A. L t T H U Y ^ T C A N N H6
1. Phirong trinh tong quat cua^ircmg thang
Ta da biet giao tuyen cua hai mat phang phan biet cat nhau la mot dudng thang. Vay trong khdng gian Oxyz ta xem ducmg thang (d) la giao cua hai
mat phang phan biet (a) va (a') wdi:
( a ) : Ax + By + Cz + D = 0 (a'): A'x + B'y + C'z + D = 0 'Ax + By + Cz + D = 0
A'x + B'y + C'z + D'=0 (1)
A'+B'+C^^ 0, A ' +B'' +C'' ^ Ova A : B : C ^ A' : B': C
He phuong trinh (1) la phuong trinh tdng quat ciia ducmg thang.
2. Phuong trinh tham so cua ducmg thdng
• Dinh ly: Trong khong gian Oxyz cho ducmg thang A di qua diem M() (xo; Yo; Z()) va nhan vecto a (a; b; c) 0 lam vecta chi phucmg. Di^u
kien can va du de diim M (x; y; z) nam tren A la c6 mot s61 sao cho:
Do do (d):
x = x^+at
y = yo+bt trong do a^ + b^ + c^ 0 (2)
Z = ZQ+Ct
• Djnh nghla: He phuong trinh (2) la phuong trinh tham so ciia ducmg thang A, trong do t la tham s6'.
Khi do A di qua diem (XQ; yo; z^) va vecto chi phuong la a (a, b, c)
3. Phucmg trinh chinh tic cua ducmg thang
^ - ^ 0 y-yo 2 - ^ 0 (3)
a b c
Dinh nghla: Phuomg trinh (3) v6i a, b, c 0 duoc goi la phuomg trinh
chinh tic cua ducmg thang
4. Vi trf tuong doi gifia ducmg thang va mat phdng
Cho duomg thing A:
x = XQ+at
y = yo+bt
Z = ZQ+Ct
va mat phang ( « ) : Ax + By + Cz + D = 0
De tim so giao di^m cua A va (a) ta giai he phuong trinh g6m cac phuong trinh cua A va (a) va cho ta phuomg trinh
(Aa + Bb + Cc) t + Axo + Byo + CZo + D = 0 (*) M6i nghiSm ciia phuong trinh(*) ^n t umg vdi 1 giao di^m cua A va (a):
1) Ne'u Aa + Bb + Cc 7i 0 thi phuong trinh (*) c6 nghiem duy nha't =>
A cat (a).
2) Ne'u Aa + Bb + Cc ^ 0 va AXo + Byo + Cz^ + D ^ 0 thi (*) v6 nghiem,
Icliido A / / ( a )
3) Neu Aa + Bb + Cc = 0 va AXQ + Byo + CZ^ + D = 0 thi (*) v6 so nghiem Idii do A c ( a ) .
5. V| tri tirong doi cua hai duomg thang
I.. ^
Trong Ichong gian Oxyz cho ducmg thang A di qua M va c6 vecto chi phuong a , duomg thang A' di qua diem M' va c6 vecto chi phuomg a', ta c6:
1) A va A' cheonhau o 2) A va A' cat nhau o • a,a a,a MM' =0 a,a ^ 0 3) A// A' <=> 4) A = A' ^ a,a a,MM a,a . = 0 ^0 a,MM' = 0 6. Khoang each
1) Khoang tic mot diem din mot ducmg thang
Cach 1: Mu6'n tim khoang each tut mdt diem M de'n duomg thang (A) ta lam nhu sau:
• Viet pt mat phang (a) qua M va vu6ng goc vdi duomg thing A • Tim toa do giao di^m H ciia (A) va (a)
• Tinh d6 dai MH, do chinh la khoang each tir M de'n A ky hieu la d(M, A)
Cdch 2: Sit dung c6ng thirc d(M, A) = M H = (Mo € ( A), a vecto chi phuomg ciia (A).
2) Khoang cdch giita dudng thing vd mat phang song song
De tinh khoang each gifia duomg thing A va mat phing (a) ta lam nhu sau:
• La'y m6t diim tuy y MQ e A
• Tinh khoang each d(Mo, a) tiif Mo den A
Khoang each nay chinh la khoang each giua A va (a) va duoc ky hieu la d(A, a).
3) Khoang cdch giQa hai ducmg thang cheo nhau
Cdch I. Di tinh khoang each gifia hai ducmg thing cheo nhau A va A' ta
lam nhu sau:
• Lay m6t di^m M tuy y trdn A' r6i tfnh khoang each tiir M d6n a. Khoang each nay chmh la khoang each gifla hai ducmg thang cheo nhau A va
A' ky hidu la d(A. A')
a,d MM'
a,a Cdch 2: S\x dung cong thiic: d( A ;A') = h =
(M e A, a la vecta chi phuofng ciia A. M ' e A', a'la vecto chi phuong ciia A ' .
B. CAC V I DU V i du 1. V i du 1.
Viet PT tong quat cua ducmg thang biet PT tham so: x = 2 - t <y = 2t z = - l + 2t Giai: Ta c6: x - 2 = - t ^ = t » 2 z + 1 = t x-2 _y 2 z + 1 _ y 2x + y-A-Q [y-z-\ 0 V i du 2.
Vie't phucmg trinh tdng quat ciia ducmg thang biet PT chinh lie:
x-\ z - 3 Giai: Ta c6: V i du 3. 2 3 x-\ 0 2 3 0 ^ z - 3 = 0 3 x - 2 > ; - 7 = 0 z - 3 = 0
Viet phuong trinh tham s6' (tiir do suy ra PT chmh iic) ciia ducmg thang ' ;c + ; ; - z + 3 = 0 biet PT t6ng quat: l 2 x - ; ; + 5 z - 4 = 0 Giai: Ta c6: + x + y - z + 3 = 0 2x - y + 5z - 4 = 0 (1) (2) 1 4 3x + 4 z - 1 = 0 <:> x = - - - z 3 3 1 4 Datz = t,thayvao(1): - - - t + y - t + 3 = 0<::> y = 10 7 + - r 3 3 Ta CO PT tham s6': 1 4 x = 1 3 3 10 7 y = + -t ^ 3 3 z = t
I Duong thang nay di qua didm ( - ; - — ; 0) va vector chi phuong 3 3
4 7 1
V i du 4. Viet phuong trinh ducmg thang la giao tuyeii ciia 2 mat phang: