L *J 15 52 2 1J Do do dudng thang can tim co phuong trinh:
7 1 Mat phang (Q) qua d, vuong goc vdi (P) co VTPT
Mat phang (Q) qua d, vuong goc vdi (P) co VTPT
n = [ u , n 7 ] = (2; l ; - 3 )
Duong thang d' ciia VTCP u * = [ n , n~p* ] = (4; - 5 ; 1) Tu do suy ra phuong trinh ciia hinh chieu d'.
Vi du 9: Cho dudng thang d va mp(P) co phuong trinh: 2 x = —h t 3 d: < v = + t z = t (P): x - 3 y + z - l = 0. 239
a) Viet phucmg trinh ducmg thang d' la hinh chieu vuong eoc cua d tren mp(P).
b) Viet phucmg trinh duong thing d, la hinh chieu song song ciia d tren mp(P) theo phuong Oz.
Giai
a) Duong thang d di qua diem A[-;-— ;O] va co vecto chi phuong
u = (1; 1; 1). Goi (Q) la mat phang di qua d va vuong goc voi mp(P) thi giao tuyen d = (P) n (Q) la hinh chieu \oiong goc ciia d tren (P).
Mat phang (P) co vecto phap tuyen mj= (1; -3; 1). Mat phing (Q) co vecto phap tuyln h^ = [u , h~p~ ] = (4; 0; -4) hay (1; 0; -1). Vi (Q) chiia ducmg thang d nen cung di qua diim A, do do (Q) co phuong trinh
2 x z = 0 hay 3x - 3z - 2 = 0. 3 Ta co (P): x-3y + z- l = 0. Dat z = t thi x = - +1, y = • 3 3 + - t. 9 3
Vay phucmg trinh ciia duong thang d' la:
x = — + t 3 y = — + —t.
9 3
b) Goi (R) la mat phang chiia d va song song hoac chiia Oz thi d| la giao
tuyen ciia mp(R) va mp(P). z
Mat phang (R) di qua A\^;~;0 j va co vecto phap tuyln la nR = [ u , k ] = ( 1 ; - 1 ; 0).
Mat phang (R) co phuong trinh la 3x - 3y - 13 = 0. Ta co (P): x-3y + z- l = 0.
Daty = tthix= —+ t,z = + 2t.
3 3
o
Vay di co phuong trinh tham s i la:
x = — + t 3 y = t
z = -^ + 2t 3
Vi du 10: Viet phucmg trinh hinh chieu ciia (A2) x - 7 = y- 3
1 2 - 1 theo
phucmg (A,): x - 3 -7
z - 1 len mat phang (a): x + y + z + 3 = 0. Giai
Hinh chieu A la giao tuyen ciia (a) voi (P), trong do (P) la mat phang chua (A2), song song voi (A|).
Vi (P) chua (A2) nen di qua A(7; 3; 9) va co VTPT n = [u7,u^] = (8; 4; 16) hay (2; 1;4).
Do do (P): 2(x - 7) + l(y - 3) + 4(z - 9) = 0 hay 2x + y + 4z - 53 = 0. [x + y + z + 3 = 0 Cac diem thuoc giao tuyen (A2) co toa do thoa man:
2x + y + 4z - 53 = 0 Dat z = t thi x = 56 - 3t, y = -59 + 2t
Vay phuong trinh tham so cua hinh chieu:
x = 56 - 3t y = -59 + 2t. z = t
Vi du 11: Cho dudng thang A va mp(P) co phuong trinh: (A) y - 2 _ z - 3 (P): 2x + z - 5 = 0.
1 2 2
Viet phucmg trinh dudng thang di qua giao diem A ciia A va (P), nam trong (P) va vuong goc vdi A.
Giai
A dang tham so: x = 1 + t, y = 2 + 2t, z - 3 + 2t.
The x, y, z vao (P) thi duoc t = 0 nen A ( l ; 2; 3).
