Một số phương pháp giải các chủ đề căn bản hình học 12 phần 2

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 DANG TOAN CAC DAI LlTtfNG HINH HOC • - Khodng each giita hai diem A(xi, yi, zi) va Bfxi, yi, zj): AB = ^{x,-x,y- +{y, - J , ) - + ( z , - z , ) ' - Gck giita hai vecia u = (x,y,z) va v = (x',y',z): x.x'+y.y'+z.z' COS(M, V ) = • + y ' +z -\^x"+y"+z" - Goc cua tarn gidc ABC: cos A = cos{AB, - Dien tich tarn gidc ABC: = - The tich lit dien A BCD: V = The tich hinh hop ABCD.A AC) [AB, AC] [AB, AC ] AD 'B'CD':V= \AB, AD] AA' Bai toan 1: Tinh cosin cua goc giiia hai vecta u va v moi truong hop sau: a) li = ( ; 1; 1); v -(2; 1;-1) b) u =3i + 4T; v - - +3k Gidi xx'+y.y'+z.z' a) cos(u,v) = yjx- +y~ +7? x'- +y'- +z'' b)Tac6 u = ( ; ; ) , v = ( ; - ; ) c o s ( u , v ) = Baitoan2: Chocacvecta: u = i - j ; v = i + ( j -8V13 65 - k); w = i - k +3 j a) Tim cosin cua cac goc ( v , i ), ( v , j ) va ( v , k ) b) Tinh cac tich v6 huong u v , u w , v w Gidi a) u = ( ; -2; 0), v = (3; 5; -5), w = (2; 3; -1) Va cac vecta dom v i T = ( ; 0; 0), f = (0; 1; 0), k = (0; 0; 1) nen cos(v,i) = v.i cos(v,j) = V.J V59 169 b) Ta CO u V = x.x' + y.y' + z.z' = -7 Tuang tir thi dugc u w = -4, v w =26 Bai toan 3: Cho vecta u y khac Chung minh rSng: cos~( u i ) + cos'^( u , j ) + cos^( u , k ) = vai / ; / va la cac vecta dan vi Gidi Cac vecta dan vi: / =(1;0;0); } = (0;l;0)va k =(0;0;1) „., ^ , ^ r, u.i Gia su u = (x; y; z) ta co: cos(u,i) = X + z^ Do d6cos"(u,i) = ~ X" •> -> • + y" + z" Tuong tir: cos"(u.j) = — ; cos^(u,k)= , ^\ x~+y+z x"+y"+z Tu suy dieu phai chimg minh Bai toan 4: Cho hinh binh hanh ABCD vai A(-3; -2; 0), B(3; -3; 1), C(5; 0; 2) Tim toa dinh D va tinh goc giiJa hai vecta AC va B D Gidi Taco BA =(-6; I ; - ! ) BC = ( ; 3; 1) Vi toa cua hai vecta khong ti le nen ba diem A, B, C, khong thang hang Goi D(x; y; z) Tu giac ABCD la hinh binh hanh va chi x +3 =2 AD = BC « x= -l « y = l VayD(-l;l;l) y+2=3 z =1 z =l Ta CO AC = (8; 2; 2), BD = (-4; 4; 0), do: cos( A C , BD) = -32 + V72.V32 • V a y ( A C , B D ) = 120° Bai toan 5: Tim toa diem M : a) Thuoc true Ox cho M each dku A ( l ; 2; 3) va B(-3; -3; 2) b) Tren mat phSng (Oxz) cho M each d^u ba dikm A ( l ; 1; 1), B ( - l ; 1; 0), C(3;l;-1) 170 Gidi a) M thuoc Ox nen M(x; 0; 0) Ta CO M A = M B » o MA^ = MB^ (1 - x)^ + ' + 3^ = (-3 - x)^ + (-3)^ + 2^ » X = -1 Vay M ( - : ; 0) b) M thuoc (Oxz) tren M ( x ; 0; z) Ta c6: M A = M B = M C A M ' = BM" AM' =CM- • ( x - l ) ' + l + ( z - l ) - = ( x + l ) - + l + z' [ ( x - l ) ' + l + ( z - l ) ' = ( x - ) " + l + (z + l ) ' x =— VayM 4x-4z = z= — Bai toan 6: Cho tam giac A B C c6 A ( - ; 1; 0), B(0; 3; -1), C ( - l ; 0; 2) 4x-2z = l a) Chung minh tam giac A B C c6 goc B nhon b) T i m toa diem H la hinh chieu cua A tren canh BC Gidi a) Ta CO B A = (-2; -2; 1), BC = (-1; -3; 3) Nen cosB = BA.BC 11 B A BC 3Vl9 > => goc B nhon b) H(x; y; z) thuoc BC nen B H = t B C Do Ta CO X = -t,.y - = -3t, z + = 3t => x = -t, y = - 3t, z = -1 + 3t A