Nối tiếp nội dung phần 1 tài liệu Cẩm nang luyện thi Đại học - Hình học, phần 2 giới thiệu các kiến thức chương 3 - Phương pháp tọa độ trong không gian bao gồm: Tích có hướng của hai vectơ và ứng dụng, lập phương trình mặt phẳng, phương trình đường thẳng,... Mời các bạn cùng tham khảo.
Trang 1Cam nang luy?n tht VH tltnh hqc - Nguyen Tat Thu
Giaihe taduac H ^ 2 9 _ 1_
15'15' 3 3) Gpi I(x;y;z) la tam duong tron ngoai tiep tam giac ABC Ta c6
, G I ^;15 30 - i - ; 0 nen HG = 2GI, tuc la ba diem G
H, I nam tren mpt duong thang
Bai 3.1.6 Cho tam giac deu ABC c6 A(5;3;-l),B{2;3;-4) va diem C nam trong
mat phSng (Oxy) c6 tung dp nho hon 3
1) Tim tpa dp diem D biet ABCD la tu di^n deu
2) Tim tpa dp diem S biet SA,SB,SC doi mpt vuong goc
Huong dan giai
«-ty it\Tin miv UVVH Kharig Vi(t
Tam giac ABC la tam giac deu nen ABCD la tu di?n deu khi va chi khi
P = BD = CD = AB = 3V2 Ta c6 h? phuong trinh
' ( x - 5 ) 2 + ( y - 3 ) 2 + ( z + l ) 2 = ( x - 2 ) 2 + ( y - 3 ) 2 + ( z + 4)2 ( x - 5 ) 2 + ( y - 3 ) 2 + ( z + l ) 2 = ( x - l ) 2 + ( y - 2 ) 2 + z 2 ' • ( x - 5 ) 2 + ( y - 3 ) 2 + ( z + l)2=18
AS.BS = 0
• BS.CS = 0 « < ! CS.AS = 0
Trang 2Liim nang luyfn ini VH ninn nyt - j T j j w y t ) x„m
Bai 3.1.7 Trong khong gian Oxyz, cho hinh hop chu nhat ABCD.A'B'C'D'
A = 0 , B e 0 x , D e C ) y , A ' 6 0 z va AB = 1, A D = 2, A A ' = 3
1) Tim tpa do cac dinh ciia hinh hpp
2) Tim diem E tren duong thSng DD' sao cho B'E 1 A'C
3) Tim diem M thupc A ' C , N thupc BD sao cho M N 1 BD,MN 1 A ' C
do tinh khoang each giua hai duang th^ng cheo nhau A ' C va BD
Huong dan giai
1) Taco A(0;0;0),B(1;0;0), D(0;2;0), A'(0;0;3)
Hinh chieu ciia C len (Oxy) la C, hinh chie'u ciia C len Oz la A nen
C(l;2;0)
Hinh chieu ciia B',C',D' len mp(Oxy)va true Oz Ian lug^ la cac diein
B,C,D va A ' nen B'(l;0;3), C'(l;2;3), D'(0;2;3)
2) Vi E thuoc duong thSng DD" nen E(0;2;Z) , suy ra B'E = ( - l ; 2 ; z - 3 )
Ma A ^ = (l;2;-3) nen B'E 1 A'C o B ^ A ^ = 0
o - l + 4 - 3 ( z - 3 ) = 0 o z = 4 Vay E(0;2;4)
3) Dat A ' M = x.A'C; BN = y.BD
Ta CO A M - AA' + A' M = AA' + X.A'C = (x; 2x;3 - 3x), suy ra M (x;2x; 3 - 3x)
A N = A B + B N = A B + y.BD = ( l - y;2y;0) N ( 1 - y;2y;0)
Theo gia thiet cua debai, ta c6: MN.A'C = 0
g^i 3.1.8 Trong khong gian voi h? tryc tpa dp Oxyz cho hinh chop S.ABCD
CO day ABCD la hinh thang vuong tai A, B voi AB = BC = a; A D = 2a; A s O,
B thupc tia Ox, D thupc tia Oy va S thupc tia Oz Duong thing SC va
RD tao voi nhau mot goc a thoa cosa = — ^
V30
J) Xac dinh tpa dp cac dinh ciia hinh chop 2) Chung minh rang ASCD vuong, tinh di?n tich tam giac SCD va tinh c6 sin ciia goc hop boi hai mat phMng (SAB) va (SCD)
3) Gpi E la trung diem canh AD Tim tpa dp tam va tinh ban kinh m|it cau ngoai tie'p hinh chop S.BCE
4) Tren cac canh SA,SB,BC,CD Ian lupt lay cac diem M,N,P,Q thoa
SM = M A , SN = 2NB, BP = 3PC, CQ = 4QD Chung minh rang M , N , P, Q khong dong phing va tinh the tich khoi chop MNPQ
^^k Huong dan giai
Trang 32) Taco CS = {-a;-a;2a),CD = (-a;a;0)=>CS.CD = 0=>ASCD vuong tai C
'pai Trong khong gian vdi h? tpa dp Oxyz cho hinh hpp chu nhlt
ABCD.A'B'C'D' c6 A triing voi goc tpa dp, B(a;0;0),D(0;a;0) ,A'(0;0;b) v6i (a > 0, b > 0) Gpi M la trung diem cua C C
1) Tinh the tich ciia khoi tu difn BDA'M 2) Cho a + b = 4 Tim max V^.guj^
Huong dan giai
1) Ta c6: C(a;a;0), B'(a;0;b), C'(a;a;b), D'(0;a;b) :^M(a;a;|) Suyra A ^ = (a;0;-b), A ^ = (0;a;-b), A ^ =
Bai 3.1.10 Trong khong gian Oxyz cho tu di?n deu ABCD c6 A ( 3 ; - l ; 2 ) va
G(l;l;l) la trpng tam tam giac ABC Duong thSng BC di qua M Xac djnh toa dp cac dinh con lai va h'nh the tich khoi tu difn do '
Huong dan giai
Taco: AG = (-2;2;-l), A M - ( l ; 5 ; y )
Suy ra riABC = AG A 2 A M = (-12; -24; -24)
Phuong trinh (ABC) :x + 2y + 2 z - 5 = 0
I U g ^ AG A HABC = ( 6 ; 3 ; - 6 )
Trang 4O t 2 + 4 t + i ^ = 0 o t = 2 ±
-4
73 Suy ra B 4-N/3 1 + 2N/3' , C f ^ 4 + ^/3.1-2^/3
Do D G 1 (ABC) nen phuong trinh GD: \
De lap phuong trinh mat p h i n g (a), ta c6 cac each sau:
Cach 1: Tim mpt diem M(xo;yo;zo) ma mat p h a n g (a) d i qua va mpt VTPT
n = (a;b;c) Khi do phuong trinh cua (a) c6 d a n g :
a ( x - X o ) + b ( y - y o ) + c ( z - Z o ) = 0. '^P-' '
Mpt so luu y khi tim VTPT cua mat phSng (a):
, Neu hai vec to a,b khong ciing phuong va c6 gia song song hoac n i m tren
(a) thi a A b = n la VTPT cua (a)
, Neu mat ph5ng (a) di qua ba diem phan bift k h o n g thang hang A,B,C thi
AB A AC = n la VTPT cua (a)
Neu (a)//(p) thi n^ = np , Neu A 1 (a) thi n^^ = u ^
Neu (a)//(P) thi i^//(a)
Neu A(a;0;0),B(0;b;0),C(0;0;c) voi abc^O t h i p h u o n g trinh (ABC):
Cac dang lap phuong trinh mat phSng
Dang 1: Lap phuong trinh mat phdng biet mat phdng
Trang 5Clin lui/iii Ihi nil Ilnili UOL \^uyenTafT}iH
Led gidi
1) Taco AB(-l;-2;l), AC = (-3;2;-2) AB A AC = (2;-5;-8)
Suy ra n„ = (2; -5;-8) Do do, phuang trinh ciia (a) la: ^ „
2 ( x - l ) - 5 ( y - l ) - 8 ( z - l ) = 0 hay 2 x - 5 y - 8 z +11 = 0
2) Goi Aj, Aj, A3 laa lugt la hinh chie'u ciia A len cac true Ox, Oy, Oz
Ta CO Aj(l;0;0), A2(0;l;0), A3(0;0;;)
Vi (a) di qua Aj,A,,A3 nen phuang trinh c6 dang:
x y z
- + — + — = 1 hay x + y + z - l = 0 I l l ^ ^
Dang 2 Lap phuang trinh mat phang (a)song song v&i mat
phang Cp; hoac (a) 1AB
Ta CO n^^ hoac AB la VTPT cua (a)
Chii y: Neu ((3): ax + by + cz + d = 0 thi np - (a;b;c)
Cty TNHH MTV DWH Khang Vi$t
Vi dv 3.2.2 Lap phuong trinh mat phSng (a) biet:
1) (a) diqua M(2;3;1) vasong song voim^t phing (P):2x + 3 y - z + l = 0
2) (a) la mat ph^ng trung true cua do^n AB vol A(1;4;2), B(-3;2;-1)
Chii y:
, Neu (P) chua (hoac song song) vol A B thi gia cua vec to A B se nam tren
I (hoac song song) voi (P)
, Neu (P) -L (Q) thi VTPT cua mat phSng nay se c6 gia nam tren hoac song song voi mat phSng kia
Vi dM 3.2.3 Viet phuong trinh mat phang (a) biet:
1) (a) chua A(l;l;2), B(1;-1;1) va song song voi CD trong do C(0; 3; 0), D(-2;0;-l)
2) (a) chua A, B va vuong goc voi mat phSng (p):x + y + z + l = 0 3) (a) di qua C va vuong goc voi hai mat phMng (P) va (y): 2x - y + z - 3 = 0 4) (a) di qua A, B va each deu hai diem M(3;-5;-1), N(-5;-1;7)
Dang 3: Lap phifdng trinh mat phang (a) biet (a) song song vdi hai vect(<
khong cung phi^dng a,b
Khi do n„ = a A b
Lai gidi
1) Taco AB = (0;-2;-l), CD = (-2;-3;-l)
Vi (a) di qua A, B va song song voi CD nen n^ = AB A CD = (-1;2; -4)
Phuang trinh (a): x - 2y + 4z - 7 = 0
2) Taco np=(l;l;l)
Vi (a) chua A,B va song song voi (p) nen n^ = ABAnp ==(-l;-l;2)
Phuong trinh mat phSng (a): x + y - 2z + 2 = 0 3) Taco n^^ = (2;-l;l) ,
Vi mat phSng (a) vuong goc voi hai mat phSng (P) va (y) nen n„ = np A n^ = (2;l;-3)