Goi d la ducmg thang di qua A, nam trong (P) va vuong goc vdi A. Khi do, vecto chi phuong u ' cua d phai vuong goc vdi vecto chi phucmg
u = (1; 2; 2) cua A, dong thdi vuong goc vdi vecto phap tuyen n = (2; 0; 1)
ciia (P), nen ta chon: u ' = [u , n ] = (2; 3; —4).
x — 1 v — 2 z — 3
Vay dudng thang d co phuong trinh chinh tac: = =
Cach khac: Goi (Q) la mat phang di qua A va vuong goc vdi A thi (Q) co vecto phap tuyen la vecto chi phuong ciia A nen co phuong trinh:
x _ i + 2(y - 2) + 2(z - 3) = 0 hay x + 2y + 2z - 11 = 0.
Giao tuyen d ciia (P) va (Q) la duong thang di qua A, nam trong (P) va d J_ A (vi d nam trong (Q) ma A 1 (Q)). Suy ra phuong trinh tham so ciia
dia: 1 2 . x = — + — t 3 3 17 4 , z = 1 3 3
V i du 12: Trong khong gian voi he toa do Oxyz. cho duong thang
A: = - ^ -2- = — va mat phang (P): x + 2y - 3z + 4 = 0. V i i t phuong trinh duong thang d nam trong (P) sao cho d cat va vuong goc voi duong
thang A:
Giai
Theo gia thiet dudng thang d di qua giao diem ciia A vdi (P): Toa do giao diem I ciia A vdi (P) thoa man he:
x + 2 _ y -2 _ z
T ~ " 1 " => K-3; 1; 1) [x + 2y - 3z + 4 = 0
Vecto phap tuyen ciia (P): n = (1; 2; - 3 ) , vecto chi phuong ciia A: u — (1; 1; - 1 )
Duong thang d can tim qua I va co vecto chi phuong x = -3 + t v = [ n , u ] = (1; - 2 ; -1) nen co phuong trinh <y = 1 - 2t z = 1 - t
V i du 13: Lap phuong trinh ciia dudng thang vuong goc vdi mat phang x = t
y = -4 + t d':
x = 1 - 2t y = -3 + t z = 4 - 5 f (Oxz) va cat hai dudng thang: d: •
z = 3- t Giai
Duong thang d qua A(0; -4; 3) co VTCP u = (1; 1; -1) Duong thing d' qua B ( l ; - 3 ; 4) co VTCP u = (-2; 1; -5)
Duong thang can tim la giao tuyen ciia mat phang (a) qua d, vuong goc vdi (Oxz) va (P) qua d', vuong goc (Oxz)
Ta co (a): x + z - 3 = 0, ((3): 5x - 2z + 3 = 0
Suy ra phuong trinh tham so ciia dudng thang:
3 x = — 7 y = t 18 z = — 7 242
Vi du 14: Viet phuong trinh duong thang di qua A ( l ; - 1 ; 1) va cat ca hai duong thang sau day: d:
x = l + 2t x = t y = t d': \ y = - l - 2 t '
z = 3 - t |z = 2 + f Giai
Ta co A khong thuoc d va d'.
Duong thang d di qua diem M ( l ; 0; 3) va co vecto chi phuong u = (2; 1; -1). Duong thang d' di qua diem M'(0; - 1 ; 2) va co vecto chi phuong
u ' = ( l; - 2 ; 1).
Duong thang A can tim la giao tuyen ciia hai mat phang: mp(A; d) va mp(A; d'). Mp(A; d) co vecto phap tuyen n = [ A M , u ] = (-3; 4; -2), mp(A; d') co vecto phap tuyen n ' = [ A M ' , u '] = (2; 2; 2) hay (1; 1; 1). Duong thang A co vecto chi phuong la [ n , n ' ] = (6; 1; - 7 ) di qua A nen
x = 1 + 6t y = - l + t . z = l - 7 t co phuong trinh tham so la:
T a c 6 u . n ' = 2 + 1 - 1 = 2 * 0 nen d cat mp(A; d'), do do d cat A.
Tuong tu, vi u '. n = -3 - 8 - 2 = -13 * 0 nen d' cat mp(A; d), do do d' cat A. Vay A la duong thang di qua A, cat ca d va d'.
