M: M2 =(1; 1; 2) nen hai duong thang do song song.
b)Goi d2 la ctuong thang di quaM 0va song song vdi Oy d2 cat (a) tai M2 thi: M0M2 = A x0 +B y0 +C z „+D
B
Goi d3 la dudng thang di qua M0 va song song vdi Oz, d3 cat (a) tai M3 thi M0M3 =- ^ + B y° + C z" + D
C
Ta co M0M i , M0M2, M0M3 doi mot vuong goc, do do: V M0M,M2M3 = - ± MoM1. M0M2. MoM3
b
|Ax0 + By0 + Cz0 + Df
6.|A.B.C|
V i du 26: Trong khong gian Oxyz cho hinh hop chu nhat ABCD.A1B1C1D1 vdi Ai(0; 0; 0), B i d ; 0; 0), Di(0; 2; 0), A(0; 0; 3). Goi M , N , P. Q l i n luot la trung diem cac canh A B . B | C i , CiD], D | D :
a) Chung minh rang cac diem M , N . P. Q cimg thuoc mot mat phang. Viet phuong trinh mat phang (a) chua chung.
b) Xac dinh thiet dien ciia hinh hop khi cat bdi mat phang (a). Tinh the tich ciia khoi chop co dinh C va day la thiet dien do.
c) Tim toa do diem I doi xiing vdi diem A i qua dudng thang MP Hoi diem I nam trong hay ngoai hinh hop?.
Giai
a) Taco M ( - ; 0; 3), N ( l ; 1; 0), P( 2
i 2; 0), Q(0; 2; | ) .
Ju
Phuong trinh mat phang (MNP) la: 6x + 3y + 2z - 9 = 0.
Thay toa do ciia diem Q vao phuong trinh tren. ta thay no thoa man. Vay bon diem M , N , P, Q dong phang, va phuong trinh ciia mat phang (a), la: 6x + 3y + 2z - 9 = 0.
b) Thiet dien la luc giac MENPQF co cac dinh la trung diem ciia cac canh ciia hinh hop chu nhat ta co CG; 2; 3)
Gpi h la chieu cao ciia hinh chop
C.MENPQF thi:
h = d(C,(a)) 6.1 + 3.2 + 2.3-9
16' +3' +2'
Gpi M ' , F' la hinh chieu ciia M va F len mp(AiBiCiDi) thi luc giac M'BiNPDiF' la hinh chiiu ciia luc giac MENPQF len mp(AiBiCiD,). Gpi 9 la goc giira mp(cc) va day ciia hinh hop thi:
2 7
Svr.n-MDTvii" = " Va SM.BlNPDlF. = SMENPQF. COS(p
coscp = |cos(na,k) 'M'BJNPDIF SMENPQF 3 2 M'BtNPDtF' _ _Z_1 cos cp 21 4 9 v - v 1 21 9 _ Vay VC.MENPQF - - • -j- • - - -
c) Mat phang qua A, va vuong goc voi MP co phuong trinh: 2y - 3z = 0.
Goi H la giao diem ciia during thang MP va mat phang tren, ta co: H 1 . Tir do ta tim duoc toa do didm I la: ( 1 ; ^ | ;
2'13 13
Didm T(x; y; z) nam trong hinh hop
36 24 (adsbygoogle = window.adsbygoogle || []).push({});
36
o 0 < x < 1, 0 < y < 2, 0 < z < 3. Nhung tung do ciia diem I la y - — > 2.
Vay diem I nam ngoai hinh hop.
Vi du 27: Cho hinh chop t i i giac ddu S.ABCD co canh day bang a va chieu cao bing h. Goi I la trung didm ciia canh ben SC. Tinh khoang each tir S den mat phang (ABI).
Giai
Ta chpn he toa dp Oxyz sao cho gdc toa dp la tam O ciia day, tia Ox chiia OA, tia Oy chiia OB, tia Oz chiia OS.
Khi do: A 1V2 ;0 ; 0 v B C 0 ; ^ ; 0 2 aV2 ;0;0 , S(0; 0; h).