Phuang trinh mat phSng (a): 2x + y - 3z - 3 = 0
Vi (a) each deu hai diem M, N nen ta c6 cac truang hg-p sau
T H 1: M N / / ( a ) K h i d 6 n^ = A B A M N = (-12;8;-16)
Phuong trinh (a): 3x - 2y + 4z - 9 = 0
T H 2: M N c3t (a) tai I, suy ra I la trung diem doan M N Dodo I(-1;-3;3)
Khi do n„' = A1 A BI = (-6; 2; -4)
f huong trinh (a): 3x - y + 2z - 6 = 0
Trang 6Dang 4 Viet phuang trinh mat phdng di qua m6t diem
vd lien quan den hhodng cdch hoac goc
Phuong trinh di qua M(xQ;yo;zQ) c6 dang
a ( x - X Q ) + b ( y - y g ) + c ( z - Z o ) = 0 vai + > 0
Dua vdo dieu ki$n khoang each hoac goc ta tim du<?c a = mb,c = nb Khi dc
ta chpn b bang mpt gia tri bat ki (b ?t 0) ta tim dugc a, c
V i dv 3.2.4 Viet phuang trinh mat phang (a) biet
1) (a) song song voi ( Q ) : 2x - 3y - 6 z - 1 4 = 0 va khoang each tir O den
(P) bang 5
2) (a) diqua A(0;1;2), B(2;1;3) va each O mot khoang bang 2
3) (a) di qua hai diem C ( 1 ; 3 ; - 2 ) , D ( - 1 ; 0 ; 3 ) va each deu hai diem
Phuang trinh (a): 2x - 3y - 6z ± 35 = 0
2) Vi mat phang (a) diqua A(0;1;2) nen phuang trinh C6 dang
ax + b(y -1) + c(z - 2) 0 <=> ax + by + cz - b - 2c = 0
(a) diqua B(2;1;3) nen 2a + b+ 3 c - b - 2 c = 0 O C = - 2 a
Mat khac d (0,(a)) = 2 nen ta c6
3 2
, a = — b = > b = — a , ta chon a = 3 => b =-2,c = - 6
2 3
phuang trinh mat phang (a): 3x - 2y - 6z +14 = 0
3) Vi mat phSng (a) di qua C(l;3;-2) nen phuang trinh c6 dang
= c, ta chpn c = l = > b = l,a = l Phuang trinh mat phSng (a) la:
x + y + z - 2 = 0 I
K - 3 b - 3 c , 5 c - 3 b - 3 c - 3 b , 1 3 15
H k e : , ta CO —-— = <=> b = —e,a = —^c
H ' a chon c = -12 => a = 45,b = -52
^Piuong trinh mat phSng (a): 45x - 52y - 12z + 81 = 0
4) Vi mat phSng (a) di qua E ( 0 ; - l ; l ) nen phuong trinh c6 dang
Trang 7d i e m c6 tpa dp la hai n g h i ^ m ciia h^ K h i do (a) d i qua giao t u y e n ciia ( a j ]
va ( a 2 ) <=> A , B 6 ( a )
V i d y 3.2.5 Cho ba mat phang:
( a , ) : X + y + z - 3 = 0; ( t t j ) : 2x + 3y + 4z - 1 = 0
C h u n g m i n h rang hai mat ph§ng ( a , ) va ( t t j ) dt nhau Viet phuong
t r i n h (P) d i qua A ( 1 ; 0 ; 1 ) va giao tuyen ciia ( a j ) va ( a 2 )
L a i g i a i
Ta CO n7=(l;l;l), n j =(2;3;4) Ian l u p t la VTPT cua hai mat p h i i n g (Oi) va
(cxj)-V I n j ?t kn2 nen hai mat phang (QJ) va (a2) cat nhau
x + y + z - 3 : : = 0 2x + 3 y + 4 z - l = 0 ' Xet h ^ p h u a n g t r i n h
Cho z = - 1 , ta C O he
• Cho z = - 2 , ta C O h?
x + y = 4 2x + 3y = 5 '
x + y = 5 2x + 3y = 9
x - 7
y = -3^ M ( 7 ; - 3 ; - l ) e ( a i ) n ( a 2 ) ' ' ^ ^ ^ ^ N ( 6 ; - l ; - 2 ) e ( a i ) n ( a 2 ) -
V I ( a ) d i qua giao tuyen cua hai m§t ph4ng {a^) va (ttz) nen ( a ) d i qu^
M , N '^
Ta C O A M - (6; - 3 ; - 2 ) , A M = (5; - 1 ; -2) ^ A M A A N = (4; 2; 2 )
Suy ra n ^ = ( 2 ; l ; l ) P h u o n g t r i n h mat ph^ng ( a ) la: 2x + y + z - 3 = 0
Dang 6: Phuong trinh doan chdn
Neu mat phSng (a) d i qua A(a;0;0),B(0;b;0),C(0;0;c), a b c ^ O t h i p h u a n g
t r i n h (a) c6 dang - + f + - = 1 ' - ,
'^jid^ 3.2.6 Lap p h u o n g t r i n h mat phSng d i qua diem M ( 1 ; 9 ; 4 ) va cSt cac
(a) t r y c tpa d p tai cac d i e m A, B,C (khac goic tpa dp) sao cho 1) M la true t a m cua tam giac ABC ' : 2) Khoang each t u goc tpa d p O deh m|t phSng ( a ) la Ian nhat
3) O A = OB = O C 4) 8 0 A = 120B + 16 = 3 7 0 C va x ^ > 0,Zc < 0
a b c 1) Ta c6: A M ( 1 - a; 9; 4), BC(0; - b; c), BM(1; 9 - b; 4), CA(a; 0; - c);
Diem M la true tam tam giac ABC k h i va chi k h i
M e (a)
A M B C = 0 o BM.CA = 0
1 9 4 ,
- + - + - = 1
a b c 9b = 4c
Trang 8Nen suy ra T > ^ Dau dang thuc xay ra khi ' >• •
• ^ ^ o a = 9b = 4c = 98
1 9 4 ,
a b c ^
Phuang trinh mat phang (a) can tim la x + 9y + 4z - 98 = 0
Cach 2: Gpi H la hinh chieu cua O tren mat phang (a)
Vi mat phang (a) luon di qua diem co'djnh M nen
d(0,(a)) = O H < O M = >/98
Dau dang thuc xay ra khi H = M , khi do (a) la mat phang di qua M va c6
vec to phap tuyen la OM(l;9;4) nen phuang trinh (a) la
1 (x -1) + 9(y - 9) + 4.(z - 4) = 0 « X + 9y + 4z - 98 = 0
3) Vi OA = OB = OC nen a = b = c , do do xay ra bo'n truong h^p sau:
Tu (1) suy ra 1 + - +1 = 1 a = 14, nen phuang trinh (a) la:
Vi a > 2 nen khong c6 gia tri thoa man
V^y phuang trinh mat phSng (a): 8x + 20y - 37z - 40 = 0
2) Tim tpa dp diem thupc m|t p h i n g
Diem M(xQ;yo;zo)€(a):ax + by + cz + d = O o a x o + byo + cZo + d = 0
H G (P) Hinh chie'u H ciia M len mp(P) dupe xac djnh boi -j _
M H = t.n„
Vi dv 3.2.7 Trong khong gian Oxyz cho ba diem A(3;3;3),B(1;2;4),C(0;3;2) vam|tphSng (a):x + y + z - 3 = 0
1) Tim tpa dp hinh chie'u cua A len (a)
j ) J i m tpa dp diem M thupc (a) sao cho M A + MB nho nhat
^' ^oi mSi diem K(x;y;z) ta d?t f(K) = x + y + z - 3
CO f(A) = 6, f(B) = 4 => A,B nSm cung phia so voi m^t ph5ng ( a ) A' la diem dol xung voi A qua ( a ) , suy ra A ' ( - 2 ; - 2 ; - 2 ) ;
do, voi mpi diem M e ( a ) , ta c6: M A + MB = M A ' + MB ^ A ' B
Trang 9Cam nang nil - Nguyen Tat ITtu
Dau "=" xay ra khi M = A' B m ( a )
Vi A'B = (3;4;6), phuang trinh A'B:
V i d\ 3.2,8 Viet phuang trinh mat ph5ng (a) biet:
1) (a) di qua hai diem A(1;1;1),B(2;-1;3) va song song voi OC v6i
C ( - l ; 2 ; - l )
2) (a) di qua M ( l ; l ; l ) , vuong goc voi (P): 2x - y + z - 1 = 0 va song song vol
• 2 1 - 3 •
3) (a) vuong goc vai hai mat ph3ng (P):x + y + z - l = 0 , (Q):2x-y + 3z-4 = 0
va khoang each tu O den (a) bang \/26
4) (a) di qua BC va each deu hai diem A , 0
Phuong trinh (a): x + 4y + 2z - 7 = 0
3) Taco iv[ = (l;l;l), n^ = (2;-l;3) Ian lugtla VTPTcua (P) va (Q)
Vi (a) vuong goc vai hai mat phang (P) va (Q) nen (a) nhan vec to
n = n^An^ = (4;-l;-3) lamVTPT
Suy ra phuang trinh (a) c6 dang : 4 x - y - 3 z + d = 0
Meit khac: d(0,(a)) = %/26 nen ta c6: = V26 => d = ±26
N/26 Vay phuang trinh (a): 4x - y - 3z ± 26 = 0
4) Vi (a) each deu A,0 nen ta c6 cac truang h^xp
T H l : A O / / ( a )
sau
Ta c6: B C = (-3; 3; -4), OA = (1; 1; 1) =^ B C A OA = (7; -1; - 6 ) Phuang trinh (a): 7x - y - 6z + 3 = 0
Vi dv 3.2.9 Lap phuang trinh mat phing (P), biet:
1) (P) di qua giao tuyen ciia hai mat phSng (a):x-3z-2=0; (P):y-2z+l = 0
va khoang each tu M 0;0;— den (P) bang—•==
\j 6V3 2) (P) di qua hai diem A(1;2;1),B(-2;1;3) sao cho khoang each tu
C ( 2 ; - l ; l ) den (P) bang hai khoang each tu D(0;3;1) den (P)
Lai gidi
1) Gia su (P): ax + by + cz + d = 0
Taco A(2;-1;0),B(5;1;1) la diem chung cua (a) va (p)
Vi (P) di qua giao tuyen ciia hai mat phang (a) va (P) nen A , B € (P)
Trang 10d = - — a => b = a;c = - — a Suy ra p h u o n g trinh (P) la:
_ x - 1 _ y + 1 _ z + 2 _ ^ , x - 2 _ y + 2 _ z - 1
• 2 ~ 1 ~ - 1 ' ^' 1 ~ 2 " - 4 •
1) Viet p h u o n g t r i n h mat phang (P) d i qua A va d j
