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Đây là tài liệu bồi dưỡng hoc sinh giỏi khối 12 với chuyên đề phương pháp tọa độ trong không gian. Tài liệu tổng hợp những kiến thức lý thuyết và đưa ra các dạng bài tập nâng cao giúp các m học sinh 12 có thể có những kiến thức chuyên sâu về phần phương pháp tọa độ trong không gian.
C h u o n g I I I : P H U O N G P H A P T O A T R O N G K H O N G D O G I A N §1 H E T O A D O T R O N G K H O N G G I A N A KIEN THUC C O B A N Toa khong gian Ba vecto don v i i , j , k tren true Ox, Oy, Oz : i = ( ; ; ) , ] = ( ; 1;0), k = (0; 0; 1) M M(x, y, z) hay M = (x, y, z): >T>~ OM = x i + y j + z k a(x, y, z) hay a = (x, y, z): a = x i + y j + z k • Hai vecta: u = (x, y, z) va v = (x', y', z') thi: u ± v = (x ± x' ; y ± y'; z ± z') ; k u = (kx; ky, kz) u v = xx' + yy' + zz' ; | u Vx + y + z 2 x.x + y.y + z.z' cos(u, v) >/x + y z Vx' +y' z' 2 + 2 + Hai diem A ( x i , y i , z\) va B(x , y , z ) thi: 2 AB = (x - x i ; y - y i , z - z\) AB = yj( 2 - x ^ + ( y - y i ) +(z - z r 2 x M chia AB theo ti so k * 1: M x i - k x _y - k y l-k l-k kz., l - k Tich co hirotig: cua a = (x, y, z) va b = (x', y', z') la vecto: n = [a , b ] = y z z y' z' z x X y x' J X ' y' • Ket qua: - Vecto [ a , b ] vuong goc vdi a , b - Do dai cua vecto [a , b ] : | [a , b ] | = | a | | b | sin(a , b ) - vecto a , b cung phuong: - vecto a , b , c dong phang: [a, b ] = [a, b ] c =0 - vecto a', b , c khong dong phang: 154 [a , b ] c * Dien tich va the tich Dien tich tam giac ABC: S=-|[AB,AC]| The tich tii dien ABCD: V = — | [ AB AC AD | The tich hinh hop ABCD.A'B'C'D': V = | [ A B , AD ] AA~' | The tich hinh lang try A B C A ' B ' C : V = | | [ A B , ADJ.AAT' | Phirong trinh mat cau: Mat cau (S) tam I(a, b, c) ban kinh R: (x - a) + (y - b) + (z - c) = R hay: x + y + z + 2Ax + 2By + 2Cz + D = 0, A + B + C - D > 2 2 2 2 2 co tam I ( - A , - B , - C ) va ban kinh R = \ / A + B + C - D Chu y: - Tam duong tron I ngoai tiep tam giac ABC khong gian: f l A = IB = IC 2 [ I e (ABC) Tam K mat cau ngoai tiep tii dien ABCD: KA = KB = KC = KD D A N G 1: TOA D O DIEM, VECTO Ba vecto don v i tren true Ox, Oy, Oz : i = ( ; ; ) , j = ( ; 1;0), k = (0; 0; 1) Hai diem A ( x i , y i , Z\) va B(x , y , z ) thi: 2 AB = (x - x i ; y - y i , z - Zi) 2 AB = f x - x ) 2 +(y - y ^ 2 + (z - z ^ 2 ^ yi - y i k - Diem M chia A B theo ti k * 1: M , z k z 2 l-k l-k l-k Toa trung diem M cua doan A B , tam G cua tam giac ABC tam E cua t i i dien ABCD voi toa dp A(x y zO, B(x , y , z ), ys, Z ) D(X4 y4 , z ): +x + x +x +x i + x + x X = l X = + y + y yi + y E y = y i + y + y + y.4 G v M y = y + z +z + z +z f z Zl + z zz= i z= Z— 1; 1( 2 x x X l 2 2 Y l Z l 3 3 z 4 Phep toan ciia hai vecto: u = (x, y, z) va v = (x , y' z') thi: u + v = (x + x' ; y + y'; z + z') k u = (kx; ky, kz), u - v = (x - x' ; y - y'; z - z') u v = xx' + yy' + zz' ; 155 [u , v ] = f y Vy ' z z' j Z X z x' x x' y s y' > - Quan he cac vecta Vecta [ a, b ] vuong goc v o i a, b Vecto a vuong goc voi b » a b = Hai vecto a, b cung phuong b = k a Hai vecto a, b cung phuong [ a, b ] = Ba vecta a, b , c dong phang [ a , b ] c = B6n diem A , B , C, D dong phang [ A B , A C ] A D = Ba vecto a, b , c khong dong phang [ a , b ] c # Bon diem A , B, C, D khong dong phang [ A B , A C ] A D * Chu y: Ung dung toa dp khong gian de giai cac bai toan hinh khong gian co dien, quan he song song, vuong goc, dp dai, goc, khoang each, vi tri tuong doi, V i du 1: Cho ba vecto a = (2; - ; 3), b = (0; 2; - ) , c = (1; 7; 2) a) T i m toa dp ciia vecto e = a - 4b - c b) Tim toa ciia vecta f = a - — b + 3c Giai a) e = a - 4b - 2c = (2 - - 2; -5 - - 14; + b) f = a - - b +3c = (8 + + ; - - - + 21; 12 + 3 V i du 2: Tim toa dp cua vecto m cho biet: a ) a + m = a v a a = ( ; - ; 1) 4) = (0; -27; 3) 1 KK - +6) = ( 1 ; - : — ) 3 V b) a + m = b va a = ( ; ; - l ) , b = ( ; - ; 3) Giai a) a + m = a => m = a = (0; -6; 3) b) a + 2m = b = > m = a + — b = —; ;2 2 { 2 ) V i du 3: Cho hai bp ba didm: 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 bp ba diem nao thang hang? Giai Taco CA = ( ; 3; 0), CB = ( ; 1; 1) V i cac toa dp khong tuong xiing ti le nen khong co s6 k nao de CA = kCB , suy A , B, C khong thang hang Ta co t T T = (10; - ; 0), G B = (5; - ; 0) => C^A' = C^"' Do A', B', C thing hang 156 V i du 4: Tinh tich vo huong ciia hai vecto moi truong hop sau: a) a = (3; 0; -8), b = (2; - ; 0) b) a = (1; - ; 2), b = (4; 3; -5) c) a = (0; a) N/2 ; S ) , b = (4; S ; - ) Giai a.b = 3.2 + 0.(-7) + (-8).0 = b) a b = 1.4 + (-5).3 + 2(-5) = - c) a b = 0.4 + N/2 73 + V3(-%/2) = V i du 5: Cho ba vecto: a = ( ; - ; 1), b = ( ; ; - l ) , c = (3; ; - ) Tinh: a) (a b ) c , a (b c ) b) a b + b c + c a , a c + b - c Giai 2 : 2 a) Taco: a.b = 1.4 + ( - l ) + l ( - l ) = D o d : ( a b ) c = c = (9; 6;-3) Ta co b c = 4.3 + 0.2 + (-1)(-1) = 13 Do a ( b c ) = 13a = ( ; - ; 13) b) Taco i = 3, b = 17, c = 14 nen a b + b c + c a = 3b + c + a = (77; 20; - ) va a c = 2 2 2 b4a.c + b -5c 2 = -53 Vi du 6: Cho I u 1=2 | v I = , goc giua hai vecto u va v bang — Tim k de vecto p = k u + 17 v vuong goc voi vecto q = u - v Giai l-i ^ ,_ | - _ 2n Ta co | u | = 2, I v | = 5, cos(u v ) - cos— — - Do p lq p.q = 0(ku +17v)(3u - v) = o k u - 17 v 2 + (51 - k)u v - 3k.4 - 17.25 + (51 - k)2.5— = 17k - 680 = k - 40 Vi du 7: Cho ba diem A(2; 0; 4) B(4; S ; 5) va C(sin5t; cos3t; sin3t) Tim t de A B vuong goc voi OC Giai Ta co AB = (2; 73 ; 1), OC = (sin5t; cos3t; sin3t) Hai dudng thang A B va OC vuong goc vdi va chi khi: A B OC = 2sin5t + 73 cos3t + sin3t = sin5t + — cos3t + — sin3t = sin5t + sin(3t + — ) = 157 Jl o sin5t = sin(-3t ) Vay t = -— + -k*.t=— + kn, k e Z 24 V i du 8: Xet su dong phang cua ba vecto a) a = (-3; 1; - ) b = (1; 1; 1), c' = (-2; 2; 1) b) a = (4; 3; 4), b = (2; - ; 2), c = (1; 2; 1) Giai a) T a c [ a , b ] = ^ ] J ~ j = (3; 1;-4) Do [a, b ] c = -6 + - = -8 * Vay vecto khong dong phang b) Taco [ a , b ] = (10; 0;-10) => [ a , b ] c = Vay vecto dong phang V i du 9: Cho a = ( : - ; 1), b = ( ; 1;2), c" = ( ; ; ) v a 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 Giai a) Taco [a", b ] = ( - ; - ; 1)=> [a", b ] c = - * Vay vecto khong dong phang b) Gia sir d = m a + nb + pc m + 4p = f m = -2 < - m + n + 2p = [AB A C ] = (-3;-3; 3) Va AD = ( ; - ; - ) = > [ A B , A C ] AD = 0: dpcm V i du 11: Chung minh cac tinh chat sau day cua tich vo huong a)[a,a]=6 c) [ k a , b ] = k [ a , b ] = [ a , k b ] 158 