Ta co giao diem M ciia SO va A I chinh la trpng tam tam giac SAC nen M(0; 0; ^)
Mat phang (ABI) cung chinh la mat phang (ABM). Vay mp(ABI) co phuong trinh la:
_?l_ + ^- + ^ = l
aV2 aV2 h
2 2 3
Do do, khoang each tir S toi mat phang (ABI) la: 2ah
d(S; (ABI)) =
V j du 28: Cho hinh chop S.ABCD co day ABCD la hinh vuong canh a, canh ben SA vuong goc voi mat phing (ABCD) va SA = 2a. Goi M , N lan
luot la trung diem ciia cac canh SA, SD.
a) Tinh khoang each tii A den mp(BCM); khoang each giua hai duong
SB va CN.
b) Tinh cosin ciia goc hop boi 2 mat phang (SCD), (SBC).
c) Tinh ti so the tich ciia hai phan hinh chop duoc chia boi mat phang (BCM).
Giai
Chon he true Oxyz sao cho goc O la diem A, tia Ox chiia AB, tia Oy chiia
AD va tia Oz chiia SA. Khi do A(0; 0; 0), B(a; 0; 0), C(a; a; 0), D(0; a; 0),
S(0; 0; 2a), M(0; 0; a), N(0; - ; a) 2 a) BC = (0; a; 0), BM = (-a: 0; a)
=> [BC , BM ] = (a2; 0; a2)
Do do, mat phang (BCM) co mot vecto
phap tuyen la ( 1 ; 0; 1), suy ra phuong
trinh mat phang (BCM) la:
l(x - a) + l(z - 0) = 0 <=> x + z - a = 0. Vay khoang each tir A den mp(BCM) la:
d(A, (BCM)) (adsbygoogle = window.adsbygoogle || []).push({});
V i2 + 12
a 72
Taco: BS = (-a; 0; 2a), CN = (-a; - - ; a), SC = (a; a; -2a) Suy ra [ B S , C N ] 2 2 a
a ;-a ; — 2
2 A.
[BS , C N ] . SC = a3 - a3 - a3 = - a3
Vay khoang each giua hai duong thang SB va CN la: ' rB S , C N_
d(SB, CN) = .SC BS, CN
2a 3
b) V i [SC , SD ] = (0; -2a2; -a2) nen mp(SCD) co vecto phap tuyen n = (0; 2; 1). Vi [SB , SC ] = (2a2; 0: a2) nen mp(SBC) co vecto phap tuyen n ' = (2; 0: 1).
Goi cp la goc giua hai mat phang (SCD) va (SBC)
Ta co: coscp 1
n . n l v"5.v5
c) Vs ABCD = - a2 2a
° S
2
— a3 V i M la trung diem ciia SA, suy ra
d(S, (BCM)) = d(A. (BCM)) a
72 310
Hinh chop S.ABCD bi mp(BCM) chia thanh hai phan, trong do co mot phin la hinh chop S.BCNM.
Hinh chop nay co duong cao bing d(S; (BCM)) = va day la hinh v 2
l a 3 72" a2
thang B C N M co dien tich bing - (a + — )aV2 = — - —
e 1 3V2a
Suy ra: VS.BCNM = - • — — a
T V, S.BCMN _
/2 •* "SABCD
Vay ti so the tich giua hai phan cua hinh chop S.ABCD chia boi mp(BCM) la |
Vi du 29: Cho khoi lap phuong ABCD.A'B'C'D1
canh bing 1. a) Tinh goc tao boi cac duong thang A C va A'B. (adsbygoogle = window.adsbygoogle || []).push({});
b) Goi M , N , P lan luot la trung diem cua cac canh A'B', BC, DD'. Chimg minh A C vuong goc voi mat phang (MNP).
c) Tinh the tich tir dien AMNP.