2) C h u n g m i n h rang d j va d j cat nhau Viet p h u o n g trinh mat p h l n g (Q)
Suy ra d j va d2 cat nhau tgi E ( 3 ; 0 ; - 3 )
Cty TNHH MTV DWH Khang Vift
y f 3.2.11 Trong khong gian Oxyz cho ba d u o n g thang
2) Gpi (P) la mat phang chua d j va d3 Lap p h u o n g trinh mat p h i n g (Q)
chua d , va tao v o i mat phang (P) mpt goc o thoa cos(p = —
Tu A B = Vl3=>(x + l ) 2 + 4 x ^ + ( x - 2 ) 2 = 1 3 o x = - l , x = |
* Voi x = l = > A B = ( 0 ; 2 ; - 3 ) , ta c6 u = ( 2 ; 3 ; - l ) la VTCP cua d j va
A ( - l ; l ; 0 ) e d 2 A € ( a ) Suy ra n = A B , u = (7;-6;-4) laVTPTcua (P)
Trang 11Cam nan)i hiycii Ihi 1)11 lliiih hoc \i;ii\fen TatThu
Vi mat phSng (Q) di qua d j nen phuang trinh (Q) c6 d^ng:
ax + by + cz + a - b = 0 (1)
vol a^ + b^ + > 0 va 2a + 3b - c = 0 o c = 2a + 3b
l4a-3b-2cl 9lb Mat khac coscp =
Phuang trinh (Q) la: 5x - 2y + 4z + 7 = 0
V i dy 3.2.12 Trong khong gian Oxyz cho duong thang A c6 phuang trinh
y i l ^ £ ± 3 vadiem M(l;2;0)
1) Vie't phuang trinh mat phang (a) di qua M , song song voi A va (a) tao
vai ba tia Ox,Oy,Oz mgt tu di^n c6 the tich bang 8
2) Viet phuong trinh m|t phing (P) di qua M, vuong goc voi (P):x+y+z-3=0
va tao voi A mpt goc 9 thoa cos 9 = J 31
Vi (a) di qua M va song song vai A nen ta c6:
Mat khac, (P) 1 (P) nen ri^.n,^ = 0 « a + b + c = 0<=>c = - a - b
•^en phuang trinh ciia (p) la: x - 2y + z + 3 = 0 = 0
407
Trang 12-a — y •
V i d v 3.2.13 Trong khong gian Oxyz cho hai mat phSng (P):x+2y+2z-3 = 0,
m|it phSng (Q): 2x - y + 2z - 9 = 0 va duong thang A : ^ = =
1 1 -2 1) Gpi (a) la m|it phang phan giac ciia goc hpp boi hai m | t phang (P) va (Q)
Tim giao diem ciia duong thang A va mat phang ( a )
2) Viet phuong trinh mat phang (p) d i qua giao tuyen ciia (P) va (Q), dorig
thai each E(8;-2;-9) mpt khoang Ion nha't
T u do ta CO duoc hai diem M la: Mj(4;0;0) va M 2 ( 3 ; - l ; 2 )
2) Ta CO A ( 3 ; - l ; l ) va B(-4;l;9) la hai diem thupc giao ciia (P) va (Q)
Do do (P) d i qua giao tuyen ciia hai mat phang (P) va (Q) khi va chi khi
A,B€(P)
(x = 3 - 7 t
Ta CO AB = (-7; 2; 8) nen phuong trinh A B :
Gpi K la hinh chieu ciia E len AB, suy ra
Gpi H la hinh chieu ciia E len mat phang (p), khi do: d(E,(P)) = E H < E K
Suy ra d(E,(P)) Ion nhat khi va chi khi H = K hay (p) la m^t phSng di q"^
K va vuong goc voi E K
Phuong trinh (p): 2x - y + 2z - 9 = O (ta thay (p) = (Q))
V i d\ 3.2,14 Trong khong gian Oxyz cho ba diem A ( l ; 2; I ) , B ( - l ; O; -2)'
C(2; 0; 0) Tim tpa dp tam duong tron ngoai tiep va true tam tam giac A B C J ^
Ana
CtyTNHHmv DWH KhanxViit Led gidi
Taco AB = ( - 2 ; - 2 ; - 3 ) , A C = ( l ; - 2 ; - l ) => AB A AC = (-4;-5;6) Phuong trinh (ABC): 4x + 5y - 6z - 8 = 0
Gpi I(x;y;z) la tam duong tron ngoai tiep tam giac ABC Ta c6:
1 5 4 ' 7 7 ' 154
Gpi H ( x ; y ; z ) la tr^c tam tam giac A B C Ta c6:
AH.BC = 0 BH.AC = 0 (*)
^3x + 2 z - 5 = 0
x - 2 y - z - l = 0 o 4x + 5 y - 6 z - 8 = 0
IChung minh A , d j , d 2 C u n g nam trong mpt mat phang T i m tpa dp cac
pinh B,C cua tam giac ABC biet duong cao tir B nam tren d j va duong phan giac trong goc C n i m tren d j
Trang 13Cam nana l,n,cn Ihi DIl lln.h ho. N^micnTatlhu
Lai gidi
Duong thang d, di qua M(l;2;4) vac c6 VTCP = (1;1;1) •
Duong thang di qua N(0;3;2) vac c6 VTCP U2 ={V,-V,2)
Gpi I la giao diem cua d p d 2 1(1; 2; 4)
U i , U 2 = (3;-l;-2) va di qua I nen
Mat phing (a) chua dj,d2 c6 n„ =
phuong trinh : 3 x - y - 2 z + 7 = 0
Ta tha'y A e (a) V|y A,dj,d2 cung thuQC mp (a)
Xac dinh diem C:
Gpi (P) la mp di qua A va vuong goc voi dj=:> (P):x + y + z + l = 0
Co C = (P) n (d2) nen tpa dp diem C la nghi^m ciia h^ phuong trinh
Suy ra phuong trinh BC:
Tpa dp diem B la nghi?m cua h? phuong trinh :
I _ yj-6 ^ z+_4
' ^Bl Vay B
Cty TNHH MTV DWH Khang Viet
Vi dv 3.2.16 Trong khong gian Oxyz cho duong thing
x-4m4-3 _ y - 2 m - 3 _ z - 8m - 7
voi m «£ ^ ' 4'2
2 m - l m + 1 4m + 3
Chung minh rSng khi m thay doi thi duong thSng d^ luon nkm trong
mpt mat phang co dinh Vie't phuong trinh mat phang do
I Lai gidi
Duong thang d^ diqua A(4m-3;2m + 3;8m + 7) va c6 VTCP
u = (2m - l;m + l;4m + 3) Gia su d^ c (a): ax + by + cz + d = 0 voi mpi m, khi do ta c6:
a(2m -1) + b(m +1) + c(4m + 3) = 0
" a(4m - 3) + b(2m + 3) + c(8m + 7) + d = 0
f (2a + b + 4c)m - a + b + 3c = 0
|(4a + 2b + 8c)m - 3a + 3b + 7c + d = 0 '2a + b + 4c = 0
-a + b + 3c = 0
Vm
Vm b-lOa
c = -3a ,
d = -6a
2a + b + 4c = 0 -3a + 3b + 7c + d = 0
Ta chpn a = 1 => b = 10,c = -3,d = -6 Vay d^ c(a):x + 10y-3z-6 = 0
Vi du 3.2.17 Trong khong gian Oxyz cho bon diem A(l;l;l),B(-l;0;-2), C(2;-l;0),D(-2;2;3)
1) Chung minh rang A,B,C,D la bon dinh ciia mpt tu di^n va tinh the tich
hi dif n ABCD 2) Lap phuong trinh mat phang (a) song song voi AB,CD va c5t hai duong thang AC,BD Ian lupt tai hai diem M,N thoa
l,AM^
Gpi G la trpng tam ciia tu dif n ABCD, (P) la mat phSng di qua G cat cac
Ccinh AB,AC, AD Ian lupt tai B',C',D' Viet phuong trinh mat phang (P)
^„^'t tu dign AB'C'D' CO the'tich Ion nhat n. 1
= AM^-l.i
Lai gidi
^ c6: A B = (-2; -1; -3), A C = (1; -2; -1), A D = (-3; 1; 2)
ra A B A A C = (-5;-5; 5) ^ ( A B A A C ) A D = 20 ^ 0
Trang 14Do do A, B,C, D la bon dinh cua tu di#n
Vi (a) song song vol A B , C D nen n^ =(3;9;-5)
Vi M N , A B , C D Ian lugrt nam tren ba mat phing song song gom : M^t
phang di qua A B va song song voi C D , mat ph3ng di qua C D va song song
voi A B va mat phang (a) nen theo dinh ly Talet dao trong khong gian ta
Ta CO phuong trinh AC:
AM^ - 1 o AM^ - AM^ = BN^ = 5AM^ ^ AM^ = 6
x = l + t
y = l - 2 t = > M { l + t ; l - 2 t ; l - t )
z = l - t Suy ra AM = (t; -2t; -t) ^ AM^ - 6 o t = ±1
AD AD' AD'
4 = ^ ^ ^ 3 3 1
AB' A C AD' \
AB AC AD AB' A C AD'
miv wvH Kltang Vift
* Bai tap van dyng Bai 3-2'l' Viet phuang trinh mat phang (a), biet:
x = 2t 1) (a) di qua diem A(2;3;-l) va duong thing dj : • y = 1 - 1
[z = 2 + t 2) (a) chua hai duong thing dj va dj : ^^-^ = ^ = 3) (a) chua dj va song song voi Oy
:0 Huong dan giai
1) Duong thing dj di qua B(0;l;2), VTCP Ui' = (2;-l;l)
Suy ra A B = (-2;-2;3), n„ = A B A U J =(1;8;6)
Phuong trinh (a): x + By + 6z - 20 = 0
2) Duong thing 62 di qua C(-l;2;0), VTCP u^ = (-2;l;-l)
Suy ra BC = ( - l ; l ; - 2 ) => n„ = U j A BC = (1;3;1)
Phuong trinh (a): x + 3y + z - 5 = 0
^) Oy CO VTCP ic = (0;1;0) Suy ra = u^ A k = (-1;0;2) Phuong trinh (a): x - 2z =: 0
3.2.2 Trong khong gian Oxyz cho 4 diem:
A ( l ; l ; l ) , B(-l;2;0), C(-2;0;-l),D(3;-l;2)
Viet phuong trinh mat phing (ABC) Tim tpa do trvrc tam tam giac ABC
^) Vi^'t phuong trinh mat phing (a) di qua AB va song song voi CD
Viet phuong trinh mat phing (P) di qua AB va each deu C, D '