Cach khac:
- Ta co the tim giao diem B cua d' va (A; d), duong thang A la duong thang qua A va B.
- Lay diem M ( l + 2t; t; 3 - t) nam tren d va diem M'(t'; - 1 - 2t'; 2 + t') nam tren d'. Ta tim gia tri ciia t va t1 sao cho diem A, M , M ' thang hang, tiic la A M va A M ' cung phuong: [ AM . A M ' ] = 0.
Vi du 15: Viet phuong trinh ciia duong thang nam trong mat phang y + 2z = 0 va cat hai during thang:
x = l - t fx = 2 - t '
y = t , d2: < y = 4 + 2t z = 4t z = l
Giai
Ta tim cac giao diem ciia hai ducmg thang da cho voi mat phang y + 2z = 0 Tham so t img voi giao diem M j ciia ducmg thang di voi mat phang tren la nghiem ciia phucmg trinh: t + 2 4t = 0 => 9t - 0 => t = 0.
V a y M i ( l ; 0 ; 0).
Tuong tu, giao diem ciia ducmg thang d2 voi mat phang tren la M2(5; -2; 1) ung voi f = - 3.
Duong thang phai tim qua M i va M2 co VTCP
fx = 1 + 4t u = M1M2 = (4; - 2 ; 1) nen co PT tham so: y = -2t
z = t
V i du 16: Viet phuong trinh duong thing (A) di qua A ( - l ; 2; -3), vuong goc . i i . , x + l y - 2 z + 3 voi (a): 6x - 2y - 3z + 8 = 0 va cat duong thang d: = -
Giai
(A) nam trong mat phang (P) di qua A, co vecto phap tuyen n = (6; - 2 ; -3),
(P): 6(x + 1) - 2(y - 2) - 3(z + 3) = 0 <=> 6x - 2y - 3z + 1 = 0.
Giao diem ciia (d) voi (P) la B ( l ; - 1 ; 3).
Do do duong thang (A) la duong thang qua A va B co phuong trinh chinh tac: x + = ——- = z + 3
H & 2 - 3 6 x = — + t 3 6 'A A B
V i du 17: Cho duong thang d: 11 t va (P): x - 3y + z - 11 = 0. z = t
Viet phuong trinh duong thang di qua goc toa do O, cat d va song song voi mp(P).
Giai
Goi (P') la mat phang di qua goc toa do O va song song voi mp(P) thi (P1) co phuong trinh: x - 3y + z = 0. Giao diem I ciia ducmg thang d va mp(P') co toa do I ^ ; 8 ; i5- } , voi t = ^
3 0 1 = (37; 24; 35) la x = y _ z 37 ~ 24 35
Vi du 18: Viet phuong trinh duong thang song song voi duong thang di va cat ca hai duong thang d2 va d3, biet phuong trinh ciia di, d2 va d3 la:
f x = -4 + 5t' x - l y + 2 z - 2 di :. x = l y = -2 + 4t z = 1 - t ; d3 : :-7 f 9t Giai
(0; 4; - 1 ) , cac phuong trinh Duong thang di co vecto chi phucmg u i
ciia d2 va d3 duoi dang tham so:
x = l + t fx = -4 + 5t' y = -2 + 4t d3 : • y = -7 + 9t' z = 2 + 3t [z = t '
Tren ducmg thang d2 lay diem M2( l + t; - 2 + 4t; 2 + 3t) va tren ducmg thang d3 lay diem M3( - 4 + 5 f ; -7 + 9t'; t').
Ta co M2M3 = (-5 + 5t' - t; -5 + 9t' - 4t; - 2 + t' - 3t) Hai vecto M2M3 va u i cung phucmg khi va chi khi:
-5 + 5 t ' - t = 0 -5 + 9 t ' - 4t -2 + t ' - 3 t <=> 4 - 1 Do do M2 ( 1 ; - 2 ; 2) va M2M3 t = 0 f = 1 ( 0 ; 4 : - l )
Vay duong thang A di qua M2 va M3 co phuong trinh:
x = 1
y = -2 + 4 t .
z = 2 - t
V i M2 £ dt nen A diing la during thang can tim.