Giai
a) Ta chon he true Oxyz sao cho goc O la dinh A' cua hinh lap phuong, tia Ox chira A'B', tia Oy chua A'D' va tia Oz chua A'A. Khi do A'(0; 0; 0). B'(l; 0; 0), D'(0; 1; 0), A(0; 0; 1), C(l; 1; 1), B ( l ; 0; 1), D(0; 1; 1), C'(l; 1;_0) Do do: AC" = ( 1 ; 1;-1), A ' B = ( 1 ; 0: 1) => A C ' . A T i = 0 => A C 1 A'B b) Ta co M( — ; 0; 0), N ( l ; - ; 1), P(0; 1; - ). ( - - 1 ) V 2 ' } M N => MN 1 AC. MP = ( - - ; 1; i ) = > V 2 2 => MP 1 A C . Vay A C 1 mp(MNP) Taco: MA = ( - - ; 0 ; v 2 1 6 M N AC' = 0 MP . A C V AMNP [ M N M P ] . MA | = = 0 > M P ] " ( i _ 1 9 6 8 \ "* •» ^ / / j \ N / i l i i ' / l , '/ A' / / / 3 _ 3 3 4 ' 4 ' 4 _3_ ~ 16 '
V i du 30: Cho hinh lap phuong ABCD.A'B'C'D' co canh bing 1. Goi M , N . P lan luot la trung diem ciia cac canh B'B, CD va A'D'
a) Tinh khoang each giua cap duong thang A'B, B'D va cap duong thing PI, A C voi I la tam ciia day ABCD.
b) Tinh goc giua hai duong thang MP va C N , goc giua hai mat phing
(PAI) va (DCCD1).
Giai
a) Ta chon he tuc Oxyz sao cho g6c toa do la A, tia Ox chua A B , tia Oy chua A D va tia Oz chua AA'. Khi do
A(0; 0; 0), B(l; 0; 0), D(0; 1; 0), A'(0; 0; 1), C(l; 1; 0), D'(0; 1; 1), B'G; 0; 1), C'G; 1; 1). Suy ra A^B =(1; 0;-l) ; B^T3 =(-1; 1; -1), A^B' = (1;0; 0) = > [ A ' B . B ' D ] = (1;2: l)nen: d(A' B, B'D) A B, B D .A'B [ A ' B , B ' D ] | 76 T a c 6 : P ( 0 ; l ; l ) , I ( I;i ; 0 ) ^ i P =(_ I ; 0 ;l ) AC' = (1; 1; 1), AP =(0; A; 1) 2
Suy ra d(PI, A C ) |[IP,AC'].AP| ^
IP, AC'] 28
b) T a c 6 M ( l ; 0 ; \ ) , N ( | ; 1; 0) =>MP = ( - 1 ; i - A ) 2 2 2 2 NC'=(-;0; 1)=>MP.NC' =0 =>MP ± NC
Mat phang (PAI) co vecto phap tuyen-
Mat phang (DCCD') co vecto phap tuyen la AD = (0; 1; 0) Gpi (p la goc giua hai mat phing tren thi: coscp = -X C. BAI LUYEN TAP
n.AD
n . L A D
Bai 1: Trong khong gian Oxyz cho bdn didm A(5; 1; 3), B(-5; 1; - l ) C(l- -3- 0)
(BCD) ^' T™ l°' ^ 0113 A' ^ c"a AWmat phing
DS: A'(l; -7; -5).
Bai 2: Trong khong gian Oxyz, cho A(0; 0; 1), B(-l; -2; 0) va 3 Tim hinh chieu H cua A len duong thing BC.
DS: H J L . _ H . _ J L 19' 19' 19
x — 2 v + 3 z
Bai 3: Cho duong thang (A): —-— = - - = — Tim diem M ' doi ximg cua
M(l; 3; 5) qua (A).
DS:M'(-5; - 1 ; 3).
Bai 4: Cho tii dien OABC co A(3; 0; 0), B(0; 4; 0), C(0; 0; 5)
1) Tim toa do H la hinh chieu ciia O tren mat phang (ABC) 2) Tim toa do K la hinh chieu ciia A tren duong thang BC.
R 1200 900 720>| f 100 80 (adsbygoogle = window.adsbygoogle || []).push({});
I 769 769 769 J ;
I. 41 41
Bai 5: Cho cac diem A(4; -6; 3), B(5; -7; 3). Tim diem C tren duong thang d
qua A va vuong goc voi mat phang (P): 8x + H y + 2z - 3 = 0 ma tam
giac ABC vuong tai B.
D S : c f - i ; - J ^
I 8 3 3
Bai 6: Cho A(0; 0; -3), B(2; 0; -1). Tim diem C tren mat phing
(P): 3x - 8y + 7z - 1 = 0 sao cho tam giac ABC la tam giac deu.