H BC nen A H BC = (-t + ) ( - l ) + (-3t + 2)(-3) + (z + 1)3 = ^ t = Vay hinh chieu H U 11 24 14 ' ' Bai toan 7: Cho ba d i l m A ( ; ; 0), B(0; 0; 1), C(2; 1; 1) a) Chung minh A , B, C khong thang hang b) Tinh chu v i , dien tich va dp dai duong cao cua tam giac A B C ve tir dinh A Tinh cosin cac goc cua tam giac A B C Gidi a) Ta CO B A = ( ; 0; -1), BC = (2; 1; 0), toa dp hai vecta khong t i le nen chiing khong cung phuong Vay ba diem A , B , C khong thang hang 171 b) AB = +0' + ( - ! ) ' = V2 , BC = V2' + ' +0' = VS AC = V l ' + ' +1' = V3 Vay chu vi tam giac ABC bang V2 + V3 + Vs Vi BC^ = AB^ + AC'^ nen tam giac ABC vuong tai A do dien tich: S=iAB.AC=^ 2 BC Vi tam giac ABC vuong tai A nen: — Ta CO SABC = " BC.hA = ^ hA = = cosA = 0, cosB = = ,cosC = =^ BC V5 BC V5 Bai toan 8: Trong khong gian toa Oxyz, cho tam giac ABC c6 A ( l ; ; - ) , B ( ; - l ; ) , C(-4; 7; 5) a) Tinh dien tich va dai duong cao hA ciia tam giac ve tu dinh A b) Tinh dai duong phan giac cua tam giac ve tir dinh B Gidi a) Ta CO A B = (1; -3; 4), AC = (-5; 5; 6), BC = (-6; 8; 2) [ A B , A C ] = (-38;-26;-10) Vay SABC 2S AB,AC = - V ' + ' + ' =V554 2V554 V277 va hA BC Vi04 Vl3 • b) Goi D(x; y; z) la chan duong phan giac ve tir B: ^ D A ^ BA ^ V26 ^ Ta CO DC BC Vi04 Vi D nam giiia A, C nen X= — 2(l-x) = x + DA = - - DC A B / / C D Vay A B C D la hinh thang [AB, AC]| + i | [ A D A C ] I = 3^1046 Bai toan 11: Cho bon d i l m A ( 1; 0; 0) B(0; 1; 0), C(0; 0; 1) va D(-2; 1; -2) a) Chung minh rang A , B, C, D la bon dinh cua mot hinh t u dien b) Tinh goc giiia cac duang thang chua cac canh doi cua tu dien c) Tinh the tich t u dien A B C D va dai duong cao cua t u dien ve t u dinh A Gidi a)Tac6: A B = ( - ; 1;0), A C = ( - l ; ; 1) A D = ( - ; l ; - ) nen | A B A C ] = V0 0 -1 -1 1 -1 -1 \ = (l;l;l) / Do [ A B , A C ] A D = -3 + - = -4 ^ 0, suy ba vecta A B , A C , A D khong dong phang Vay A , B C, D la bon dinh cua mot hinh tu dien b) Taco C D = ( - ; l ; - ) BC = ( ; - ; 1), B D = (-2; 0;-2) 173 Goi a, y Ian lugt la goc tao bai cac cap duomg thang: A B va C D , A C va BD, A D va BC thi ta c6: cosa = cos(AB,CD) = c o s p = cos(AB,CD) cosy = cos(AB.CD) + l + 0| V2.Vr4 + SA/T? 14 0-2 V2.V8 = 3Vi7 -1-2 ^/2.^/l4 c) The tich t u dien A B C D la V = 14 AB,AB AD _ ~ 3V 3V Do dai duang cao ve tir dinh A la: hA S 2V3 BC, B D Bai toan 12: Cho t u dien A B C D c6: A ( - l ; 2; ) , B(0; 0; 1), C(0; 3; ) , D(2; 1; ) a) l i n h dien tich tarn giac A B C va the tich t u dien A B C D b) rim hinh chieu cua D len mat phSng ( A B C ) Giai a ) T a c A B = { ; - ; 1), A C = ( ; ; ) , A D n e n j A B A C ] = (-1; ; ) ^ S A H C = ^ V a ( A B , ACJ.AD = ^ = > V A i c n = =(3;-l;0) [AB, AC] [AB.ACJ.AD V6 =- b) Goi H(x; y; z) la hinh chieu D tren mat phdng ( A B C ) t h i : A H = ( X + 1; y - 2; z), D H = ( X - 2; y - 1; z) Ta c6: X AB,AC A H = X DH.AC = X - DH.AB = X = 2y + z = + y= E(5;-|:3) Bai toan 4: Cho b6n di§m A(-3; 5; 15), B(0; 0; 7), C(2; - ; 4), D(4; -3; 0) Chung minh