^' Vig't phuong trinh mat phing (P) di qua BC va each A mpt khoang Ion nhat
^ Huong dan giai
' "^aco AB = (-2;l;-l),AC = (-3;l;2)=^riABC =(3;7;1)
Trang 15Cam nangTIyfit thi E)IJ H ' " " '«?<• r^g^
Phuong trinh (ABC): 3x + 7y + z -11 = 0
GQi H(a;b;c) la trvc tarn tarn giac ABC, ta c6:
Phuong trinh (a): 2x + y - 3z = 0
3) Phuong trinh (P) c6 dang: a(x -1) + b(y -1) + c(z -1) = 0 vol +c^ *0
Do B g (a) nen suy ra -2a + b - c = 0 => c = -2a + b
Nen ta viet lai phuong trinh (p) nhu sau: ax + by + (-2a + b)z + a - 2b = 0
Vi (a) each deu C D nen d(C,(a)) = d(D,(a))
<=> la - 3b| = |b| o a = 4b, a = 2b
• Voia =4b, ta chQn b = 1 => a = 4 Phuong trinh (P):4x + y - 7 z + 2 = 0
• Vol a = 2b, ta chpn b = 1 => a = 2 Phuong trinh (p): 2x + y - 3z = 0
Gpi H la hinh chieu cua A len mat phang (P), ta c6 A H = d(A,(P)),
va A H < AK nen d(A,(P)) Ion nhat khi va chi khi H = K
Hay (P) la mat ph3ng di qua K va vuong goc voi AK
Phuong trinh (P): l l x - 4y + 5z +19 = 0
Bai 3.2.3 Trong khong gian Oxyz cho ba duong thSng
, X y z - 1 x - l _ y + l _ z + 2 , x + 1 y z + 1
c-ty TNHH MTV DWH Khang Vift
1) Viet phuong trinh mat phSng (a) di qua A(1;2;3),B(-1;0;2) va cat d ^ d j
Ian lu(?t t^i C D sao cho CD = V38 2) Viet phuong trinh mat phSng (P) song song va each deu hai duong thJing
di va dg
3) Viet phuong trinh mat ph3ng (P) di qua O va c^t d 2 , d 3 Ian lugt t^ii hai
diem M , N sao cho M N = N/I4 dong thoi M N song song voi m^t phSng
(Q):2x + y + z - l = 0 va X N < - ^
Huong dan giii
1) Ta C O duong th^ng dj c6 VTCP u^ = (2;-1;-2), duong thang d j c6 VTCP
Suy ra n^ = AB A CD = (8;-13;10)
tPhuong trinh (a): 8x - 13y + lOz -12 = 0
f/oi x = 3=oCD = (-5;2;3).Suy ra n ^ = AB A CD = (-4;ll;-14)
Trang 16Cam nang luycit th! Till Iliiih hoc - iwguyen i
Bai 3.2,4 Lap phuong trinh mat phSng (a) biet
1) (a) qua hai diem A(l;2;-l),B(0;-3;2) va vuong goc vai m^it phang
( P ) : 2 x - y - z + l = 0
2) (a) each deu hai mat phSng ((3): x + 2y - 2z + 2 = 0, (y): 2x + 2y + z + 3 = 0
3) (a) qua hai diem C(-l;0;2),D(l;-2;3) va khoang each tii goc tQa dp toi mat
phang (a) la 2
4) (a) song song voi mat phang (Q): x - 2y - 2z - 3 = 0 va khoang each giiia
hai mat phang la 3
5) (a) di qua E(0; 1; 1) va d(A,(a)) = 2;d(B,(a)) = y , trong do A(l;2;-1),
B(0;-3;2)
Huong dan giai
1) Taco AB(-l;-5;3),n(p)(2;-l;-l) nen AB, ri ( P ) = (8;5;11)
Mat phing (a) qua A, B va vuong goc voi mat phang (P) nen
^1^+2^+(-2)2 72^+2^+12
X + 2y - 2z + 2| = |2x + 2y + z + 3
x+2y-2z+2=2x+2y+z+3 x+3z+l=0
I x + 2y - 2z + 2 = -2x - 2y - z - 3 3x + 4y - z + 5 = 0
Vay CO hai m$t phSng (a) can tim la
r (a): X + 3z +1 = 0 hoac (a): 3x + 4y - z + 5 = 0
3) Mat phing (a) di qua diem C(-l;0;2) nen c6 phuong trinh dang A(x + l) + By + C(z-2) = 0, A^+B^+C^ >0
Vi (a) qua D(l;-2;3) nen 2A -2B + C = 0 C = 2B-2A (1)
A-2C|
Taco d(0,(a)) = 2 nen - = 2 (2)
V A V B V C ^ The (1) vao (2) roi binh phuong, riit gpn ta thu dug-c
Vay CO hai mat ph5ng thoa man 2x + y - 2z + 6 = 0, 2x - 5y - 14z + 30 = 0 \
4) (a) song song voi mat ph^ng (Q): x - 2y - 2z - 3 = 0 nen c6 phuong trinh
(a): X - 2y - 2z + D = 0 Lay diem N(3; 0; 0) 6 (Q)
3 + D Taco d((a),(Q)) = d(N,(a))o = 3
« | D + 3 = 9 C ^ D = 6;D = -12
Vay CO hai mat phang can tim x - 2y - 2z + 6 = 0, x - 2y - 2z -12 = 0
Mat phang (a) qua E(0; 1; 1) c6 phuong trinh dang
Trang 1711
11 '-56B + 1 4 c f
3
la 6x + 2 y - 3 z + l = 0 |
+) Voi B = = - ^ C thi chpn C = 227 =:> B =-34, A = 26 phuong trinh (a) la
26x - 34y + 227z -193 = 0 Vay c6 hai mSt phSng can tim la I
6x + 2 y - 3 z + l = 0, 26x-34y + 227z-193 = 0
Bai 3.2.5 Trong khong gian Oxyz cho m|t cau (S):x^+y^+z^-4x+6y-2z-28=fl
1) Viet phuong trinh mat phJing (P) tiep xiic voi mat cau (S) va song song
hai duong thang dj; d2
2) Viet phuong trinh mat phing (a) di qua dj va t^o voi dj mpt goc (p
N /42
cos(p = - ^
3) Viet phuong trinh mat phing ((3) chua dj va cat (S) theo mpt duong tron'
ban kinh bang — •
l i V M H M i V DVVH Kliang Vi(t
Huong dan giai Ivlat cau (S) CO tam 1(2;-3; 1), ban kinh R = N/42 puong thiing d, di qua Mi(-5;1;-13), VTCP u^ = (2;-3; 2) puong thang dj di qua M2(-7;-l;8), VTCP u^ = (3;-2;l) ,) Vi mat phSng (P) song song voi hai duong thSng d p d j nen ta c6:
Hp =ui A U j =(1;4;5) Suy ra phuong trinh (P) c6 dang: x + 4y + 5z + m = 0
M|t khac, (P) tiep xiic voi (S) nen d(I, (P)) = R
^ W n g trinh (a): 25x + 16y - z + 96 = 0
^' (P) chua d2 nen phuong trinh c6 dang:
ax + by + cz + 7a + b - 8c = 0
^oi +b^ + ^0 va 3a-2b + c = 0=>c = 2 b - 3 a
T h p , 124 N/6 '^o gia thiet bai toan, ta c6: d(I,(P)) = 42
Trang 18-<=> l9a - 2b - 7cL ^ ^ ^ 3 30a - 16b = ^edOa^ - 12ab + Sb^)
Phuang trinh (P) la: 379x + 670y + 203z +1699 = 0
Bai 3.2.6 Trong khong gian vol hf true tpa dp Oxyz, cho 2 diem A(2;0;]),
B(0;-2;3) va mat ph^ng ( P ) : 2 x - y - z + 4 = 0 Tim tpa dp diem M thuoc
(P) sao cho M A = M B = 3
Huong dan giai
Gpi E la trung diem AB ta c6: E(l;-1;2), AB = (-2;-2;2)
Phuang trinh mat phang trung true (Q) eiia A B c6 phuong trinh:
x + y - z + 2 = 0
Vi Goi M(a;b;e) suy ra: I 2 a - b - e + 4 = 0 M A = M B nen suy ra a + b - c + 2 = 0 ': M € (Q) M € (P) nc = 3 + - a b = l + i a (Q) 2
2 Mat khae: MA^ = 9 (a - 2)^ + -^a + 1 2 + (|a + 2)2=9.j
Giai ra ta dupe a = 0,a = -—
V$y CO hai diem thoa yeu cau bai toan la: M(0;1;3), M f_6 4 12] 7 ' 7 ' 7
Bai 3.2.7 Trong khong gian voi h^ true tpa dp Oxyz, cho mat cau (5) '
phuong trinh x^+y^ + z ^ - 4 x - 4 y - 4 z = 0 v a diem A(4; 4; 0) Viet phu^"^^
trinh mat phSng (OAB), biet B thupc (S) va tam giac OAB deu
Huong dan giai
Xet B(a;b;c) Vi tam giae AOB deu nen ta c6 h^: ' '"'^ '
• B(0;4;4), taco
x - y + z = 0 OA, OB
OA, OB
= (16;-16; 16) nen phuong trinh (OAB):
= (l6;-16;-16) nen phuong trinh (OAB):
• B(4;0;4),tac6
x - y - z = 0 Bai 3.2.8 Trong khong gian h^ toa dp Oxyz, cho duong thSng
X — 2 y +1 z A: = —— = — va mat phang (P):x + y + z - 3 = 0 Gpi I la giao diem ciia A va (P) Tim tpa dp diem M thupe (P) sao cho MI vuong goc voi A
va MI = 4V14
Huong dan giai
Taco A eat (P) tai I(l;l;l)
Diem M(x;y;3-x-y)€(P) => MI = (l - x ; l - y ; x + y -2) Duong thing A eo a = (l;-2;-l) laVTCP
Ta CO : MI.a = 0
MI^ =16.14 i ( l - x ) 2 + ( l - y ) 2 + ( _ 2 + x + y)2= 16.14 y = 2 x - l
x = -3
y = -7 hoac fx = 5 y = 9
CO hai diem thoa yeu cau bai toan: M(-3;-7;13) va M(5;9;-ll)
3.2.9 Trong khong gian tpa dp Oxyz, cho duong thang A : " ~ ~ - ~" = ~Y'
mat phang (P):x-2y + z = 0 Gpi C la giao diem ciia A voi (P), M la
^'^'m thupc A Tinh khoang each tu M den (P), biet MC = \/6 , ) , iv
Trang 19Cam tiattg luyftt thi tJH Htnh hqc - Nguyen lai imi ^
H u o n g d a n g i a i