D S : C ( 2 ; - 2 ; - 3 ) , C ( - | ; - | ; - | )
Bai 7: Trong khong gian Oxyz, cho mat phang (a): x + y + z - 4 = 0 va ba diem:
A(3; 0; 0), B(0; -6; 0), C(0; 0; 6). Goi G la trpng tam ciia tam giac ABC.
1) Tim toa do hinh chieu vuong goc H ciia diem G tren (a).
2) Tim diem M thuoc (a) sao cho |MA + MB + MC | nho nhit.
DS: 1) H(2; - 1 ; 3) 2) M(2; - 1 ; 3).
Bai 8: Tim diem doi xiing ciia M(3;0;-l) qua duong thang
(j. x - l _ y + 4 _ z - 3
' 1 ~ 2 ~ 3
H D : diing quan he vecto.
Bai 9: Tim diem doi xiing ciia A(2; 3; 2) qua mat phang
(P): x + 2 y- 3 z + l = 0
H D : diing quan he tuong giao.
Bai 10: Cho tii dien ABCD voi A(2, 3, 1), B(4, 1, -2), C(6, 3, 7), D (-5, -4, 8).
Tim hinh chieu D len mat phang (ABC).
DS:
7 7 7
Bai 11: Tinh khoang each tir diem A ( l ; 2; 1) den duong thing (d):
* = I _ _ i = z + 3 3 4
DS: d(A;(d)) 347
26 (adsbygoogle = window.adsbygoogle || []).push({});
Bai 12: Trong khong gian Oxyz, cho A(2; 0; 0), B(0; 0; 8) va diem C sao
cho AC = (0; 6; 0). Tinh khoang each tu trung diem I ciia BC den duong
thang OA.
Bai 13: Cho tii dien SABC voi cac dinh S(-2; 2; 4), A(-2; 2; 0), B(-5; 2; 0),
C(-2; 1; 1). Tinh khoang each giira hai canh doi SA va BC.
DS: d=^0
10
Bai 14: Cho hai duong thang song song
/ J N X + 7 y - 5 z - 9 , f A v x y + 4 z + 18
(di) = = va (d2): - = =
v
' 3 - 1 4 3 - 1 4
Tinh khoang each giua (di) va (d2)
DS: d((d,);(d2)) =25
Bai 15: Cho duong thang (d): ^ 1 = J L = 111 v a hai diem A(3; 0; 2),
3 2 1
B( l ; 2; 1). Ke AA', BB' vuong goc voi duong thang (d). Tinh do dai doan thang A'B'.
DS: 1 1
x + 1 v + 2 z 4
Bai 16: Cho d: = = va mat phang (P): x + y + z - 3 = 0. 1 2 2
Tim toa do giao diem ciia d voi (P) va tinh goc giua d va (P). DS: M ( l ; 2; 0), sina = - i - .
3v3
Bai 17: Trong khong gian Oxyz, cho tii dien ABCD. Tinh goc va khoang each giua hai canh A B va CD biet A(3; - 1 ; 0), B(0; - 7 ; 3), C(-2; 1; -1) vaD(3; 2; 6).
DS: A B 1 CD va: d(AB; CD) = -j¥L.
V162 Bai 18: Cho duong thang (d) va mat phang (P):
(d): x = 1 + 2t, y = 2 - t, z = 3t (P): 2 x - y - 2 z + l = 0
1) Tim toa do cac diem M thuoc duong thang (d) sao cho khoang each tir moi diem do den mat phang (P) bang 1.
2) Xac dinh toa do ciia diem K doi ximg voi 1(2; - 1 ; 3) qua during thang (d) DS: 1) Mi(9; -2; 12), M2(-3; 4; -6) 2) K(4; 3; 3)
Bai 19: Cho AQ; 1; 0) va B(3; - 1 ; 4). Tim diem I tren duong thang (d): x + 1 = -— - = z + - sao cho IA + IB nho nhat.
BS: 1(1; - 1 ; 2). (adsbygoogle = window.adsbygoogle || []).push({});
Bai 20: Cho A ( - l ; 3; -2) va B(-9; 4; 9) va mat phang (P): 2 x - y + z + l = 0. Tim diem K e (P) sao cho A K + B K nho nhat.
DS: K ( - l ; 2; 3).