Cach khac: Viet phuong trinh mat phang (a) di qua d2 va song song voi
di, phuong Uinh mat phang (P) di qua d3 va song song voi di. Hai mat
phang do cat nhau theo giao tuyen A la duong thang can tim, neu A khong triing voi d|.
Vi du 19: Lap phuong trinh ciia duong thang A di qua diem A(3; - 1 ; -4) cat true Oy va song song voi mat phang y + 2x = 0.
Giai
Ta co diem A o ngoai mat phang y + 2x = 0.
Phuong trinh mat phang (a) di qua diem A(3; - 1 ; -4) va song song voi mat phang y + 2x = 0 co dang y + 2x + D = 0, D * 0. V i diem A(3; - 1 ; -4) thuoc mat phang do nen ta tinh duoc D = -5.
Vay mat phang (a) co phucmg trinh la: y + 2x - 5 = 0. True Oy cat mat phang (a) tai diem M(0; 5; 0).
Vay phucmg trinh ducmg thang A M la ducmg thang can tim: x - 3_y+ l _ z + 4
-3 ~ 6 ~ 4
V i du 20: Viet phucmg trinh chinh tac cua ducmg vuong goc chung cua 2 ducmg thang: : t = 3 6- va d' 2 + C 1 - t ' 2 - t Giai d qua A(0; 3; 6) taco VTCP u = ( 1 ; 0 ; 1) d'quaB(2; l ; 2 ) c o VTCP u ' = ( l ; - l ; - l ) Taco u . u ' = 0 , [ u , u j . A B = 2* 0
nen 2 duong thang d, d' cheo nhau va vuong goc nhau.
Hai duong thang vuong goc nhau nen duong vuong goc chung la giao tuyen cua mat phang (P) qua d, vuong goc d' va (Q) qua d' vuong goc d. (P) di qua diem A va co vecto phap tuyen u ' co phuong Uinh:
x - (y - 3) - (z - 6) = 0 hay x - y - z + 9 = 0.
Tuong tu, (Q) di qua diem B va co vecto phap tuyen u nen co phucmg trinh: x- 2 + z - 2 = 0 h a y x + z - 4 = 0.
Suy ra phuong trinh cua duong vuong goc chung cua d va d' la: fx = t x _ y - 5 z - 4 T ~ 2 - 1 y = 5 z = 4- 2t t
V i du 21: Lap phuong trinh duong vuong goc chung cua hai ducmg thang:
(di): x - 7 y - 3 va (d2): z - 1
1 2 - 1 x
*" -7
Giai (di), (d2) co vecto chi phuong lan lugl la:
u, = ( 1 ; 2 ; - 1 ) , XT2 = ( - 7 ; 2; 3)
[ u j , u2 ] = (8; 4; 16) nen vecto chi phuong cua ducmg vuong goc chung A la u7= (2; 1; 4) Mat phang (P) chua (d2) va song song voi u4 la:
5x +34 - l l z - 38 = 0 ; (P) n (di) = M(7; 3; 9)
Duong thang (A) can tim di qua M , co vecto chi phuong : x - 7 _ y - 3 _z - 9
2 ~ 1 ~ 4
V i du 22: Viet phuong trinh tham so ciia duong thang vuong goc chung ciia AC va BD biet A(4; 1; 4), B(3; 3; 1), C ( l ; 5; 5), D ( l ; 1; 1).
Giai
PT ducmg AC la (d|): x = 4 - 3t, y = 1 + 4t, z = 4 + t co VTCP
uT = (-3; 4; 1).
PT ducmg BD la (d2): x = 3 - 2k, y = 3 - 2k, z = 1 co VTCP
u~2 = (-2; - 2 ; 0)
Goi duong vuong goc chung la (A) qua E thuoc di, F thuoc d2:
E(4 - 3t; 1 + 4t; 4 + t); F(3 - 2k; 3 - 2k; 1) FE = Q - 3t + 2k; - 2 + 4t + 2k; 3 + t). Ta co: F E .U l= 0 [26t + 2 k - 8 = 0 (FE.u2 = 0 t + 4k - 1 = 0 <=> i t = A 17 17 Suy ra E 53.37.73 1 7 ' 1 7 ' 1 7 , F 4 5 4 5 . 1 7 17'17'17
Duong vuong goc chung (A) co vecto phuong FE = f—;- — ;—! hay(l;-l;7)
U 7 17 17 J
45 45 nen co PT la: x = — + t, y = t , z = 1 + 7.t
17 17
qua mat phang (P): x + 2y + z - l = 0.