x = l + 2 t Cach 1: P h u o n g t r i n h t h a m so' ciia A : < y = t , t e R
Vay b = c = ^ la gia trj can t i m
Bai 3.2.11 T r o n g k h o n g gian toa dp O x y z , cho hai m a t phang (P): x + y + z - 3
M a t p h i n g (P) thoa m a n y e u cau bai toan t r o n g hai t r u o n g h p p sau:
T r u o n g h p p 1: (P) d i qua A , B song song v a i C D
Ta CO A B = ( - 3 ; - l ; 2 ) , C D = (-2;4;0), suy ra n = rAB,CDl = (-8;-4;-14) la
VTPT cua (P) P h u a n g t r i n h (P): 4x + 2 y + 7z - 1 5 = 0
T r u o n g h g p 2: (P) d i qua A , B v a cat C D tai I , suy ra I la t r u n g d i e m cua
CD D o d o I ( 1 ; 1 ; 1 ) = > A I = ( 0 ; - 1 ; 0 ) ^ Vec t o p h a p t u y e n cua m a t p h a n g (P): n = A B , A I ==(2;0;3)
P h u a n g t r i n h ( P ) : 2x + 3z - 5 = 0 Vay ( P ) : 4 x + 2 y + 7 z - 1 5 = 0 hoac ( P ) : 2 x + 3 z - 5 = 0 Sai 3.2.13 T r o n g k h o n g g i a n v o i h f tpa d p O x y z V i e t p h u a n g t r i n h m a t p h i n g (P) qua A ( 0 ; 0 ; 3 ) , M ( l ; 2 ; 0 ) va c i t cac true Ox, O y Ian l u p t tai B, C sao cho tarn giac A B C c6 t r p n g t a m thupc d u o n g t h i n g A M
Trang 20Vi G e A M nen - = - = - a = 2;b = 4
3 6 3
X y z Khi do, phuang trinh (P) CO dang: — + — + — = 1 ;
2 4 3 Bai 3.2.14 Trong khong gian voi tpa dp Oxyz cho A(2;5;3) va duoria
o
X — l y z — 2
thang d : ^ = Y = —^— • Tim tpa dp hinh chieu vuong goc cua A len d va
viet phuang trinh mat phang (P) chua duong thang d sao cho khoang
each tu A den (P) Ion nha't
Huong dan giki
• Gpi H ' la hinh chieu cua A len mp(P)
Khi do, ta c6: A H ' < A H d(A, (P)) Ion nha't
o H = H ' c:> (P) 1 A H
Suy ra A H = (1; -4; 1) la VTPT cua (P) va (P) di qua H
Vay phuong trinh (P): x - 4 y + z - 3 = 0
Bai 3.2.15 Trong khong gian voi h? tpa dp Oxyz, cho ba diem A(0;l;2)
ciia m p ( A B C ) Phuong trinh mp( A B C ) :x + 2 y - 4 z + 6 = 0
Gpi H(a;b;c) la true tam tam giac A B C
= > H e ( A B C ) = > a + 2 b - 4 c + 6 = 0 (1) ^'
Taco: CH = ( a ; b - l ; c - 2 ) , BH = ( a - 2 ; b + 2 ; c - l )
^, | C H l A B ^ | A B C H = 0 ^ | 2 a - 3 b - c + 5 = 0 [ B H I A C BH.AC = 0 |2a + b + c - 3 = 0
2) Giasu M(a;b;c)€(P)^2a + 2b + c - 3 = 0 (3)
^ " - 2 b - 4 c + 5 = -4a + 4 b - 2 c + 9
-4a + 4b - 2c + 9 = 4a - 2c + 5
Tu (3) va (4) ta tim dupe: a = 2;b = 3;e -7
Vay M(2; 3;-7) la diem can tim , , 3 ,
Bai 3.2.16 Tim m,n de 3 mat phang sau ciing di qua mpt duong thang:
(p):x + my + n z - 2 = 0, (Q):x + y - 3 z + l = 0 va ( R ) : 2x + 3y + z - l = 0
Khi do hay viet phuang trinh mat ph^ng (a) di qua duang thiing chung do
va tao voi (P) mot goc cp sao cho cos(p = —^2=
V679
Huong dan giai 'x + y - 3 z + l = 0 2x + 3y + z - l = 0
* Cho z = l = ^ x = 6,y = -4=> A ( 6 ; - 4 ; l ) £ ( Q ) n ( R )
* Cho z = 0=>x = -4,y = 3 = > B ( - 4 ; 3 ; 0 ) e ( Q ) n ( R )
Ba mat phang da cho cung di qua mpt duang thang <=> A, B G (P)
I-4m + n = -4 f m = 2 3m = 6
Ta c6: n = ( l ; 2; 4) la VTPT ciia (P)
Vi (a) di qua A nen phuong trinh ciia (a) c6 dang:
a(x-6) + b(y + 4) + c ( z - l ) = 0 ,f
J^o B e (a) nen ta c6: c = -10a + 7b
Suy ra V = (a; b; -10a + 7b) la VTPT ciia (a)
Trang 21« V97 |39a - 30b| = 23^3(l01a2 + SOb^-140ab)
« 3.97(13a - 10b)' = 23^ (lOla^ - 140ab + SOb^)
1 Viet phu-ong trinh mat phSng (a) di qua M va A
2 Viet phuang trinh mat phSng (P) di qua M, song song vol A va each A
mgt khoang bang 3
Huong dan giai i
1 Duong thang A di qua A(0; 0; 1), VTCP u = ( l ; 1; 4) ' Ta CO A M = (0;3;-3):^ A M A U = (15;-3;-3) Suy ra n = ( 5 ; - l ; - l ) la mpt
VTPT ciia (a)
Phuong trinh (a): 5x - y - z +1 = 0
2 Gia su n = (a;b;c) la mQt VTPT ciia (P), phuang trinh (p):
gai 3.2.18 Trong khong gian Oxyz cho diem M(l;2;3) Mat phSng (a) di qua
M va cat cac tia Ox,Oy,Oz Ian lugt t?i A,B,C Viet phuong trinh mat
I phSng (a) biet: J V ' ^
I Tu dien OABC CO the tich nho nhat w; ( j ^ ,
2 Khoang each tii- O den mat phang (a) Ian nhat ,^
Huong dan giai
Goi A(a; 0; 0), B(0; b; 0), C(0; 0; c) voi a, b, c> 0 j
X V z Phuang trinh (a): — + ^ + - = 1
Hi a b c
WLDO (a) di qua M nen ta c6: - + - + - = 1 (*)
H B a b C
Hfa CO the tich tiV dien OABC la: V = -abc
Mat khac, tu (*) ta suy ra: 1 = i + - + - > 33^11
a b c Vabe • abc > 6.27 => V > 27 Dang thucxay ra khi
Trang 22Cam uang luyen thi DH Httth hQC -Nguyen Tat tnu
Bai 3.2.19 Trong khong gian Oxyz cho d u a n g t h i n g d ^ c6 p h u o n g trinh
X - 3m - 1 _ yj2]_ _ , mi 0 ; l ;
-2 m 1 - m 3m + 1
C h u n g m i n h rang k h i m thay doi, d u a n g thang d ^ luon nam trong mgt
mat phSng (a) co d j n h Viet p h u a n g trinh mat phSng do
Huong dan giai
Bai 3.2.20 Trong khong gian Oxyz cho 3 diem A ( l ; l ; - 1 ) , B ( l ; l ; 2 ) , C ( - l ; 2 ; - 2 )
va mat phang (P) c6 p h u o n g trinh x - 2 y + 2z + l = 0 M a t phang ( a ) d i qua
A, vuong goc v o i mat phang (P), cat d u o n g thang BC tai I sao cho IB = 2IC •
Viet p h u a n g trinh mat phSng ( a )
Huong dan giai
Goi mat phang ( a ) c6 phuang trinh la ax + by + cz + d = 0 v o i a;b;c khong
Vay t i m duoc 2 m p ( a ) la 2x - y - 2z - 3 = 0 hoac 2x + 3y + 2z - 3 = 0
Bai 3.2.21 Trong khong gian Oxyz, viet p h u o n g trinh m a t p h l n g (P) chua
x = t
duong thang (d) : <y = -2 + t sao cho giao tuyen ciia mat phang (P) va mat
z = - 6 + 2t cau (S): x^ + y^ + z^ + 2x - 2y + 2z - 1 = 0 la d u a n g tron c6 ban k i n h r = 1
Huong dan giai
D u o n g t h i n g d d i qua A ( 0 ; - 2 ; - 6 ) va c6 u = (1;1;2) la VTCP
Phuang trinh (P) c6 dang: ax + by + cz + 2b + 6c = 0
Vi d c ( P ) nen u.n,^ = 0<=>a + b + 2c = 0=>a = - b - 2 c Mat phMng (P) c6 dang: (b + 2c)x - by - cz - 2b - 6c = 0 Mat cau (S) c6 tam I ( - l ; 1; -1), ban kinh R = 2
cat (S) theo m p t d u a n g tron giao tiep (C) c6 ban k i n h r = 1
Trang 23Vay, CO 2 mat phSng (P): "(P,): X + y - z - 4 = 0
( P 2):7x-17y + 5 z - 4 = 0 '
Bai 3.2.22 Trong khong gian Oxyz cho mat phang ( a ) : 2 x - y + z - 5 = 0 Viet
phuong trinh mat phang (P) qua giao tuyeh ciia (a ) va mat phiing (xOy) va
(P) tao voi 3 mat phang tQa dp mot tu di^n c6 the tich bSng
36
Huong dan giai
Phuong trinh mat phang (xOy): z = 0
Phuong trinh mat phang (P) thupc chum xac dinh boi (ot) va (xOy) c6
Huong dan giai
1) Ap dyng bat dang thuc Co si ta c6:
AD
4 = ^ ^ ^ > 3 3 ,
AB' A C A D '
AB.ACAD AB'.AC'.AD'
AB'.AC'.AD' ^ 27 AB.ACAD "64
3
Trang 24§ 3 PHLTOfNG T R l N H O L T d N G T H A N G
De lap p h u a n g t r i n h d u a n g t h i n g A, ta can t i m m p t d i e m M(xo;yQ;zQ) rrig
A d i qua va m o t VTCP u^(a;b;c) K h i d o p h u a n g t r i n h ciia A c6 dang:
X = XQ + a t
y = Yo + b t , t e M
Z = ZQ + ct
1 Cac dang toan ve viet phuong trinh duong thang
Dang 1: Viet phuang trinh duong thang di qua hai diem A , B