Bai 21: Trong khong gian Oxyz cho A ( l ; 1; 2), B(2; 4; 3). Tim toa do cac diem
M e (Oxz) va N e (Oxy) dk AM + M N + NB co gia tri be nhat.
DS: M l | ; 0; l j . N(3; 1; 0). Bai 22: Cho A (-4, 4, 0), B (2, 0, 4),
den AB. DS: 7 l 3
C (1, 2, -1). Tinh khoang each tir C
z 3
Bai 23:Tinh khoang each tir M(3, 2, 4) den A : = -
2 DS:
2
Bai 24: Tinh khoang each giira 2 duong thang
x + 1 y - 2 z- 3 , x + 1 y + 2 z + 3
= z = va = =
1 1 2 1 1 2
H D : 2 duong thang song song.
Bai 25:Tinh khoang each giua 2 mat phang
x + 2y + 2z - 1 = 0 va x + 2y + 2z + 7 = 0
H D : 2 mat phang song song
Bai 26: T u dien ABCD co A (1, 2, -1), B(0, 1, 3), C(2, 0, 4), D ( - l , 2, 0). Tinh goc, khoang each giira A B va CD
H D : diing cong thuc.
Bai 27:Tinh goc giua duong thang. mat phang:
1) a: x--72y + z- 4 = 0 va (3: x - Vzy - z + 5 = 0
x = 1
2) d:<y = 2 + t v a a : x + z + 4 = 0
z = 3 + t
DS: 1) 60° 2) 30°
Bai 28: Cho tir dien A( l , 1, 3) ,B(6, - 1 , 0), C(0, 2, 3) ,D(-1, 3, 7).
Tinh cos cua goc hop boi duong thang A D va BC; (ABC) H D : xac dinh cac vecto chi phuong. vecto phap tuyen
x- 2 y + l z (adsbygoogle = window.adsbygoogle || []).push({});
sin a - 1
Bai 29:Chung minh duong thang d: mot goc khong doi
DS: 45°
Bai 30: Tim dieu kien de 2 mat phang sau vuong goc 5x + y - 3z - 2 = 0 va 2x + my - 3z + 11 = 0 BS; m = -19
Bai 31: Tu dien ABCD vai A(2, 3, 1), B(4, 1, -2), C(6, 3, 7), D (-5, - 4 , 8). Tinh goc giua 2 duong thang AB, CD va ban kinh mat cau ngoai tiep.
11 _ V447
DS: cos(AB,MN) ;R
3V323' 4
Bai 32: Cho A (3, 5, -4), B (-1, 1, 2), C(-5, -5, -2) va M (5, 1, 5), N (4, 3, 2), P (-3, - 2 , 1). Goi G, E, F la trong tam tam giac A B C , MNP va tu dien MABC. Tinh tan goc EGF
DS: tan a = V2050
392
Bai 33: Cho A (4, 3, -1) va a: 3x + 2y + z - 12 = O.Tim M e a d e A M be nhat.
H D : M la hinh chieu cua A len a
Bai 34:Cho A (1, - 2 , 11), B ( - 1 , 1, 2) va a: x + 2y - 2z + 1 = 0. Tim M e a
de M A + M B be nhat.
H D : xet vi tri tuong doi ciia A, B voi a
Bai 35: Trong khong gian vdi he toa do Oxyz, cho 3 diem A(0; 1; 1), B( l ; 0; 0), CQ; 2; -1).
1) Viet phuong trinh mat phang (a) qua A , B, C.
2) Viet phuong trinh mat phing (p) qua D(0; 1; 0) biet rang giao tuyen
x - l y + 2 z - 1 ciia (a) va (P) la d: _ DS: 1) 3x + y + 2z - 3 -2 -2 0 2) 2x + y + z - 1 = 0. i i y 2 Z + 6
Bai 36: Lap phuong trinh mat phang (Q) chiia dudng thang d: - =- — = va 2 1 4 vuong goc vdi mat phang (P): 2 x - y + 2 z- 3 = 0
DS: x + 2y - 4 = 0.
Bai 37: Trong khong gian vdi he toa do Oxyz, cho hai dudng thang cheo nhau
fx = - t
x - l y - 7 z - 3 (di):
1 va (d2): : -4 - 2t
1 + 1
Chiing minh (di) _L (d2) va lap phuong trinh cac mat phang chua dudng
thang nay va vuong goc vdi dudng thang kia.
