15 10 a =2+ va B 15 ;b = i - 10 10 15 15 va phuang trinh (d): 8x + 9y - 25 = Cau 8a Goi 1(1+t; l+2t; -2t) e d la tam ciia mat cku (S) cSn tim Do (S) tiep xuc vai (a) va mat phang (Oxy) nen: d(I, (a)) = d(I, Oxy))c^ 2(l + t ) - ( l + 2t)-4t + l 2t t - l = 3t « t = - l h o a c t = Vai t = -1 thi (S) c6 tam 1(0; 1; 2) va ban kinh R = nen (S) c6 phuang trinh x' + ( y + l ) ^ + (z-2)^ = 1 ^rix Vai t - - thi (S) IT'• CO ,^6 2^ tam I - ; - ; — va ban kinh R = — nen (S) c6 phuang trinh V5 5 V ^ 2^^ + + z+ X — 5 Cau 9a Dat z = x + iy, ( x, y e R) 25 Ta c6: z z + 3(z - z) = x^ + y^ + 3.2iy = x^ + y^ + 6yi Do do: z z + 3(z - z) = - 3i 338 + y" = x=± Vl5 6y = -3 Vl5 i hoac z = Vay z = , 2 Vl5 i ' 2 Cau 7b Nhan xet A ^ d Goi I(x, y) la tarn hinh vuong A B C D V T C P cua d la: u=(-l;2) Ta CO I d va A I ± u nen t i m dugfc n> c Goi B(a; b) V i I B = l A va B e d nen tim dugc B 1 va D 5'5 B 19^ 5' 19 va ngugc lai "5'5 ;D I 5'5 Do phuomg trinh cac canh hinh vuong A B C D la: ( A B ) : 3x - y = 0; ( A D ) : x + 3y - 10 = 0; (BC): X + 3y - = 0; (CD): 3x - y + = hoac ( A B ) : X + 3y - 10 = 0; ( A D ) : 3x - y = 0; (BC): 3x - y + = 0; (DC): x + 3y - = Cau 8b d d i q u a A ( l ; ; - ) v a c VTCP u = ( - ; ; - l ) A d i q u a B ( - l ; 0; l ) v a c V T C P n = ( ; l ; - ) Ta CO [ u V ] A B ^ nen d va A cheo Duong vuong goc chung IJ c6 VTCP: a = [ u , v ] = (-5; -8; -14) Mat p h i n g (P) chua d va IJ c6 V T P T np = [ u , a ] = (-50; -23; 31) va di qua A ( l ; 0; -2) nen c6 phuomg trinh: -50(x- l ) - ( y - ) + 31(z + 2) = hay 50x + 23y - 31z - 112 = Mat phang (Q) chua A nen IJ c6 V T P T n^ = [ V , a ] = (-30; 66; -27) va di qua B ( - l ; 0; 1) nen c6 phuomg trinh: -10(x + 1) + 22(y - 0) - 9(z - 1) = hay lOx - 22y + 9z + = Cau 9b Dat: z = x + y i , x, y e R va M ( x ; y ) la d i l m biku dien s6 phuc z tren mat phang phiic 339 Taco « z-i =|(l+i)z| t = V3 71 Ket hgrp nghiem, nghiem phuong trinh la: x = y + kTi ( k e Z ) C a u D K : x > - - , BPT da cho tro thanh: (X +1)- - ( x + \yj2x + 5+ 2(V2x + ) ' > (1) f Y Khix = - - t h i ( l ) « > - - + >0:dung \ J Do X = - ^ la nghiem bat phuong trinh Khix ^ - - t h i ( l ) « ( , )'-3( , V ^ x +1 Dodo: < X+ K ) + 2>0 \ f ^ yj2x +5 « V2x + - - < x < - l « - U x < - t = e;x = e=>t = e'^ Suyra:I= dx = Jx(e'+lnx) Vay: I = ln(e*+ ) - X X J dt , —= = lnt lnt — J t ' = InCe*" + ) - Cau Ap dung dinh ly duong trung tuyen tarn giac, ta c6: _ AB- + AD- BD- _ a- " =^ AC = a => BC^ = AB^ + AC^ ^ AB AC Suy SA»CD = 2S \ABC = /,':' -X Ve GH // OA (H e AB), GK SH Ta CO AB ± GH, AB SG => AB GK a GK ( S A B ) ^ G K = , GH = B " " = ^ Vay VS.ABCD = y Cau Dat a = X + 4, b = y + 5, c = z + (x, y, z > 0) Do do: (x + 4)^ + (y + 5)^ + (z + 6)^ = 90 x^ + y^ + + 12(x + y + z) - 4x - 2y = 13 Gia su X + y + z < x, y, z e [0; 1) Suy x^ < x; y^ < y; z^ < z nen: x^ + y^ + z^ + 12(x + y + z) - 4x - 2y < 13(x + y + z) < 13 (v6 ly) Do do: X + y + z > Vay: a + b + c = x + y + z + > Cau 7,a Ha A H vuong goc vai d' la durong phan giac goc B, ta c6 H(h; 2-h) Suy ra: A H - (h + 4; -h) ^ = (-1; 1) la VTCP cua d', nen h = -2 Vay H(-2; 4) Goi A' doi xumg A qua H, nen A' e BC va H la trung dilm AA' Do A'(0; 6) Ta CO B e d' B(b; 2-b), C G d o C(10-2c; c) Ma BC = 2BA nen A' trung didm BC b + 10-2c = b - c = -10 - b + c = 12 { - b + c = 10 { b = -10 c=0 Vay B(-10; 12) C(10; 0) 343 Cau 8.a Goi I la tarn mat cau, ta c6 I e A « Do do: O I = d ( I , (P)) — 2 Cau 9.a Ta c6: |z + i | = - ^ x ' + ( y + 1)' (2) Val d(0, (A'AB)) = OH va OA'H = 30° Dat OH = X OA' = 2x va O'A' = OA = Tarn giac OO'A' vuong tai O' nen a" + Vay d(00' A'B) = f 2xV , , =4x" —j= S ) aV3 X = — T = 2V2 aV3 2^2 Cau Ta c6 - + - + - > + + (a + b)^ + (b + c)^ + (a + b)(b + e) b c a b+c a+b a(a + b)(b + c) b(a + b)(b + c) e(a + b)(b + c) ^ 1 b e a « (a + b)^ + (b + c)^ + (a + b)(b + c) ^ a'e , , b"(a+b) ,2 , cb(b + c) < — + a" + ab + ac + — -+h + ab + e^ + be—^ b e a V i : (a + b)^ + (b + e)^ + (a + b)(b + c) - a^ + ac + c^ + 3b^ + 3ab + 3be T•^ ^ i , , a'e b"(a+b) cb(b + e) ' , Do ta can chung minh: — + — +— ' ^ 2b + 2bc + ab b e a a-c b-(a+b) cb(b + e) Ihatvav: — + + 348 a c b A + uA a c c2 b b + + - — + 2 c b + c a^ —+ " a ab + ( V a ^ +^ j ^ ) + 2b^ ^ ab + 2bc Dau + b' Cy + 2b^ (dpcm) xay a = b = c Cau 7a (C) c6 lam 1(1; -1) va ban kinh R = Goi M la trung diem H K , tam giac IH IHK vuong can a I nen I M ± H K va d(I, A) = GoiC (A): ax5) + by +^ c =2a (a^ + b^ Ta O P(2; e A + 5b + c^=0) Va d(I; A) = ~ « 2(a - b + c ) ' = 25(a^ + b^) (2) T u (1) va (2): 2(a + 6b)^ = 25(a^ + b^) (3) Cho b = 1, va giai (3) ta dugc a = -1 hoac a = 47 23' V a i a = -1 ^ c = -3 =^ (A): x - y + - Vai a 209 47 c = - 23 (A):47x + y - = 23 Cau 8a Goi M(a; b; c) V i M each deu ba diem A B C nen MA = MB { M A = MC (a - ) - + b" +c- = a" +(b - ) " + c" [(a - ) - + b ' + c' = a' + (b - 3)' +(c 'a-b =0 if fb = a { a - 3b - 2c = - [c = - a Do M(a; a; - a) V i M each deu diem A va mp(a) nen M A = d ( M , ( a ) ) a= ^ ( a - l ) ^ + a^ + ( - a ) - = (^ + ^^ + ^ ) ' o 6a^ - 52a + 40 = o 23 a= Chpn diem M toa nguyen la M ( l ; 1; 2) Cau 9a P(x) = (x" + x - ) ' = ^ C ^ ( x ' ) ' ' ( x - ) ' ' k =0 = 'fzc:x'^"'(-i)'"] = xzc^c:(-i)'"x"'-'' k-0 Vm = 349 Chon k, m cho 10-k-m = < m < k < ,la dugrc cac cap (k, m) la (4; 3) va (5; 2) k,m e N Vay he s6 cua la: - C^C^ + C^C; = - Cau 7b Tarn giac ABC deu c6 dinh C thupc Ox nen A, B doi xung qua Ox Goi A(xo; yo) va B(xo; -yo) thuoc (E): — + y^ = thi AC = AB = yo I nen ta c6: ^0 =2;.yo = 4yf3 Voi Xo = 2; yo = C = A hoac C = B (loai) Vay hai diem A, B la: 4V3 ;B 4V3 7' hoac A (2 W37' O 4V3' Cau 8b Vi M nam tren duong thang d nen M(2 - m; + m; + m) Duong thang A di qua M va song song voi duong thang d c6 phuong trinh ^ x-2 +m _ y - - m _ z-4-m • ~ i ~ ^2 Vi N nam tren duong thang A nen: N(2 - m + 2n; + m+ n; + m - 2n) VaN nam tren (a) nen: (2 - m + 2n) - (3 + m + n) + (4 + m - 2n) - = n = -3 - m Suy N(-3m - 4; 0; 3m + 10) Ta CO M N = « (6 + 2m)^ + (3 + m)^ + (2m + 6)^ = "m = -2 m^ + 6m +8 = m = -4 SuyraM(4; 1;2), M ( ; - l ; ) Cau 9b DK: x > log ^4" Chia hai ve cua phuong trinh cho 3" va dat t = Suy raO < t < - I Phuongtrinh: V t - t ' + V t ' - t ' = Vi < t < nen: - (2t -1^) = (t - 1)^ ^ => 2t -1^ < 1; - (4t^ - 3t^) = (t - l)^(3t^ + 2t + 1) ^ ^ 4t^ - 3t^ < Do ^2x-i^ + V4t^ -3t'' < Nen diubingxay Vay phuong trinh c6 nghiem nhat x = ra: t = x = 350 * * Chii de 1: K H O I D A D I E N V A P H E P DC>I H I N H Chu d^ 2: K H O I D A D I E N D E U V A P H E P V I T U ' 27 Chu dg 3: T H E T I C H K H O I L A N G T R U V A K H O I H O P 46 Chu dg 4: T H E T I C H K H O I D A D I E N vA 58 Chu da 5: M A T cAu, Chu da 6: M A T T R U - H I N H T R U - K H O I T R U KHOI CHOP K H O I CAu 94 115 Chii de : M A T N O N - H I N H N O N - K H O I N O N 124 Chii da 8: T O A N C U C T R I K H O N G G I A N 135 Chii de 9: T O A D O K H O N G G I A N 159 Chu da 10: P H U O N G T R I N H M A T cAU 181 Chii de 11: P H U O N G T R I N H M A T P H A N G 192 Chii de 12: P H U O N G T R I N H D U C N G T H A N G 204 Chii de 13: T U ' O N G G I A O C U A D U C J N G T H A N G , MAT PHANG VA MAT CAU Chu de 14: G O C vA K H O A N G C A C H T O A DO Chii da 15: C A C H I N H K H O I V A L T N G D U N G T O A D O PHU L U C : CAC DE O N T H I T O N G HOP 224 266 308 335 351 nhiasachhongan corn v n Email: n h a s a c h h o n g a n GUI ' TRAC NGHI$M \ ^ 1^ \ / ^ / X- \ - L e T h a i T o - VTnh L o n g - D T : 9 - Duang / - T a y N i n h * D T : - 19Truang Chinh - Budn - 129 Phan \ \ y y C h u Trinh - Oa Nang 3852971 - 518 Cach ^ y- y chlm bia va c h i l : , NS HONG AN" - NS N h a T r a n g - C a m R a n h * D T : 3854496 - NS D o n g Ho - Rach Gia - K i e n G i a n g Phong '• djnh sach chinh pham, • DT: 3821317 - 25 LS L9i - Tp Thanh H a * DT: 3857099 - N S Lien SUPng, 127 Tran Qudc Tuan - T r a Vinh - 278 LeHong y M aThupt *DT: 3953408 - 01 Hai B aTrtfng - B u d n MS T h u $ t *D T : (0500) \ ^ ISBN: 978-604-939-542-0 - Q u y NhOn * DT: 3823453 M a n g T h a n g Tann - T h u D a u M o t - B D - V T h j S a u , / T o n DuTc T h a n g - L o n g X u y e n - 6H a n T h u y e n - TP H u e - 78 B a c h D i n g - D a N i n g - D T : ( 1 ) 8 75bnb 935092 / / Gia: 72.000d / / ^ / ... (x2, y2, zi) c a) [ a , a ] = = (0; 0; ) = Yi b) [ a , b ] = yi Yi X2 Y2 ^2 = (yiZ2 = -(Y2Z1 Y2 V YI X| / - y Z i ; Z | X 2- Z2Xi; X i y - Y1Z2; X2yi) Z2X1 - Z X ; X y i - xiy2) Z2 Z2 ^2. .. OA' Do A H = — A B = 25 OA' AB' 27 36 25 '25 25 ;0 H 48 36 25 25 48 Vay phuong trinh mat cau can tim: X- 25 ;0 y- 36 25 + z' = — Bai toan 8: Cho dilm A ( l ; - ; 2) va B (2; 0; 1) Tim quy tich... yi X2 X3 Zj + X, y2 y3 X3 X, z y2 X2 = [c,a] + [c,b] b) a [ b c ] = X X|| z, z y2 x + z, +yi Z] yi Z3 y3 X3 Z3 X3 Z, +y3 y2 Z2 X| Z, X2 y2 y, X3 Xi +Z3 Xj yi X2 =[a.b]c y2 Bai toan 9: Trong