Ta CO A B la VTCP va A la diem d i qua
V i dv 3.3.1 Trong khong gian Oxyz cho ba diem
A ( l ; l ; l ) , B ( - 2 ; 0 ; 0 ) , C ( 0 ; 3 ; - 2 ) 1) Viet p h u a n g t r i n h d u o n g thang AB
2) Viet p h u a n g t r i n h d u a n g thSng t r u n g tuyen A M cua tam giac ABC
V i du 3.3.2 Viet p h u o n g t r i n h d u o n g thang A biet:
1) A d i qua A ( l ; - 2 ; 1) va song song v o i d u o n g thang d :
Vi A 1 (a) nen n = ( 2 ; l ; - 3 ) la VTCP cua d u o n g thang A
p h u a n g t r i n h d u o n g thang A la: - — ^ = ——^ =
2 1 3
tDang 3: Viet phuang trinh duong thang A , biet A cdt
Vmphucmg trinh duang thing A diquaAcdt d va vuong goc voi d'
Trang 25-(Jiim uaiig luyen thi VH Htnn woe - JVj^wyc"
i x - 1 y - 2 Z + 4
Phuong trinh duong thang A : = — - - ^ j - •
Vic? ^/nmng frin/i duang thang di qua A cat d va song song vai (a)
Goi M = A n d = ^ M ( x o + a t ; y o + b t , Z o + c t ) Dua vao AM.n„=0 ta tin^
duoc t •
x — 4 y z — ]
Vi dy 3.3.4 Trong khong gian Oxyz cho duong thang d : — ^ = — =
va mat phSng (a): x - y + 4z - 1 = 0 Viet phuong trinh duong thSng A di
qua A(1;2;1) CM d va song song voi (a) •
z = l - t
• ViC'l yhitang trinh duang thang A di nam trong (a) cat d va vuong goc (hoac cat) d'
Gpi M = A n d = : > M = d n ( a ) , suy ra tpa d p M = n^ A U ^
V i du 3.3.5 Trong khong gian Oxyz cho mat phang (a): 2x - y - z +1 = 0 va
0 , x + 1 y - 1 z - 2 x - 1 y - 3 z - 1
hai duong thang dj : — - = ^ — = — j - va d2 =
1) Viet phuong trinh duong thSng d di n^m trong (a) cMt dj va vuong §oc
Viei phifang trinh duang thang A di qua M va cat hai duang thang d va d' ' ''
Goi A = A n d, B = A n d ' Tham so hoa hai diem A, B theo t va t ' D^ra vao A,M,B thang hang hay M A = kMB ta tim dugc t , t '
Vi du 3.3.6 Trong khong gian Oxyz cho hai duong thang
Lot gidi
Goi M = A n d , N = A n d ' , t a c6 M ( 1 + t ; - l + 2 t ; - t ) , N ( 4 - 3 t ' ; 3 - t ' ; - 2 + 4t') Suy ra A M = (t + 2;2t - 5;-t + 6), A N = (5 - 3 t ' ; - l - t'4 + 4t')
4 - 1 4
' V/e? phuong trinh duang thSng A cat hai duang thing d, d' va vuong goc vai (P) hoqc song song vai d j
Goi A = A n d , B = A n d ' Tham so hoa hai diem A, B theo t va t '
Dira vao dieu kien AB cung phuong voi np (hoac Uj^ ) ta tim dugc t , t ' '
m • •• •
Trang 26cam nang luy^n thi DH Hinh H Q C - Nguyen Tat lUiT
V i d v 3.3.7 Trong k h o n g gian Oxyz cho hai d u a n g thang
1) Viet p h u a n g trinh d u a n g t h i n g A cat hai d u d n g thang d , d ' va vuong
goc v o i mat phang ( a ) : y + 4z - 2 = 0
2) Viet p h u a n g t r i n h d u o n g t h i n g A cat hai d u o n g thang d , d ' va song
Viel phuang trinh duang thing A di qua A, cdt d va tao vol d' mgt goc (p
Gpi B = A n d , tham so hoa tga dp B theo t D u a vao
t i m dupe t
c o s( A B , U d ) = cos(p t
L t y ii\nn mi VTJWH^Jtang Vift
V i d v 3.3.8 Trong khong gian Oxyz cho hai d u o n g t h i n g
Vi A tao v o i d ' m o t goc (p nen ta c6
liang 4: Viet phuang trinh duang thdng A vuong goc voi hai
vecta khong ciing phuang a, b ,
K h i d o = a A b
Vict phuang trinh dudng thang A di qua M va vuong goc voi hai duang thang d d'(Hogc song song vai hai mat phang (P) va (Q),
Ta CO = u j A u j (hoac u ^ = A n g )
Trang 27V i d v 3.3.9 Viet p h u o n g t r i n h d u o n g thiing A biet A
1) D i qua A ( l ; 2 ; l ) va v u o n g goc v o l hai d u o n g thSng
( a ) : x + z - l = 0, (p):3x + y - 2 z - l = 0
3) D i q u a C ( 2 ; 2 ; - l ) v u o n g goc v o i d j va song song v o i ( p )
L o i g i a i
1) T a c o u ^ = ( l ; - 2 ; 2 ) , u ^ = ( - 3 ; l ; - l ) la VTCP cua hai d u o n g thSng d^va d j
V i A v u o n g goc v o i hai d u o n g thJing d j , d 2 nen = U ; ^ A U J = ( 0 ; - 5 ; - 5 ) ,
y - 2 + t , t e M
P h u o n g t r i n h A :
z = l + t 2) T a c o n „ = ( l ; 0 ; l ) H p = ( 3 ; l ; - 2 )
V i A song song v o i hai m a t p h i n g ( a ) va (p) nert = n „ A H p = ( - l ; 5 ; l )
V i d\ 3.3.10 T r o n g khong gian Oxyz cho hai mat phang (P):x + y - z - 3 = 0'
( Q ) : 2x + y - 1 = 0 C h u n g m i n h rang hai mat phSng (P) va (Q) cat nhau
Viet p h u o n g t r i n h d u o n g t h i n g d la giao hiyen cua (P) va (Q)
Lot gidi
T a c o n p = ( l ; l ; - l ) , n Q = ( 2 ; l ; 0 )
V i n p ^ k n ^ nen hai m|it phang (P) va (Q) c^t nhau
' i i d la giao tuyen ciia hai mat phling (P) va (Q) nen :
pang 5: Viet phuang trinh duang vuong goc chung d ciia hai
duang thdng cheo nhau a va b /
Go! M = d n a , N = d n b T h a m s o h 6 a t p a d p c u a M , N t h e o t v a t '
M N u = 0 Dua vac h?;
M N u I, = 0 de xac d i n h t va t'
Vi dy 3.3.11 T r o n g khong gian Oxyz cho hai d u o n g t h i n g cheo nhau
J x - 1 _ y - 2 _ z - l x - 4 _ y - 2 z - 1
1 - 1 - 1 ' ' - 1 - 2 ~ 4 • Viet p h u a n g trinh d u a n g v u o n g goc chung ciia a va b
Lai gidi
Taco a = ( l ; - l ; - l ) , b = ( - l ; - 2 ; 4 ) la V T C P c u a hai d u a n g t h i n g a va b Goi M N la d u o n g vuong goc chung cua hai d u a n g thang a va b v a i
M e a , N 6 b Taco M ( l + a ; 2 - a ; l - a ) , N { 4 - b ; 2 - 2 b ; l + 4b)
Trang 28Tpa d p giao d i e m M thoa m a n
+) M e A nen M(XQ + at; + bt; ZQ + ct)
+) M e (a) nen A{\Q + at) + B(yo + bt) + C(zo + ct) + D = 0
Ta t i m dupe t t u do suy ra tpa dp cua M
• D u o n g thang A song song mat phang (a) k h i va chi k h i
fi.u = 0
M g ( a ) <=i>
A a + Bb + Cc = 0
Axg +Byo +Czo H-D^^O'
D u o n g thang A n a m trong mat phSng (a) k h i va chi k h i
n.u = 0
M e (a)
Aa + Bb + Cc = 0
AXQ +Byo + Czo + D = 0'
V i d\ 3.3.12 Xet vj t r i t u o n g doi giua d u o n g thang d va m p ( a ) T i m tpa dp
giao d i e m cua c h i i n g neu c6
3 V i tri tuong doi giira hai duong thang
Xet hai d u o n g thang c6 p h u o n g t r i n h
' Neu u , = k.U2 va M , e A, t h i hai d u o n g thSng t r u n g nhau
• N e u u j = k.U2 va A2 t h i hai d u o n g t h ^ g song song
• N e u u j ^ k.U2 t h i xet h ^ p h u o n g trinh
Trang 29Cam nang luyett thi DH Hinh hqc - Nguyen lat inu
Taco = ( - 2 ; - 2 ; - 2 ) , u7 A i i ^ = (-l;-2;0) => M ^ M " (uj A i^) = 6 t 0
Suy ra dj va cheo nhau ^
2) Duong thSng dj di qua M , ( l ; 2 ; - 3 ) va c6 iI7 = (l;-2;2) la VTCP
Duong thang di qua M2(-3;5;-6) va c6 u^ = ( 3 ; - l ; l ) la VTCP
' Mat khac g d2 nen suy ra d^ va d2 song song voi nhau
4 Bai toan lien quan den khoang each
+) Khoang each tu diem M(xo;y,);Zo) den duong thing A , :
d(M, Ai) =
+) Khoang each giua hai duong thSng song song:
d(A,, A 2 ) - d ( M 2 , A i ) =
Vi du 3.3.14 Tinh cae khoang each sau
1) Khoang each tu A(3;2;l) den duong thang A:
2) Khoang each giiia hai duong thang
x + 1 y - 2 _ z
A, : ^ - y ± l i ± 2 ^ x - 2 ^ y - l _ z - 3
- 1 3 ^ ^ ^ ^ " -2 3) Khoang each giua duong thSng ^ • ^ = ^ = ^ mat phang