b) [a , b ] = - [ b , a ] Giai Yi y a) [ a , a ] i Zj z b) [a, b] z i z x i x' x y Xj y : l Zj Xj yi i x = (0; 0; 0) = : yi y ) = (yi - Y2 i ; zix - z xi ; xiy - x y ) = - ( y z i - yiz ; z Xi - Z!X ; x y - x y ) z z x x 2 z 2 2 y yi x A x 2 z 2 '"2 |z i' k z 2 x i x x i : x y yi = - [ b , a] Ket qua [ a , a ] = - [ a , a ] => [ a , a ] c) k [ a , b ] = yi Ik'" y i z ky y z i z ,k z kz x z kz x z i x x , k x x 2 yi y ) kxj k y j 2 kx x i x x x y = [ka, b] Tucmg tu: k [ a , b ] = [ a , k b ] V i du 12: Chung minh cac tinh chat sau day cua tich vo huang a) [ c , a + b ] = [ c , a ] + [ c , £ ] b) a [b , c ] = [a , b ] c c) I [a b ] | = | a b | -(a.b) Giai 2 y yi + y a) [c, a + b] y yi z z 3i z l Z y z - [a.b]c z 2 j : z x z l x z = [c, a] y z z z y i Jy i |y y yi b) a [ b c ] = Xj z i x + x i l z z x -1 + [c, b] x y x +yi Z, 3 y l l l +y +z x X, y Z 2 x x X X Z 3 Z 2 i |z '|z Z 2 x 3 x ; yf + y y j , Jy y yi x Z x X c) VP = |a | | b | - ( a b ) = | a | | b | - | a P I b | W a = Ia | | b (1 - cos a) = | a | | b sin a 2 2 2 = |[a.b]| 2 2 2 = VT V i du 13: Trong khong gian cho ba vecto a , b , c tirng doi khong cimg phuong Chung minh rang dieu kien can va dii de vecto tong: a + b + c = la [ a , b ] = [ b , c ] = [ c , a ] Giai Tira + b + c = - ^ a = - ( b + c ) = > [ a , - b - c ] = D o d [ a , - b - c ] = [ a - b ] - [ a c ] = =>[c, a ] = [a b J Tuong tu ta cung c o [ b , c ] = [ a , b ] Vay: [ a , b ] = [ b , c ] = [ c , a ] Nguoc lai, tir [ a , b ] = [ b , c ] => [ b , a + c ] = Mat khac, [ b , b ] = = > [ b a + b + c ] = = > b cimg phuong voi vecto a + b + c Chung minh tuong tu ta cung co vecto a ciing phuong voi vecto a + b + c Nhung a va b khong cimg phuong, vay a + b + c = V i du 14: Cho diem M(a; b; c) a) Tim toa hinh chieu ciia M tren cac mat phang toa va tren cac true toa b) Tim khoang each tir diem M den cac mat phang toa do, den cac true toa Giai a) Goi M i ( x ; y; 0) la hinh chieu ciia diem M(a; b; c) tren mp(Oxy) thi: M M j = (x - a; y - b; -c) V i MM I = va M M ^ T = nen x - a = l y - b = VayM,(a;b;0) Tuong tu, hinh chieu ciia M tren mp(Oyz) la Mi(0; b; c) hinh chieu ciia M tren mp(Oxz) la M (a: 0; c) Gia sir M ( x ; 0; 0) la hinh chieu ciia M(a; b; c) tren true Ox thi x MM X = (x - a; - b ; - c ) V i M M i = nen x = a, do M (a; 0: 0) V x Tuong tu, hinh chieu ciia M(a; b; c) tren true Oy la M ( ; b; 0), hinh chieu ciia M(a; b; c) tren true Oz la M (0; 0; c) y z b) d(M; (Oxy)) = M M , = V ( a - a ) + ( b - b ) + ( c - ) 2 = |c | Tuong tu d(M; (Oyz)) = | a | , d(M; (Ozx)) = | b | Ta co d(M; Ox) = M M , = x/(a - a) + (b - 0) + (c - 0) =x/b +c Tuong t u d(M; Oy) = Va + c 160 2 d(M; Oz) = Va + b 2 2 V i du 15: Cho hai diem A ( x i ; y i ; z\) va B(x ; y ; z ) T i m toa diem M chia doan thang A B theo ti s6 k * Giai Voi diem M(x; y; z) ta co: MA = (xi - x; y, - y; z\ - z), 2 MB = (x - x; y - y; z - z) Diem M chia doan A B theo ti so k * va chi 2 x = —l-k yi - ^ MA =kMB Yi ~ Y = k ( y - y) o l-k Zj - z = k(z - z) Z =— l-k V i du 16: Trong khong gian Oxyz cho ba diem A ( l ; 2; 4), B(2; - ; 0), C(-2;3;-l) a) Gpi (x; y; z) la cac toa dp cua diem M nam tren mat phang (ABC) Tim su lien he giua x, y, z b) Tim toa dp cua diem D biet rang hinh ABCD la