DS: (a): x + 2y - z - 12 = 0 va (p): 2x + y + 4z = 0. (adsbygoogle = window.adsbygoogle || []).push({});
Bai 38: Trong khong gian Oxyz, cho hinh lap phuong ABCD.A'B'C'D' vdi
A(0; 0; 0), BQ; 0; 0), D(0; 1; 0), A'(0; 0; 1). Goi M , N la trung diem ciia
AB, CD.
1) Tinh khoang each giua 2 dudng thang A'C va M N .
2) Viet phuong trinh mat phang chua A'C va tao vdi Oxy goc a ma
J_
s
DS: 1) d(A'C, MN)
cosa =
Bai 39: Viet phucmg trinh mat phang each deu hai ducmg thang cheo nhau
(d): x = 2 + t, y = 1 - t, z = 2t va (d'): x = -2t' y = 3, z = 1+ t'
BS: x + 5y + 2z - 12 = 0.
Bai 40: Trong khong gian voi he toa do Oxyz cho duong thang d la giao
tuyen cua 2 mat phang (P) va (Q) lan luot co phuong trinh:
(P): x - 2 y - z - 2 = 0 ; (Q): x + 2y - 4 = 0
Tim diem doi xung cua goc toa do O qua duong thang d.
104 32 -44'
2 11
DS: O'
21 21 21
Bai 41: Trong khong gian Oxyz, cho diem A(0; 1; 2) va 2 duong thang x = 1 + t
j x y - l z + 1 . , di: - = - = ; d2: ^ y = - l- 2 t
2 1 - 1 2 y
z = 2 + t
1) Viet phuong trinh mat phang (P) qua A, song song voi di va d2.
2) Tim M thuoc di, N thuoc d2 sao cho A, M , N thang hang.
DS: 1) x + 3y + 5z - 13 = 0 2) M(0; 1; -1), N(0; 1; 1)
Bai 42: Trong khong gian Oxyz cho 2 duong thang:
x = 2t fx = 1 + t '
A, : < y = -2 + 3t ; A2 : j y = 2 + t (adsbygoogle = window.adsbygoogle || []).push({});
z = 4t [z = 1 + 2t
1) Viet phuong trinh mat phang (P) chua Ai va song song A2
2) Cho diem M(2; 1; 4). Tim toa do diem H thuoc A2 sao cho doan M H co do dai be nhat.
DS: 1) 2 x - z = 0 2 ) H ( 2 ; 3; 3 )
x = - t
Bai 43: Viet phuong trinh mat phang (Q) di qua duong thang (d): < y = 2t -1 va
z = t + 2
tao voi mat phang (P): 2x - y - 2z - 2 = 0 mot goc nho nhat.
DS: x + y - z + 3 = 0.
Bai 44: Trong khong gian voi he toa do Oxyz. cho diem A(2; 5; 3) va dudng ]. x _ 1 _ Z = z ~ 2
1 thang d:
2 1 2
1) Tim toa do hinh chieu vuong goc cua A tren dudng thang d.
2) Viet phuong trinh mat phang (a) chua d sao cho khoang each tir A den (a) ldn nhat.
DS: 1) H ( 3 ; l ; 4 ) 2) x - 4y + z - 3 = 0. [x = l - 2 t
Bai 45:Cho I (2. - 1 , 1) va d: <y = - 1 -t . Lap phuong trinh mat phang a qua |z = 2t
I va vuong goc d.
Bai 46: Lap phucmg trinh mat phang chua ducmg thang: d
,. x-l y + 3 z-1
song song d : — j — = — — = ——
HD: VTPT n = [u, u~'].
Bai 47:Lap phuong trinh mat phang chua 2 duong thang:
5t 2 - 2t va 1 - 2t x = l + t y = 2 - t va \ z = 2t x = -5 y = - t z = 3 + 2t t
H D : 2 ducmg thang song song.
Bai 48: Trong khong gian voi he toa do Oxyz, cho diem H ( l ; 2; -1) va
ducmg thang d: y - 3 Lap phuong trinh duong thang A di
1 3 2
qua diem H, cat ducmg thang d va song song voi mat phang (a):