Do do hai duong thSng Aj va Aj cheo nhau
U j A U ; x/38 '
3) Duong thing A di qua M ( l ; l ; - 2 ) v a u = ( - 2 ; l ; 3 ) la VTCP
Matphing (a) c6 n = (l;-4;2)la VTPT
Ta CO u.n = 0 va M g (a) nen A / /(a)
Suy ra d(A,(a)) = d(M,(a)) = 1 - 4 - 4 + 1
Vl + 16 + 4 V2T '
5 Tim tpa dp diem M thupc duong thing d thoa tinh chift T Tham so hoa tpa dp diem M theo t Dua vao tinh chat T ta tim dupe t
Chu y: M e d : ^ K + a t ; y o + b t ; z o + c t ) 3.3.15 Trong khong gian Oxyz cho duong thing
^ x - 1 y + 1 z - 2 , , _ ^, A: = - = va diem A(4;3;2)
2 3 1
^) Tim tpa do diem M thupc dyong thing A sao cho A M = ^/l05
^) Tim tpa dp diem A' doi xung voi A qua A
^) Tim tpa dp diem D thupc A sao cho khoang each tu D den ^_^) : x - 2 y + 2z + 2 = 0 bjng 1
jg^o M e A nen M( 1 + 2 t ; - l - 3t;2 - t) =^ A M = (2t - 3;-3t - 4;-t)
Trang 30Cam nang luyfti thi DH Hinh hqc - Nguyen lat IHu
Do do A M = N/T05 <=> A M ^ = 105 » (2t - 3 ^ + (St + 4)^ + = 105
14t2 + 12t - 80 = 0 « ^ t = 2,t = - —
7 Vay CO 2 d i e m M thoa yeu cau bai toan la: M ( 5 ; - 7 ; 0 ) va M ' 3 3 5 3 3 4
, 7 ' 7 ' 7 2) Goi H la h i n h chieu cua A len A ^
D o H la t r u n g d i e m cua A A ' nen A '
V
3) T a c o D € A nen D ( 1 + 2 t ; - l - 3t;2 - t)
l + 2 t - 2 ( - l - 3 t ) + 2 ( 2 - t ) + 2l Suy ra d ( D , ( a ) ) =
1) Viet p h u o n g t r i n h d u o n g thang A cat hai d u o n g thMng d i , d 2 Ian lug^
tai M , N thoa M N = 3\/3 va M N song song v o i m p ( a ) : x - 2y + 3z = 0
2) Viet p h u o n g t r i n h d u o n g thang d d i qua O va cat d j tai A sao cho mat
cau t a m A va d i qua O tiep xiic v o i mat phang (P): 5x + y - 4z - 1 6 = 0
m = - l —
n = l
^ m = - l rn = - 3
n = l n = - 5 , suy ra M ( 0 ; -3; 2), M N = (5; - 2 ; 1)
Vi mat cau t a m A d i qua O tiep xiic v o i (P) nen ta c6 : O A = d(A,(P))
Ma OA^ = (3 + 2af + (a - 3)^ + (a - if = 6a^ + 2a + 22
Chung m i n h rang hai d u o n g thang A j , A 2 cheo nhau Viet p h u o n g trinh
d u o n g v u o n g goc chung cua hai d u o n g thang do
2) Viet
p h u o n g t r i n h d u o n g thang A d i qua O, cat Aj va v u o n g goc v o i A2
3 de dang c h u n g m i n h duc^c A i , A 2 la hai d u o n g thSng cheo nhau
A la d u o n g v u o n g goc chung ciia hai d u o n g t h i n g va A j
Trang 31-rmn iinux hn/ett miT}IlTmi7niW-TW>y^» larinu
D o do a = 0, b = 0 nen A(2; - 1 ; 0), B(3; 2; - 1 ) , AB(1; 3; - 1 )
x = 2 + t Phuong trinh d u o n g v u o n g goc chung can t i m la y = - l + 3t ( t e R )
z = - t 2) Goi M = A n A , = > M ( 2 - m ; - l + m ; 2 m ) = > O M = ( 2 - m ; - l + m ; 2 m )
V i A 1 A2 nen M O u ^2 = 0 <=> 2(2 - m ) - (-1 + m ) - 2 m = 0 o m = 1
Suy ra u.\ O M = (1;0;2)
x = t
y = 0 , t € R Phuong trinh A :
z = 2t
V i d u 3.3.18 Trong khong gian toa dp Oxyz cho cac diem A(8;2;0),
B(3;-13;-5), C(-3;3;5), D ( 2 ; - 2 ; 0 ) Goi H la diem tren dogn C D ma
3 H D = 2HC va P, Q t h i i t y la trung diem cua A D , BC T i m to? dp diem
M tren d u o n g th^ng AB sao cho cac d u o n g thang H M , PQ cat nhau
Do do H M va PQ c3t nhau
Vay M (11; 11; 3) la diem can t i m
V i dM 3.3.19 Trong khong gian toa dp Oxyz cho d i e m M ( 4 ; 3 ; l ) , d u o n g
x - l y + 2 z + 3 thang d : = - = va mat phang (P): x + 2y - z + 3 - 0 Lap
M N A u '
- S u y ra d ( M , A) = — f ^ ^ i —
A u ' nho nhat khong doi nen d ( M , A ) nho nhat k h i va chi k h i
(Say P = ( 7 b - 3 c - 1 8 ) ^ + { 7 a - c - 2 7 ) ^ + ( 3 a - b - 9 ) 2 nho nhat v o i a,b,c
h 6 a ( l )
A p d u n g bat d3ng thuc: x^ + y^ > - 2 x y ta c6: •' (7b - 3c -18)^ + 34^ > -2.34(7b - 3c - 1 8 ) ; ,1 , h iV
^ (7a - c - 27)^ +16^ > -2.16(7a - c - 27)
Trang 32Dang thuc xay ra khi , ta chon a = 2, b = - 1 , c = 3
Cach 2 Ta c6 u"' = n A u = (1; 3; 7) la VTCP cua A
Do do phuong ciia duong thing A khong doi
Goi H, K Ian lup-t la hinh chieu cua M len (a) va A
1 2 2 "^^^ phang (P) c6 phuong trinh x - y + z + l = 0
Hay viet phuong trinh duong thang d biet khoang each tu d den (P)
Gpi M la giao diem cua A va d , suy ra M ( - 2 +1;-2 + 2t;-4 + 2t)
Vi d each (P) mot khoang bang nen d//(P) va d(M,(P)) = - ^
( « ) - ^ y — n = 0 Vi.^ phuong trinh du6ng thing A
h a ^ u o n g H.ang A , A , va mat phing (a) ,an l u t t , i A , B , M L a
L&igidi
' " ^ A ( U a ; - 2 + a;l),B(3 + 2b;4 + b ; 1 3 b ) Goi M(x; y; z), tir A M = 2MB, ta c6:
Trang 33Cam nang luyftt thi DH llitili hor V^'KI/I'K I at THu
Matkhac AlAj =>AB.Uj =0<=>2a-3b-8 = 0 (2)
thing A tao voi hai duang thang dpdj mpt tam giac can tai I va c6
dien tich bang 42
2) Viet phuong trinh duong phan giac ciia goc tao boi hai duong thang
cij di qua diem Mi(3;3;3) c6 Uj =(2;2;1) la VTCP;
Phuong trinh A:
7 ' 7 ' 7 >AB: '±._± _ 1 ^ 21' 21' 21
5 5 7 3'3'3
^' ^ ^13 10 16^
3'3'? ;B 7 ' 7 ' 7 >AB = 32 23 13 21'21'21
Trang 34—4 5 1
Vi du 3.3.23 Trong khong gian Oxyz cho hinh chop S.ABC c6 S(2;l;2),
A(3;0;-l), day ABC la tarn giac vuong can tai A va SA l ( A B C ) Hinh
chieu vuong goc cua A len SB, SC Ian lug-t la hai diem M,N va mat
phSng (AMN) CO phuang trinh 2y + 3z + 3 = 0 Gpi D la diem doi xung
voi A qua trung diem E cua BC
1) Viet phuong trinh duong thang SD
2) Viet phuang trinh duong thiing BC
hoi gidi S
1) Tir giai thiet ta suy ra dugc ABDC la hinh vuong
Ta c6: B D 1 (SAB) B D 1 A M , lai c6
A M I SB nen suy ra A M l ( S B D )
hay ta suy ra dugc AM 1 A D
Hoan toan tuong tu, ta chung minh
du(?c A N 1 SD, suy ra SD 1 (AMN)
Vay phuong trinh SD : y = l + 2t
z = 2 + 3t
Cty TNHH MTV DVVHKhang Viit
I^Taco SA =(!;-!;-3) nen phuong trinh m|it phSng (ABC) la:
[d(A,(a)) = 76
^ [a = 6 (do x^ > 0) [b = - l
!,^«c>ng trinh A : ^ = Z z i l _ £ ± 3 0
re
>A(9;13;-30), AB = (-10;-11; 31)
Trang 35V i d v 3.3.25 Trong khong gian Oxyz, cho hai duong thang
d j diem P sao cho tam giac MPN c6 dien h'ch hang 2
2) Tim A , B e d i va C D e d j sao cho "ABCD la t u dign deu
2) Goi M, N Ian lugt la trung diem cua AB va CD
Ta chung minh duoc M N la duongvuong goc chung ciia d^ va d2 •
- l y in/tinmTV uvvti rMang-yj^
Suy ra M - H , N - P nen N ( | ; | ; ^ ) , va M N = GQI a la dp dai cac canh ciia tu dien ABCD, ta c6:
va D
3 ' 6 ' 3
f 14 + 72 7 - 7 2 10-721
_Trong do A, B va C, D c6 thehoan vj cho nhau
Vi dy 3.3.26 Trong khong gian Oxyz bon diem A(3;2;l), B(5;-2;3),
||< C(2;l;0), D(-4;5;2) Hai diem M, N Ian lupt d i dpng tren cac duong
B thMng AB, CD sao cho A M = C N Chung minh r^ng trung diem I cua
M N luon nam tren mpt duong thSng co djnh Viet phuong trinh duong thang do
Lai gidi
Ta co: A B = (2; -4; 2), A C = (-1; - 1 ; -1), C D = (-6; 4; 2)
Suy ra A B A A ( : - ( 6 ; 0 ; - 6 ) , (AB A ACJ.CD =-48 ^ 0 ^
Do do A B va C D la hai duong thSng cheo nhau '