hai duong thang A B va CD cat nhau, hay tim toa giao diem Gidi Taco: AB =(3;-5;-8) AC =(5;-6;-11) AD = (7; -8; -15) CD = (2; -2; -4) Do l A B , AC I = (7; -7; 7) =^ [ A B A C ] AD = nen AB, CD dong phang hon nxxa A B , CD khong cung phuong, do duong thang AB va CD cat Goi M(xvi; yvi ZM) la giao diem cua AB va CD Dat M A = k M B , MC = k ' M D Ta c6: lex Ic X X» Xv, = — (1 1-k X = ~ 1-k' |A ~ 1-k 2-4k' 1-k' 176 ^YA-ky,, ^Yc-k'yp ^ 1-k ^ 1-k' 1-k' 15-7k _ 1-k' Giai duac k' = — nen M 11 ^ - l + 3k' 1-k -kz^ ^ y^.-k'Zp ^ 1-k 1-k - k' - J ' ;ii Bai toan 5: Cho hai diSm A ( ; 5; 3), B(3; 7; 4) T i m diSm C(x; y; 6) d l A , B , C liiang hang Giai x-2 =k Ta CO A , B, C thdng hang A C = k A B « • y - = 2k 3= k x =5 « y = ll k =3 Vaydigm C(5; 11; 6) Bai toan 6: Cho tarn giac A B C vuong a A biSt A ( ; 2; -1), B(3; 0; 2), C(x; -2; 1) a) Tim tarn va ban kinh ducmg tron ngoai tiep tarn giac A B C b) T i m dai duong cao ciia tam giac A B C ve t u dinh A Giai a) Tam giac A B C vuong tai A nen A B ^ + AC^ = BC^ Ma AB^ = + + - 14, AC^ = (x - 4)^ + 16 + = (x - 4)^ + 20 BC^ = ( x - 3)^ + + = (x - 3)^ + => X = 18 => C(18; -2; 1) Tam ducmg tron ngoai tiep I la trung diSm cua BC r2i 3^ Nen I — ; - l ; ' 2? / va ban kinh R = BC V230 2 b) Tam giac A B C vuong tai A , duong cao A H nen A H B C = A B A C AH = AB.AC 6V4830 BC 115 Bai toan 7: Cho hai di^m A ( ; 0; -1), B(0; -2; 3) a) T i m toa diem C e Oy de tam giac A B C c6 dien tich bang VTl va thoa man OC < 1.' ' b) T i m toa diem D e (Oxz) de A B C D la hinh thang c6 canh day A B Giai a) Goi C(0; y; 0) ^ A B = (-2; -2; 4), A C = (-2; y; 1) 177 Taco: SABC = V u ^ | [ A B , A C ] | = « VTl^V(2+4y)- +36+(2y+4)' ^VTl 20y^ + 32y + 12 = o y = -1 (loai) vi OC < hoac Y = - | (nhan) VayC(0;-|;0) b) Goi D(x; 0; z) G (OXZ) => D C = (-x; -1; -z) A B C D la hinh thang va chi A B , D C cung huong - X -1 - z > < : ^ x = l , z = -2 -2 -2 VayD(l;0;-2) Bai toan 8: Cho tu dien ABCD c6: A(2; 1; -1), B(3; 0; 1), C(2; - ; 3) va D thuoc true Oy Tim toa dp dinh D biet VABCD = Gidi Goi D(0; y; 0) thuoc true Oy Ta eo: A B ^ ( M ; 2), A D = (-2; y - ^ ) , ^ C ^ ( ; -2; 4) = > [ A B , A C ] - ( ; -4; -2)=>[AB, A C ] AD =-4(y - 1) - =-4y + Theo gia thiet VABCD = » - [ AB, AC ] AD =5 o - 4y + = 30 o y = -7; y = Vay eo dikm D tren true Oy: (0; -7; 0) va (0; 8; 0) Bai toan 9: Trong khong gian Oxyz, eho digm A ( l ; 0; 3), B(-3; 1; 3), C ( l ; 5; 1) va M(x; y; 0) I Tim gia tri nho nhat T MA ] I + MA+ MC Gidi Goi I la trung diem eua BC: =:>I(-1;3;2)=^ MB + MC = 2MI T = 2(MA + MI) ZA - > va z, = > => A va I nam ve eung phia doi vai mp(Oxy) va M(x; y; 0) thuoe mp(Oxy) nen lay doi xung I ( - l ; 3; 2) qua mp(Oxy) J(-l; 3; -2) ^ M I = MJ ^ T = 2(MA + MJ) ^ 2AJ = V38 r ^ Dau = xay M la giao diem eua doan MJ voi mp(Oxy) la M — ; - ; \ 5 y VayminT = 2V38 178 ... (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. .. 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... 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