hinh binh hanh Giai a) AB = (1; - ; - ) , AC (-3; 1; -5), A M = (x - 1; y - 2; z - 4) Xj - x = k(x - x) k y = 2 Ta co M nam uen mat phang (ABC) [ A B , AC] A M = 19(x - 1) + 17(y - 2) - 8(z - 4) = 19x + 17y - 8z - 21 = b) V i ABCD la hinh binh hanh nen AB = DC : = -2 - x D -3 = - y » y =6 D D -4 = - - z Vay D(-3; 6; 3) n V i du 17: Cho A(2; l ; 3), B(4; 0; 1), C(-10; 5; 3) a) Chung minh rang: A, B, C la ba dinh ciia mpt tam giac b) Tim chan duong phan giac ngoai ciia goc B ciia tam giac ABC Giai r a) BA = ( - ; - l ; X B C = ( - ; 5; 2) Ta co BA va BC khong ciing phuong nen A B, C la ba dinh ciia mpt tam giac b) Gpi BE la ducmg phan giac ngoai ciia goc B, do: BA EA EA BC " EC ^ EC ~ 15 " V i vecto E A , EC ciing huong nen E chia doan AC theo t i k = — E(5;~;3) 161 V j du 18: Cho hinh hop ABCD.A'B'C'D* biet A ( x y ; z ), C(x ; y ; z ), B'(x' ; y' ; z' ), D'(x' ; y' ; z' ) Tim toa cua cac dinh lai Giai Goi Q = AC n BD, Q' = A ' C n B'D' thi Q, Q' la trung diem cua AC, B'D' nen: f i + yi+y ,z + z l ' "* i; 2 x 4 x x 3 x n 3 V /x' +x I 2 _ y + y , z +z ' ' 4 Tu AA*' = BB ' = CC"' = DTI' = QQ?, suy ( x + x ' + x ' " yi + y ' + y - y 2 J I f x ' + x - - HX _ y' + y - y i + y z ' + z ' - z + z ) ) 2 J I z + z + z — z l 2~ yi + y + y '2-y'4 J 2 { A C B x 2 X + 4 X + X X l 3 X X : 3 ( i+ -x' -f x y -y2+y4 l 3~ 2 I V i du 19: Cho hinh hop ABCD.A'B'C'D' co cac diem A ( l ; 0; 1), B(2; 1; 2), D ( l ; - ; 1) va C'(4; 5; -5) Tim cac diem lai Giai Ta co ABCD la hinh binh hanh nen: x - 2=0 x =2 y = » < y = Do do: C(2; 0; 2) BC = AD x x D z + z z + z c c c c z - 2=0 c L z =2 c Va A A ' = BB' = DD' = CC' = (2; 5; -7) Nen A'(3; 5; -6), B'(4; 6; -5), D'(3; 4; -6) V i du20: a) Tim toa diem M thuoc true Ox cho M each deu hai diem A(l;2;3)vaB(-3;-3;2)' b) Tren mat phang (Oxz) thi diem M each deu ba diem A ( l ; 1; 1), B ( - l ; 1; 0), C(3; 1; -1) Giai a) M thuoc Ox nen M(x; 0; 0) Ta co MA = MB o MA = M B o (1 - x) + + = (-3 - x) + (-3) + o x - - Vay M ( - l ; 0; 0) b) M thuoc (Oxz) tren M(x; 0; z) Ta co: M A = M B = M C 2 (AM = B M 2 A M = CM 162 2 ((x-l) + + (Z-1) 2 2 = ( X + 1) +l + z (x - ) +1 + (z - ) = (x - 3) + + (z + l ) 2 2 » { 4x + 2z = 4x-4z = x= VayNlf-jO;-6 ! o - J v Vi du 21: Cho tam giac ABC co A(-2; 1; 0), B(0; 3; - ) , C ( - l ; 0; 2) a) Chung minh tam giac ABC co goc B nhon b) T i m toa diem H la hinh chieu ciia A tren canh BC Giai a) Taco BA = ( - ; - ; 1), BC = ( - l ; - ; 3) BA.BC 11 Nen cos B = > => goc B nhon BA BC 3V19 b) H(x; y; z) thuoc BC nen B H = tBC Do x = - t , y - = -3t, z + = 3t => x = - t , y = - 3t, z = - + 3t Ta co A H BC nen A H BC = (-t + ) ( - l ) + (-3t + 2)(-3) + (-1 + 3t).3 = Vay hinh chieu H f - — ; - —; — 3 Vi du 22: Cho b6n diem A(-3; 5; 15), B(0; 0; 7), C(2; - ; 4), D(4; - ; 0) Hoi hai duong thang A B va CD co cat hay khong Neu chiing cat nhau, hay tim toa giao diem Giai Ta co: AB = (3; - ; - ) , AC = (5; - ; -11) v AD - (7; - ; -15), CD = (2; - ; -4) Do [ A B A C ] = (7; - ; 7) => [ A B A C ] AD = nen A B CD dong phang, hon nua A B , CD khong ciing phuong do duong thang A B va CD cat Goi M ( X M ; >M, ZM) la giao diem ciia A B va CD Dat MA = k M B , MC = k'MD Taco: k x A kx - 4k l-k 1-k' l-k' i-k - l + 3k y - y _y y i-k