Trang 36Han nua A C A B = 0 nen AC la
[ A C B D = 0
duong vuong goc chung cua hai
duong thang A B va CD
Goi K la trung diem do^n thang
AC, suy ra K 5 3 1 ; Aj la duong
„ 2 ' 2 ' 2 ,
thang di qua K va song song voi AB;
Aj la duong thang di qua K va song
song voi CD
Gpi E la hinh chie'u cua M len A j , F la hinh chieu cua N len A2
Khi do MENF la hinh binh hanh nen I la trung diem cua EF
Dong thoi KE = A M = CN = KF nen KI la duong phan giac A cua goc tao
boi hai duong thang Aj, A j Ma A co djnh nen I luon nam tren duong
thang c6' djnh A
D a t e , ^ A B =
f
CD -CD = ^/T4'^/l4'^/T4
Vi A la phan giac cua goc tao boi hai duong thang A^, A2 nen ta c6 hai
truong hop sau:
Lap phuong trinh duong thang A , biet f f
J di qua A,B
2 A di qua B, song song voi (P) va vuong goc voi d
di qua A , cat duong thang d va tao voi mat phang (P) mpt goc (p sao Lcoscp = ^
P Huang dan giii
Duong th3ng ddiqua M(1;0;-1) va C6 U=(1;2;-3) la V T C P
^en sin 9 = n.AC 2 t - 2 ( 2 t - l ) - 3 t - 3 l
Trang 37r^^^^^^^^^^^^^ DH mnh h„c - Nxuy&n Tat Thu
1 d la giao tuyen ciia hai mat phang (?) va (Q)
2 d di O va song song voi hai mat phang (P) va (Q)
Huong dan giai
1 Xet he phuong trinh
x - t
3 3
2 Vl d song song voi hai mat phSng (?) va (Q) nen
u ^ ^3-1; _ 4) la VTC? ciia duong thang d
> , J X y _ ^ Phuong trinh
Cty TNHH MTV DWH Khang Vipt
Bai 3.3.3 Trong khong gian Oxyz cho hai duong thiing
d, :• x - 1 _ y + 1 _ _ ^ J x + 2 y z + 1 T~" -2 ~ - 1 ' ^ • 2 ~ -2 ~ 1 Viet phuong trinh duong thang A , bie't
F1 A di qua O va vuong goc voi ca hai duong thang dpdj
2 A di qua A(0;6;6), cat dj va vuong goc voi dj
3 A di qua B(-12;10;8) va cSt ca hai duong thSng d j , d 2
4 A nam trong mat phang (?): 5x + lOy - 8z -16 = 0 va cat ca hai duong thSngdpdj
Huong dan giai
I Duong thing d, di qua Mi(l;-1;0) va c6 U j = ( l ; - 2 ; - l ) la VTCP
di qua M2(-2;0;-l) va c6 u^ = (2;-2;l) la VTCP
Vi A vuong goc voi ca hai duong thang d,,d2 nen
' u = u J A U 2 = (-4; -3; 2) la VTCP ciia duong thang A
^ 1 3 4 Cach 2
Goi (a) la mat phang di qua A va d j , suy ra n,^ = U j A A M J = ( 5 ; 5 ; - 5 )
GQI (P) la mat phang di qua A va vuong goc voi d 2 Suy ra np = U 2 =(2;-2;l)
Vi A di qua A , c3t dj va vuong goc voi dj nen A la giao tuyen ciia hai matphSng (a) va (P) :
Trang 38Cam ttang luyen thi DH Hinh hqc - Nguyen Tat Thu
3 Cach 1: Gpi E , F Ian lupt la giao diem A vai dj^dj
a + kb-9k = -8 Suy ra B E = (l5;-15;-10)
Phuong trinh A : ^ ^ = ——^ = - — -
^ 3 - 3 - 2
Cach 2 Gpi (Q) la mat phang chua B va dj va (R) la mat phang chiia B va d j
Khi do A la giao tuyen ciia hai mat phang (Q) va (R)
.x + 1 2 _ y - 1 0 _ z - 8 Tir do ta tim dirpc phuong trinh A : ,
Suy ra E(-2;5;3), F(-4;2;-2) =^ E F = (-2;-3;-5)
_ x + 2 _ y - 5 _ z - 3 Phuong trinh A:
Bai 3.3.4 Trong khong gian Oxyz cho duong thSng A:
x = l + t
y = 2 - t va mat
z = l + 2t phing (a):2x + y + 2 z - l l = 0
1) Lap phuong trinh hinh chieu ciia A len mat phSng (a)
2) Viet phuong trinh duong thang Aj nam trong (a) dong thai cat va vuong
goc voi A
3) Viet phuong trinh duong thang A2 nam trong mat phang (a), cat A va t?''
voi duang thang A mpt goc (p thoa cos cp = 7J3
18
1
Hirong dan giai
Duang thSng A di qua M(l;2;l) c6 VTCP i ^ = ( l ; - i ; 2 ) ' Mat phang (a) c6 VTPT n^ =(2;1;2)
fx = 1 + t 1) Cach 1: Ta thay h^ phuong trinh
'3
y = 2 - t
z = l + 2t 2x + y + 2 z - l l = 0
x = 2
y = l
z = 3 Suyra A n ( P ) = I(2;l;3)
Gpi H la hinh chieu ciia M len (a), suy ra phuong trinh M H :
Hinh chieu ciia A len mat phang (a) chinh la I H Phuong trinh IH : ^—^ = = ^—^
^ 1 14 -8 Cach 2 Gpi (P) la mat phSng chua A va vuong goc vai (a) Suy ra np = n„ A u^ = (4;-2;-3)
Gpi A' la hinh chieu cua A len mat phang (a),tac6 A' = (a )n(P) Suy ra u^ = n^^^ A n p =(l;14;-8) va l€ A' nen phuong trinh A' la :
x - 2 _ y - l _ z - 3
1 " 14 " -8 • 2) Gpi J = AinAr=>J = An(a)=^J = I
r)u - x - 2 y - 1 z - 3
1 huong trmh Aj : = — ^ = - - ^
^) Vi A2 nam trong mat ph5ng (a) va cit A nen Aj di qua I
Gpi u^^ = (a; b;c), ta CO : u - H a = 0 <=> 2a + b + 2c = 0 <=> b = -2a - 2c xh
Trang 39Mat khac : cos(p = la - b + 2c|
18
^6(53^ + 8 a c + 5c^)
<=> 3V2 |3a + 4c| = 7^5a^ +8ac + 5c^ ^ 18(3a + 4c)^ = 49(53^ + Sac + 5c^)
<=> 83a^ - 40ac- 43c^ = 0 <:> a = c,a = - — c
83
• Ne'u a = c => b = -4a, chpn a = l = > b = -4,c = l
X — 2 y — I z — 3 Phuong trinh : — — = ^—^ = •
Bai 3.3.1. Viet phuang trinh duong thang A , biet
1) A di qua hai diem A(l;2;l),B(3;-2;-l)
2) A di qua goc toa dp O va vuong goc voi mat phang (a): 2x - 3y - z = 0
3) A di qua A va song song voi duong thang d : = — =
1 2 3 4) A la giao tuyen ciia hai mat phang
(ttj): X + y + z - 3 = 0 va ( a j ) : 3 x - 2 y + z - 2 = 0 '
5) A di qua B, vuong goc voi Ox va duong thang d (6 y 3)
Huong dan giai
1) Ta CO AB = (2;-4;-2), phuong trinh A :
z
2) Ta CO u^ - n„^ = (2; -3; -1), phuong trinh A : - = = —
3) Taco u 7 = Uj' = (l;2;-3), phuong trinh A:~1 = IZ1 = IL:1 _
4) Ta c6: ii7 = A i ^ ^ = (3; 2; -5) Ta thay M ( l ; 1; 1) e (aj) n (aj)
Phuong trinh A : ^^-^ = ^^-^ = ^—^
5) Tren Ox CO vec to don vi i = (l;0;0)
Vi A cung vuong goc vol Ox va d nen u^ = T A U j = (0;3;2)
x = 3
y = -2 + 3t, t e M
z = - l + 2t phuong trinh A:
gai 3.3-2- Trong khong gian Oxyz cho hai duong thang
4) Viet phuong trinh duong thang d c3t A p A j Ian lupt tai E,F thoa EF = 4^2
va d song song voi mat phling ( a ) : x - y + z + l = 0
Huong dan giai
Duong th5ng Aj di qua M i ( l ; l ; 0 ) , VTCP U i = ( l ; - l ; l ) Duong thang di qua M 2 ( 2 ; - 6 ; l ) , VTCP U j =(2;-3;4)
Suyra U j = ( - l ; - 2 ; - l ) , M^Mj =n;-7;l)
=> |uj A U 2 | M j M 2 = 12 0, do do hai duong thang A^ va A2 cheo nhau
Gpi M N la duong vuong goc chung ciia hai duong thing Aj va Aj ^
M e A j =:>M(l + m ; l - m ; m ) , N € A 2 => N(2 + 2n;-6 - 3n;l + 4n) Suy ra Mfsi = (2n - m + l ; - 3 n + m - 7 ; 4 n - m + 1)
Trang 40Cam nangluyen thi DJI lliiih Hoc r^^^uxien lui mu
z = -3 3) GQi C = A ' n A i = 5 C( l + t ; l - t ; t ) = ^ B C = ( t - 2 ; - t + l ; t - 5 ) '*
Vi A ' l A j nen ta suy ra BC.Uj = 0 o 2(t - 2 ) - 3 ( - t + 1) + 4 ( t - 5 ) = 0
jj Viet phuong trinh duong thSng A nkm trong (P) va cSt hai duong thiing
2^Viet phuong trinh duong thing A' nam trong (P), cat dj va vuong goc voi d j
3) Viet phuong trinh duong thing d nSm trong (P), cSt d j va tao voi dj mpt goc cp thoa cos(p = }
3v3
Huang dan giai
1) Gia su A cat dj,d2 Ian lupt tai A, B
^ u y ra A ( l - a; a; -a), B(-l + 2b; - 1 - b; 1 + 2b)
• | A e ( P ) nen taco A , B € ( P ) ^
Do do: 2 ( l - a ) + 3a-2(-a) + 4 = 0 [a = -l
2(-] + 2b) + 3 ( - l - b ) - 2 ( l + 2b) + 4 = 0 ^ | b = - l Suyra A(3;-2;2), AB = (-6; 2;-3)