V i du 23: Lap phucmg trinh duong thang d' doi xung voi duong thang d: x - 2 _ y _ z - 1
- 1 ~3~~^7~
Giai
Goi A la giao diem cua d va (P) thi A(2 - t; 3t; 1 - 7t). T h i toa do vao (P) thi t = 1 nen AQ; 3; -6).
Duong thang d di qua B(2; 0; 1). Ta tim hinh chieu H cua B len (P). Phuong trinh duong thang qua B, vuong goc voi (P) co
x = 2 + r VTCP u = nP = ( 1 ; 2 ; 1): y = 2t'
z = 1 + t '
The x, y, z vao (P) thi duoc t' = — nen H -r Do do diem doi xung B qua (P) la B
2 2 3' 3 ,f 4 4 1 3' 3' 3 Duong thing d' co VTCP A B ' = ^ - B-1 9 3 3 3 ^ , , x - l y - 3 z + 6 phucmg tnnh - 3 - = — = — hay ( 1 ; -13; 19) nen co 247
V» du 24: Cho 2 ducmg thing x - 3
(A,): y _ 1 2 - 1 » / A \ x - 7 y - 3 z - 9 r A , ,
va (A2): = = ——- 1 . Lap phucmg Uinh
-7 2 3 l 2
ducmg thang (A3) d6i xung voi (A2) qua (A|).
Giai
Lay diem M e (A2)
=> M(t + 7; 2t + 3; - t + 9)
Mat phang (P) qua M vuong goc voi (Ai):
-7(x - t - 7) + 2(y - 2t - 3) + 3(z + t - 9) = 0
Ta co (A1): x = - 7 k + 3, y = 2k + 1, z = 3k + 1 nen giao diem ciia (A,) va
-3.t (P) la I 21.t 6t 9t i ,
" s T+ g' ~ 3 i " - 1- - 3 i+ 1lu n g V O T k
31
Goi M ' doi xung vdi M qua (A,) thi I la trung diem doan M M ' nen co
M, r i l t 74t 13t \
131 ^ 31 ^ 31 J'
Vay dudng thang (A3) can tim chiia cac diem M ' nen co phuong uinh la:
x = - l + l l t
- y = - 1 - 74t.
z = -7 + 13t
V l d u 2 5 : V i
e t phuong Uinh dudng thing di qua M ( l ; - 5 ; 3) va tao vdi hai dudng thing Ox, Oy cac goc bang 60°
Giai
Goi u = (a; b; c), a2 + b2 + c2 * 0 la vecto chi phuong ciia dudng thing
can tim. Cac dudng thang Ox, Oy co cac vecto chi phuong la i = (1; 0; 0).
j = (0; 1; 0). Theo gia thiet cua bai toan thi:
lal |b| 1
/ ' ' ; = ; ; 1 ' cos60° = -
Va2 + b2 + c2 Va2 + b2 + c2 2
» a2 = b2 = - ( a2 + b2 + c2) o 4a2 = 4b2 = a2 + b2 + c2
<=> 2a2 = 2b2 = c2 Chon c=>^thia = ±l,b = ±l.
Vay co 4 trudng hop xay ra:
- Vdi u = (1; 1; 72 ) thi dudng thing co phuong trinh: x - l _ y + 5 _ z - 3
i "TJT
x - l _ y + 5 _ z - 3 1 - _ 3 r ~ ~ 7 r
- V o i u = ( - 1 ; 1; 7 2 ) thi ducmg thang co phuong trinh: x - l _ y + 5 _ z - 3
- 1 ~ 1 ~ ^ 2 ~
- Voi u = ( - 1 ; - 1 ; 7 2 ) thi duong thang co phucmg trinh: x - l _ y + 5 _ z - 3
- 1 ~ - 1 ~ ^ 2 ~
Vi du 26: Trong khong gian voi he toa do Oxyz, cho mat phang (P): x - 2y + 2z - 5 = 0 va hai d i i m A(-3; 0; 1), B ( l ; - 1 ; 3) a) Viet phucmg trinh tia A B , doan AB.
b) Trong cac duong thang di qua A va song song voi (P), hay viet phuong trinh duong thang ma khoang each tir B den duong thang do la nho nhat.
Giai
a) Vecto chi phucmg AB = (4; - 1 ; 2). Phucmg trinh tham so cua tia A B : x = -3 + 4t y = - t , t > 0 , doanAB: ^y = - t l + 2t x = -3 + 4t 0 < t < 1. l + 2t
b) Goi A la duong thang can tim; A nam trong mat phang (Q) qua A va song song voi (P).
Phuong trinh (Q): x - 2y + 2z + 1 = 0. K, H la hinh chieu ciia B tren A, (Q).
Ta co B K > B H nen A H la duong thang can tim l x - l _ y + l _ z- 3 Toa do H(x; y; z) thoa man: I 1 -2 2 •
|x-2y + 2z + l = 0
H I -1;1 1;7
9 9 9
AH 26 11
~9~' 9 Vay phuong trinh A:
x + 3
26 n y_
D A N G 2: LAP P H U O N G TRINH MAT P H A N G , MAT C A U Phuong trinh tong quat cua mat phang:
Mat phang qua M0(x0,yo, z0) va vecto phap tuyen n = (A, B, C) Ax + By + Cz + D = 0, A2
+ B2 + C2
£ 0 hay A(x - x0) + B(y - y0) + C(z - z0) = 0
Chu y: - Mot mat phang duoc xac dinh boi mot diem va mot duong thang khong qua diem do, hoac 2 ducmg thang song song hoac 2 ducmg thang cat nhau.
- Mat phang chiia 2 during thang cat nhau:
Neu (?) = mp(d,d') thi chon VTPT n = [ u d, u d ] - Mat phang chua 2 duong thang song song:
Neu (P) = mp(d, d') va d qua A, d' qua B thi chon VTPT n = [ u d, AB ] Phuong trinh mat cau:
Mat cau (S) tam I(a, b, c) ban kinh R:
(x - a)2 + (y - b)2 + (z - c)2 = R2 hay:
x2 + y2 + z2 + 2Ax + 2By + 2Cz + D = 0, A2 + B2 + C2 - D > 0 co tam I ( - A , - B , - C ) va ban kinh R = V A2 + B2 + C2 - D V i t r i tuong doi giira mat cau va duong thang:
Cho mat cau S(I; R) va duong thang A. Goi H la hinh chieu cua tam I tren A va d = I H la khoang each tu I toi A.
a) Neu d < R: duong thang A cat mat cau tai hai diem A , B Do dai day AB = 2%/R2 - d2
b) Neu d = R, dudng thang A va mat cau S(I; R) co diem chung duy nhat la H . Khi do, dudng thang A tiep xuc vdi mat cau tai diem H hoac A la tiep tuyen cua mat cau tai tiep diem H.
c) Neu d > R: dudng thang khong co diem chung vdi mat cau. Vi du 1: Cho diem A(2; 3; 1) va hai dudng thang:
fx = -2 - t
x + 5 _ y - 2 _ z y = 2 + t va d, . -
2 3 - 1 1 z = 2t z = 2t
a) Viet phuong trinh mp(P) di qua A va d i . b) Viet phuong trinh mp(Q) di qua A va d2.
Giai
a) Duong thang di di qua diem Mi(-2; 2; 0) va co vecto chi phuong u 7 = ( - l ; 1;2).
Mat phang (P) di qua A va di co vecto phap tuyen rip" = [AM7,u7] = ( - 1 ; 9; -5).
Vay mp(P) co phuong trinh:
-(x + 2) + 9(y - 2) - 5z = 0 hay x - 9y + 5z + 20 = 0.