i-k' l-k' l-k k A ~ B_ y ~ < = > l-k l-k' l-k l-k' [—3 Giai duoc k' = — nen M — ; —; 11 I 11 12 X T B k A Z k B c k z D k c 163 V i du 23: a) Cho hai diem A(2; 5; 3), B(3; 7; 4) Tim diem C(x; y; 6) dk A B, C thang hang b) Cho hai diem A ( - l ; 6; 6), B(3; - ; -2) Tim diem M thuoc mp(Oxy) cho M A + M B nho nhat Giai a) A, B, C thang hang AC = k AB 2: k = 2k = 1 Vay diem C(5; 11; 6) k = b) V i ZA = ZB = -2 => z ZB < => A, B hai phia cua mp(Oxy) Vay M A + M B nho nhat A, B, M thang hang A A M , AB ciing phuong [ A M , AB ] = Gpi M(x; y; 0) e mp(Oxy) •=> A M = (x + 1; y - 6; - ) , AB = ( ; - ; - ) Taco: [ A M , A B ] = (-8y - 24 ; 8x - 16; -12x - 4y + 12) = -8y - 24 = i8x-16 = -T y = -3 - x - y + 12 = Vay M A + M B ngin nhat M(2; - ; 0) V i du 24: Cho t i i dien ABCD co: A(2; 1; - ) , B(3; 0; 1), C(2; - ; 3) va D thuoc true Oy Biet VABCD = Tim toa dp dinh D Giai Gpi D(0; y; 0) thuoc true Oy Ta co: l J AB = (1; - ; 2), AD = (-2; y - 1; 1), AC = (0; - ; 4) => [ AB , A C ] = (0; - ; -2) => [ A B , AC] AD = -4(y - 1) - = - y + Theo gia thiet VABCD = — I [ AB, AC ] AD |=5 | - 4y + | = 30 y = - ; y = Vay co diem D tren true Oy: (0; - ; 0) va (0; 8; 0) V i du 25: Gpi G la trpng tam ciia tu dien ABCD Chimg minh rang duong thang di qua G va mpt dinh ciia tir dien ciing di qua trpng tam ciia mat doi dien voi dinh Gpi A' la trpng tam tam giac GA BCD Chiing minh rang GA' Giai Ta giai bang phuong phap toa dp Trong khong gian toa dp Oxyz, gia sir A(x ; y ; zi), B(x ; y ; z ), C(x3; y , z ), D(x ; y ; z ) thi trpng tam A' ciia tam giac BCD, trpng tam tii dien G: x 164 4 x 2 3 x +x +x y +y +y z +z + z v X, + X2 z ; 4 yi + y + y + y4 i + x A z z + +4 z z Do do: cX_f i x - x - x -x 3yi - y - y - y i -z - ~ 4 4 z GA'-f + x 2 + ~ y i + y +ys + y 12 12 + x x z 3 z Suy ra: GA = -3 G A ' => G, A , A' thang hang va z i+ z +3 12 z GA GA' + z Tuong tu thi co dpcm V i du 26: Cho tu dien noi tiep mat cau tam O va co A B = AC = A D Goi G la tam AACD, E, F la trung diem BG, A E Chung minh: OF _L BG O OD AC Giai AB = AC = A D va OB = OC = OD => OA (BCD) tai chan duong cao H voi HB = HC = HD Chon H lam goc toa do, voi he true Hx, Hy, Hz cho H A la true Hz, HB la true Hy, H D la true Hx A ( ; ; a ) , B(0; b; o j ; C ( c , ; c ; 0), D(d; 0; 0) va O(0; 0; z) suy ( i + a \ |'c +d b c a { ' '3J' I '2 '6, c d c c £ Cj + d b 12 ' ^ _ f va& OF c | c 7a 12'l2 + i 12 + d c 7a + —; 12 12 b z BG = f ^ i - A - b ; i 3 AC = (c,; c ; -a), OD = (d; 0; -z) Theo gia thiet OA = OB = OC = OD OA = OB = OC = OD v 2 (a - z) = b + z = c + 2 o a - 2az = b = c + c = d 2 +z = d + z 2 2 2 2 (1) Taco: OF.BG = o(ci + d) + c - 9b + 7a - 12az = 2 2 (2) Khai trien (2) va thay the (1) ta duoc: (2) az + cid = OD AC = : dpcm 165 D A N G 2: G O C , K H O A N G C A C H , DI$N T I C K T j H | j l C l i - Do dai cua vecto: u = (x, y, z) : I u | = \ / x + y + ^ Khoang each giua hai diem A ( x i , y i , zi) va B(X2, y , Z2): 2 AB = 7(x ) + ( y - y ) + (z X l 2 x Z l ) - Goc giua hai vecto: u = (x,y,z) v a v = (x',y',z'): , x.x'+y.y'+z.z' cos(u, v) = x/x + y + z yjx' + y ' + z' 2 2 2 - Goc cua tam giac A B C : cos A = cos(AB, AC) Chii y: Goc giua vecto tir 0° den 180° va cac goc lai giira duong thang, mat phang deu tir 