I 93 1) Dat z= t, tac6 PT tham s6 cua (d): x = cosa + tsina
97. Mat phang (P) chiia dudng thang da cho va vuong gdc vdi mp
x + 2y + 3z + 4 = 0, cd cap vecta chi phucmg u = (3;4;1) va v = (1;2;3),
di qua ( 2 ; - 2 ; l ) . P T m p (P) la: 4 1 2 3 « (x - 2) + (y + 2) 1 .3 3 - 1 + (z -1) r 3 4 1 2 = 0 1 0 ( x - 2 ) - 8 ( y + 2) + 2 ( z - 1 ) = 0 <z> 5 x - 4 y + z - 1 9 = 0 fSx - 4y - 19 = 0 Phucmg trinh hinh chieu la: <^
^ [x + 2y + 3z + 4 = 0 ..
98. (d,) di qua di^m M,(-3; -2; 6) vecta chi phuomg w, = (2; 3; 4). (dj) di qua diem M2(5; - 1 ; 20) va vectof chi phuofng = (1; -4; 1)
= 0 va M, ^ U2 . d, cat d, <=> MjMj . u, , U 2 Co M , M , =(8;1;14) M,M2 . . 3 4 4 2 + 14 2' 3 u,.u. = 8 + + 14 1 -4 "1' "2 = 8 -4 1 1 1 1 -4 = 8.19+2-14.11 =0(dpcm) 99. Viet phuong trinh mp (P) chiia (d2) va song song (d,).
Phucng trinh tham so cua (d,) va (d2):
X = t y = 4 - 1 (d^) z = - 1 + 2t ^ 2 3 y = t z = - 2 + t
Mp (P) C O cap vecto chi phuong M, = (1;-1;2) va = (-1;3;3) nen c6
vecto phap tuyen n = [w, , "21 = f - 1 2 2 1 1 - 1
V 3 3 3 - 1 >
- 1 3 / = (-9; -5; 2), di qua diem ( - ; 0; -2) nen c6 PT: -9(x - - ) -5y +2(z +2) = 0 <::> -9x -5y +2z +10 = 0. Khoang each giiia hai dudng thang cheo nhau la khoang each tir
Mi(0;4;-1) den (P): - 5 . 4 - 2 . 1 + 10 12
100. Theo gia thiet: MA = MB = MC ^ M thuoc cac mp trung true (P), (Q) ctia AB va AC M G (d) la giao tuyen (P), (Q).
Mat phang (P) qua trung diem ciia AB va nhan AB = (-2;1;-1) lam , , 3 1
vecto phap tuyen. Toa do trung di^m cua AB: (0; - ; - ) nen PT mp (P): -2x + y - - - ( z - - ) =0
2 2 <=>-2x+y-z-l =0.
Mat phang (Q) qua trung diem ciia AC: (^'~^'^) vanhan AC = (1;-4;1)
. lam vecto phap tuyen, nSn PT mp (Q):
3 3
X 4 ( y + l ) + z - - =0 » x - 4 y + z - 7 =0
2 ^ 2
Tap hop di^m M la ducmg thang (d) c6 PT: 2x - y + z + 1 = 0 x - 4 y + z - 7 = 0
01. 1) Ducmg thang ( A ) phai tim la giao tuyen ciia hai mp (P) va (Q). Mat phang (P) qua A va vuOng goc (d) nen c6 vecto phap tuyen:
n =(3;l;l)vac6PT: 3 x + y - l + Z- 1 =0
<::>3x + y + z - 2 =0.
Mat phang (Q) thu6c chiim mp: m(x+l) + n(x + y-z+2) =0 Vi no qua A nen: m + n(l-l+2) = 0
«> m = -2n. Chon m = -2; n = 1 ;
• C O PT mp (Q): -x + y -z =0.
r X - y + z = 0
PT ducmg thang ( A ) phai tim la: <^ . . l3x + y + z - 2 = 0
2) a) Dacmg thing AB c6 vecto chi phuang AB = ( 6 M ; 4 ) = 2(3;-2;2) = 2. M => d//AB, nghia la d va AB cung thuoc mot mp.
b) DS. I(2;0;4)
102. Giao diem A ciia d va (P) la nghiem cua he:
X - 1 y z + 2 2 1 - 3 2x + y + z - l = 0 =^A(2;
103. Chuyen doi PT ducmg thang sang tham s6': 5x-3y+2z-5 = 0 4x - 2y - 2z - 2 = 0 Hinh 101 Daty = t, CO PT tham s6': 9 x - 5 y - 7 = 0 <=^x= | + 9 9 y = t 9 9
Ta nhan tha'y: 4. ^ - 3 + 7.^ = 0 chiing to d// (P).
Lai CO 4 . - + 7 . - - 7 = 0 nen d e (P) 9 9
104. 1) D6i ra PT tham so: Dat y = t thi (d,):
X = - t y = t z = - 4 + 2t Daty = t t h i( d 2 ) : X = 1 - 3t y = t z = 2 - t Dudng thing (d,) Di qua diem M,(0;0;-4)
Vecto chi phuong M, = (-1;1;2)
Ducmg thang (d2)
Di qua diem (1;0;2)
Vecto chi phuong = (-3;1;-1)
Xet tich M . M j M,M, = (1;0;6) u,,U2 1 2 2 - 1 - 1 1 + 0 + 6 1 - 1 - 1 -3 -3 1 = - 3 + 1 2 9^0 => (d.) cheo (dj)
2) Trudc het, viet PT mp (P) chiia (d2) va song song (d,), mp (P) c6 cap vecto chi phuong M, va nen c6 vecto phap tuyen:
n = [u, ,u, ] =
= (-3; -7 ;2) va PT mp (P) la: -3 (x-1) - 7y + 2 (z -2) = 0 < » - 3 x - 7 y + 2 z - l = 0
Khoang each giira hai dudng thang la khoang each tu Mj den (P): 1 2 2 - 1 - 1 1 N V 1 - 1 - 1 -3 -3 1 / d = - - 8 - 1 3)DS: V 3^ + 7 ^ + 2 ^ V62 X - 9y + 5z + 20 = 0 x - 2 y - 5 z + 9 = 0
Duong thang phai tim la giao tuyen 'ciia hai mp: Q (M; d,) va R(M; d^.
Mat phang (Q) thuoc chiim mp: m (x +y) + n (x - y +z + 4) - 0. V i no di qua M (2; 3; 1) nen ta c6: 5m + 4n = 0 o 5m = -4n. Chon m = ^ , n = 5, ta c6 PT mp (Q): x - 9y + 5z + 20 = 0 . Mat phang (R) thuoc chum mp: p (x +3y - 1 ) + q (y + z - 2) = 0 .
105. DS:
V i no qua M nSn ta c6: lOp + 2q = 0 <=> 5p = - q . Chon p = 1 q = 5, ta CO PT mp (R): x - 2y - 5z +9 = 0 .
106. Co N (-3;2;5)
MN = M;0;6) MN = MN = V ( - 4 ) ' + 0 ' + 6 ' = V52 = 2.Vl3
107. Theo tinh chat hinlh binh hanh thi giao di^m hai ducmg cheo la trung diem m6i ducmg cheo, nghia la M la trung diem AC. Theo c6ng thiic toa do trung diem:
1 = \ { - . - 2))
M 2 = ^ ( y A + 3 ) ^ A ( 4 ; l ; - 2 )
CQng theo tinh chat hinh binh hanh: A B // CD ntn du5ng thang A B c6
vecta chi phucmg CD (2; 1; - 2 ) . Vay PT ducmg thang chiia canh A B la:
X = 4 + 2t y = 1 + t z = - 2 - 2 t
2. Mat phang hinh binh hanh c6 cap vecto chi phucmg:
AD M;3;-5) va CD = (2; 1 ;-2) nen c6 vecta phap tuye'n:
n = [AD, CD] = = (-1;-18;-10) 3 - 5 -5 - 4 - 4 3 v 1 - 2 9 - 2 2 5 2 1 y n =-(l;18;10).
PT mp hin.i binh hanh la x + 18 (y - 4 ) + 10(z +7) = 0 <::i>x + 18y + l O z - 2 = 0
Khoang each tiJf gdc toa do den mp hinh binh hanh dugrc tinh theo cong thiic:
- 2 7
Vi'+i8^+io^ sVn