0° den 90° Dien tich va the tich: Dien tich tam giac ABC: S = - | I [ A B , A C ] I The tich tii dien ABCD: V = - | [AB, AC ] AD | The tich hinh hop A B C D A ' B ' C ' D ' : V = | [ A B , AD ] A A ' | The tich hinh lang tru A B C A ' B ' C : V = i | [ A B , A D ] A A ' | Z V i du 1: Tinh cosin ciia goc giira hai vecto u va v moi trudng hop sau: a) u = ( ; 1; 1); v = ( ; 1;-1) b) u = 3F + f ; v =-2j +3k Giai a) cos(u, v) = xx'+ y.y'+ z.z ^x + y z Vx' + + y' _ V2 + z' b)Taco u =(3;4;0), v = (0; -2; 3) => cos(u , v ) = V i du 2: Cho cac vecto: u = i - j w=2i - k); — k + 3j a) T i m cosin ciia cac goc ( v i ), ( v b) Tinh cac tich vo hudng u v a) ;v=3i+5(j •8Vl3 j )va(v k ) u w, v w Giai u = ( l ; - ; ) , v = ( ; 5;-5), w = ( ; ; - l ) ^ Va cac vecto don vi X = (1; 0; 0), j = (0; 1; 0), k = (0; 0; 1) nen 166 v.i cos(v, i ) = 7= cos(v,k) = cos(v, j ) = 759 v.k M T R pj 59 -5 759 Iv k b) Ta co u v = x.x' + y.y' + z.z' = - Tuong tu thi duoc u w = - , v w = 26 Vi du 3: Cho hinh binh hanh ABCD voi A(-3; - ; 0), B(3; -3; 1), C(5; 0; 2) Tim toa dinh D va tinh goc giua hai vecto AC va BD Giai Ta co BA = (-6; 1; - ) , BC = (2; 3; 1) V i toa ciia hai vecto khong ti le nen ba diem A, B, C khong thang hang Gpi D(x; y; z) Tii giac ABCD la hinh binh hanh va chi x +3=2 x = -1 AD = BC y + = \y = l VayD(-l;l;l) |z =1 Ta co AC = (8; 2; 2), BD = (-4; 4; 0), do: cos(AC, B D ) 772.732 Vay ( A C , B D ) = 120° Vi du 4: Cho vecto u y khac Chiing minh rang: cos ( u , i ) + cos ( u , j ) + cos ( u , k ) = 2 Giai u.i Gia sii u = (x; y; z) ta co: cos(u, i) = Do cos (u, i ) = •x + y + z v/x + y + z 2 2 Tuong tu: cos (u, j ) 2 ; cos (u,k) x +y +z 2 x +y +z 2 Tir suy dieu phai chimg minh Vi du 5: Cho tam giac ABC vuong A biet A(4; 2; -1), B(3; 0; 2), C(x; - ; 1) a) Tim tam va ban kinh duong Uon ngoai tiep tam giac ABC b) Tim dp dai duong cao ciia tam giac ABC "e tir dinh A Giai a) Tam giac ABC 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; - ; 1) Tam duong tron ngoai tiep I la trung diem ciia BC 2 2 :: 2 2 167 21 BC /230 va ban kinh R = — = 2 b) Tam giac ABC vuong tai A , dudng cao A H nen AH.BC = A B A C AB.AC 674830 => A H = = BC 115 V i du 6: Cho ba diem A ( l ; 0; 0), B(0; 0; 1), C(2; 1; 1) a) Chung minh A , B, C khong thang hang b) Tinh chu vi, dien tich va dai duong cao ciia tam giac ABC ve tii dinh A c) Tinh cac goc cua tam giac ABC Giai Nen I ; ; A T T a) Ta co BA = (1; 0; - ) , BC = (2; 1; 0), toa hai vecto khong ti le hen chiing khong ciing phuong Vay ba diem A, B, C khong thang hang b) A B = V l + + (-1) = 72 2 BC = V2 + l 2 + = 75 AC = l + l + l = 73 Vay chu v i tam giac A B C bang 72 + N/3 + 75 V i B C = A B + A C nen tam giac A B C vuong tai A do dien tich: /R S = -AB.AC = — 2 2S 30 Ta co S B C = - B C h BC c) V i tam giac A B C vuong tai A nen: AB 72 _ AC 73 COSD = = —=• , COS C = = —== BC 75 BC 75 Cach khac: Tinh cosB theo cong thiic: BC.BA _ 2.1 + + _ 72 cosB BC.BA N/5.72 "75 2 2 2 ABC A A D V i du 7: Trong khong gian toa Oxyz, cho tam giac ABC co A ( l ; 2; -1), B(2; - ; 3), C(-4; 7; 5) a) Tinh dai duong cao h ciia tam giac ve tir dinh A b) Tinh dai duong phan giac ciia tam giac ve tir dinh B Giai a) Ta co AB = (1; - ; 4) AC = (-5; 5; 6), BC = (-6; 8; 2) A [AB Vay AC] = (-38;-26;-10) SABC = - [AB, AC] I = - 738 + 26 + 10 2S 2 2N/555 /555 ABC BC V104 726 b) Gpi D(x; y; z) la chan dudng phan giac ve tir B: 168 /555 , DA Ta co DC BA BC 726 /104 V i D nam giua A, C nen DA = — DC x = — 2(1 - x) = x + 11 O CD = 2DA \2 ( - y ) = y - o T ( - l - z) = z - z=1 y = Vi du 8: Cho hinh binh hanh ABCD co dinh A(3; 0; 4), B ( l ; 2; 3), C(9; 6; 4) a) Tinh goc B cua tam giac ABC b) Tinh dien tich hinh binh hanh ABCD Giai a) Ta co BA = (2; - ; 1), BC = (8; 4; 1) Ta co cosB = cos( B A , BC) = -j=- 75 b) Hinh binh hanh ABCD co dien tich: S = 2S BC A = 2.- I [ B A , BC ] | Taco [ B A , B C ] = (-6; 6; 24) => S = 18 72 Vi du 9: Cho b6n diem co toa A(2; 5; -4), B ( l ; 6; 3), C(-4; - 1; 12), D(-2; - ; -2) a) Chung minh ABCD la mpt hinh thang b) Tinh dien tich hinh thang ABCD Giai a) AB = ( - ; 1; 7), AC = (-6; - ; 16), hai vecto khong ciing phuong v i toa dp khong ti le suy A, B, C khong thang hang va co: A_ DC = (-2; 2; 14) = AB => A B // CD Vay ABCD la hinh thang ") SABCD = SABC + SADC D" = - | [ A B , A C ] | + - | [ A D , A C ] | = 371046 2 V i d u 10: Cho hai diem A(2; 0; -1), B(0; - ; 3) a) Tim toa dp diem C e Oy de tam giac ABC co dien tich bang T i l va thoa man OC > b) Tim toa diem D e (Oxz) &k ABCD la hinh thang co canh day A B Giai a) Gpi C(0; y; 0) => AB = (-2; - ; 4), AC = (-2; y; 1) 169 Ta co: S BC = VU A O - I [ A B , AC] | 11 o - v / ( + 4y) +36 + (2y + 4r = V l l 20y + 32y + 12 = y = -1 hoac y = -— (loai) VayC(0;-l;0) b) Goi D(x; 0; z) e (Oxz) => DC = (-x; - ; -z) A B C D la hinh thang va chi A B , DC cung huong - -z — = — = — < = > x = l , z = - - - VayD(l;0;-2) Vi du 11: Cho b6n diem A ( l ; 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 tu dien b) Tinh goc giua cac duong thang chua cac canh doi cua tu dien c) Tinh the tich tu dien ABCD va dai duong cao cua tu dien ve tir dinh A Giai a) Taco: AB = ( - ; 1;0), AC = ( - l ; ; 1), AD = ( - ; l ; - ) / 0 -1 -1 \ nen [ A B A C ] 0;i;i) V0 1 - - / _ x D o a i [ A B , A C ] AD = - + - = - * 0, suy ba vecto A B , AC, AD kho lg dong phang Vay A, B, C, D la bon dinh ciia mot hinh tir dien Taco CD = ( - ; l ; - ) , BC = ( ; - l ; 1), BD = ( - ; ; - ) Goi a, p, y lan luot la goc tao boi cac cap duong thang: A B va CD, AC va BD, A D va BC thi ta co: , — — , + •01 3N/7 cos a cos(AB,CD) 14 V2.N/14 cosp = |cos(AC,BD)| cosy = |cos(AD,BC)| = + 0-21 V2.V8 1-1-21 377 yfe.yfU 14 " A B.ADA|C=] A D I = Th£ tich tii dien ABCD la V = -1[AB AC] 6IL J I Do dai duong cao ve tir dinh A la: 3V 3V 2^3 "i—^Zi ^BCD i | | [ B C , B D ] | 21 V i du 12: Cho tir dien ABCD co: A ( - l ; 2; 0), B(0; 1; 1), C(0; 3; 0), D(2; 1; 0) a) Tinh dien tich tam giac ABC va t h i tich t i i dien A B C D b) T i m hinh chiiu ciia D len mat phang (ABC) A — = r : Giai a) Taco AB = ( ! ; - ! ; 1), AC = ( ; 1;0), AD = (3; - ; 0) TS -r-x -, I V6 n e n [ A B , A C ] = ( - ; 1;2) S c=-| l[AB, A C A B Va[AB, AC].AD = - ^ > V I [ A B A C ] AD | = i b) Goi H(x; y; z) la hinh chieu D tren mat phang (ABC) thi: AH = (x + 1; y - 2; z ) , D H = ( x - ; y - 1; z) Taco: x =— DH.AB = x -y +z DH.AC = x +y = i 2z x y AB,AC] A H = z=— Vay H 3 3, Vi du 13:Trong khong gian Oxyz, cho diem A ( l ; 0; 3), B(-3; 1; 3), C(l; 5; 1) A B CD= ± Y = va M(x; y; 0) Tim gia tri nho nhat T = | M A | + |MB + MC| Giai Goi I la trung diem cua BC: => I ( - l ; ; ) = > M B + MC = 2MI =>T = 2(MA + M I ) z = > v a z i = > = > A v a I nam ve ciing phia doi vdi mp(Oxy) va M(x; y; 0) thuoc mp(Oxy) nen lay doi xirng I ( - l ; 3; 2) qua mp(Oxy) A J ( - l ; 3; -2) => M I = MJ => T = 2(MA + MJ) > 2AJ - 738 Dau = xay M la giao diem ciia doan MJ vdi mp(Oxy) la M ' Vay minT = 2738 Vi du 14: Cho hinh lap phuong ABCD.A'B'C'D' co canh bang a Goi I , J lan luot la trung diem ciia A'D' va B'B a) Chiing minh rang IJ _L A C Tinh dai doan thang IJ b) Chung minh rang D'B mp(A'C'D), mp(ACB') c) Tinh goc giira hai dudng thang IJ va A'D Giai a) Chon he toa Axyz cho A ( ; ; ) , D(a; 0; 0), B(0; a; 0), A'(0; 0;a) Ta co C (a; a; a), B'(0; a; a), D'(a; 0; a) nen: l ( | ; ; a ) ; J(0; a; | ) 171 Taco: LJ = ( - - ; a - ; ~2 -2 2 AC' = ( a - ; a - ; a - ) = (a; a; a) a nen LJ A C ' = - - a + a.a - - a = - a + a = = ( ; a ; 2 \f \ r V a y l J l A C Doan IJ = 1— - + a2 + I 2) a a } \2 2j b) DS chung minh D'B mp(A'C'D), ta chung minh D ' B I A ' C D ' B l A ' D o D ' B A ' C = 0, D ' B A ' D = Taco D ' B = (-a; a;-a), A ' C = (a; a; 0); A ' D = (a; 0;-a) Do D ' B A ' C = 0, DTJ.AT) = Tuong tu, ta cung chung minh duoc D'B mp(ACB') c) A ' D = (a; 0; -a) Goi cp la goc giua hai dudng thang IJ va A ' D thi: •— a + a.O (-a) coscp = cos(JJ, A ' D ) IJ.A'D IJ.A'D 176 iV2 Vay cp = 90° V i d u 15- Cho hinh lap phuong ABCD.A'B'C'D' canh bang a Tren cac canh BB', CD, A'D' lan luot lay cac diem M , N , P cho B ' M = CN = D'P = ka : < k < 1) a) Tinh dien tich tam giac MNP theo k va a b) Xac dinh v i tri M tren BB' de dien tich tam giac MNP co gia tri be nhat Giai: Chon he true toa Axyz nhu hinh ve A(0; 0; 0), B(a; 0; 0) C(a; a; 0), D(0; a; 0) A'(0; 0; a), B'(a; 0; a) C(a; a; a), D'(0; a; a) a) ECM = k B T l => M(a ; 0; a - k a ) CN = kCD => N(a - ka; a ; 0) D ' P = k D ' A ' = > P ( ; a - k a ; a) => M N = (-ka; a; -a + ka), MP = (-a; a - ka; ka) nen: [ MN MP ] = ( k a - ka + a ; k a - ka + a ; k a - ka + a ) 2 2 2 SMNP=i |[MN, MP] | =^(k -k+l)vdike (0; 1) z z 172 2 2 2 b) Taco: k - k + = ( k - - ) + - > | 4 2 Dau = k = - e (0; 1) nen SMNP be nhat M la trung diem BB' V i d u 16: Cho hinh lap phuong ABCD.A,B,C,Di canh a, tren BC, lay diem M cho DTM, DA^, AB[ dong phang Tinh dien tich S cua A M A B j Giai Chon he Oxyz cho B = O, B,(a; 0; 0), C,(a; a; 0), C(0; a; 0), A(0; 0; a), A i ( a ; ; a ) , D i ( a ; a; a), D(0; a; a) Vi M G D / i / i / ' il / i BC| nen goi M(x; x; 0) Ta co L \ M = (x - a; x - a; -a) DA = (a; -a; 0) X A B = (a;0;-a) V i D j M , DA , ABj dong phang nen X 3a [ D ^ D A ^ A B j = => x = —2 3a nen MA MB = A, c>X _ _ /» _\—— / \ B X ^ (3a 3a Do M 2— , D, Ci X 3a 22 : a Vl9 Vay: S [MA, M B ] V i du 17: Lang tru t i i giac d i u ABCD.A1B1C1D1 co chieu cao bang nua canh day Diem M thay doi tren canh A B Tim gia tri Ion nhat ciia goc ATMC Giai D A Chon he true nhu hinh ve (Aixyz) Dat A M = x, < x < Ta co: M(x; 0; ) ; Ai(0; 0; 0); Ci(2; 2; 0) \ / \ \ \ \ nen M A i = (-x; 0; - ) ; MCj = ( - x ; ; - l ) A ' \ Dat a = A^MCTi thi: cosa = c o s ( M A i , M C ) X (x-l) x -2x + l >/x +l.>/(2-x) +5 2 Vx +l.v/(2-x) +5 >0 173