62. Cho hinh tru c6 cac day la hinh tron tarn O vh tarn O', ban kinh ddy bang
chi^u cao va bang a.
Tren ducmg tron day tarn O la'y diem A, trdn ducmg tron day tarn O' la'y
diem B sao cho AB = 2a.
Tinh the tfch kh6'i tir dien OO'AB.
(Trich de thi Dai hoc - Khoi A - 2006).

ChUffng in
PHl/ONG P H A P T O A D O T R O N G K H O N G G I A N
I. Ht TOA DO TRONG KHONG GIAN
A. L t THUYfeT C A N N H 6
1. Toa do cua diem va cua vecta
/. He tog. do: Trong khong
gian CO ba true toa d6 vuong goe
vdi nhau doi mot va c6 dinh hudng
Ba true nhu vay duac goi la ht
toa d6 vuong goc trong kh6ng gian
Didm O goi la g6'c toa d6
- True hoanh, dinh hudmg
duang X Ox, c6 vecta don vi
i = (l;0;0)

o

Hinh 34

True tung, dinh hudng duong y'Oy, eo vecta dan vi
} =(0; 1;0)
f True cao, dinh hudng duang z'Oz, c6 vecta dan vi
^=(0;0;1)
- Cac mat phang xOy, yOx, xOz d6i m6t vudng goc vdi nhau duac goi


0

=

(0; 0;

cho

m6t diem M tuy y, m6i diem
hoan toan xac dinh boi vecta

a) a = h «

c) a va b ciing phuong

M

d) Trong khong

OM

gian

c6 mot so k: a, = kb,, aj = kbj, a, = kbj.
Oxyz

c6 A ( a , ; a2; a,) va B(b,; b2; b,) thi

.45 = ( b - a , ; b2-a2; b j - a j ) .

35

va A (b,; bj; h^) la m Or sty duac xac dinh bcfi cong thiic:

Nguoc lai vdfi (x, y, z) ta c6 didm M duy nhat trong khong gian thoa man
OM

=

X.

a .b = a,b| + a2b2 + ajbj

i + y.j + z.k

Bp ba so (x, y, z) gpi la toa dp cua diem M doi v6i he true Oxyz da cho
va k i hieu la M ( x , y, z) hoac M = (x, y, z)
3. Toq do cua

. .

.

2) Ifng

dung

* Do dai cua vecto

vecta

hieu la u (a,, aj, a,)

cos 9 =

2 . Bieu thirc toa do cua cac phep toan vecto

S ( b i ; b2; bj) k h i do
± b = (a,± b,; a2± bj; a j i b,)

b) k. a = (ka,; kaj; k-a,) = k(a,; di^; aj) (k la so thuc)

a,b, +a2b2 +a3b3
^/af+af+a^

-^/bf+bf+b

=> a J . b <=> aib] + a2b2 + a3b3 = 0

/. Dinh U. Trong kldong gian Oxyz cho cac vec ta a (a,; aa; aj)

d) a

A ( X A ; yA; Z A ) . B ( X B ; yg; Zg) la

4. T i c h CO hudng cua hai vecto (hay tich vecto)
/) Dinh

nghia:

Tich c6 hu6ng (hay tich vecto) cua hai vecta a (ai, a2, a,)

3i)Chvtngm\nh: AD+ BC = BD + AC
b) Goi M la die'm chia dudng trung tuyen AA, ciia mat phang ABC theo
so 3: 7 (
= — ). Chung minh rang:
MA, 7

3)Apdung
J. Tick dien tick cua hinh binh hanh va thetich khoi hdp
• A B C D la hinh binh hanh SABCD = A B . A D sin A
AB,AD

• A B C D A B C ' D ' la hinh

hop

• VABCD.A'B'C'D'= T A B , A D

.AA

Ba vecta a, c, bdong phang <=> a,b .c = 0
Ba vecta a, b, c khdng d6ng phing
10 '
1
Ma DA, = -(DB
+ DC)
T>oA6 DM = — DA — DB + — DC.
10
20
20

X-^'-

-V-


Cdnh 2: Bien ddi ve phai:

= - - b+a
2

— DA ^ — DB ^ — DC
10
20
20

= - — b + a + -{c2
2

+ M^)+ —(Z)M+ MB)+—-{DM

= --(DM

4

= ld^^MN =
2

MN

0
•

20

2

2

—(a-b+c)

1
- {a^+ b+c+l{-a.b
4

+ a.c-b

c)

^

dV2


a.b = c.a = b.c = a

cos 60"= -d\
2

1) MN = MA + AB+ BN
1 r - 1
= - - b +a+- BD
2
2

1BD.

Vi du 3.
U m toa d6 hinh chie'u cua die'm A ( l ; - 3 ; -5) trdn:
1) mp Oxy;

2) mp Oxz

3) mp Oyz;

4) True hoanh;

5) True tung

6) True cao.
A

1


chieu ciia A la A^iO; 0; -5).
Vi du 4: Cho A(-3; 2; -1). Tim toa do diem doi xiing cua A qua gdc toa

Giai he: P'^-y^^ ^
• [x + 2 y = - 4

1^ = 2
[y=-3

c , (2; - 3 ; 0)

Vi du 7.
Cho tii dien ABCD c6 A ( l ; -2; -1), B(-5; 10; -1), C(4; 1; 11), D(-8; -2; 2).
ViS't phuang trinh mat ciu ngoai tie'p tii dien ABCD.
Giai:
Goi I(x; y; z) la tarn mat c&u ngoai tie'p tii dien ABCD, thi ta phai c6:

d6, qua eac true toa do, qua cac mat phang toa d6.
IA = IB = IC = ID
Giai: Qua gdc toa do: toa do diem doi xiJng ciia A la: (3; - 2 ; 1).
IA=IB
Qua true hoanh x' Ox: toa dd di^m ddi xiing ciia A la: (-3; 2; -1).
Qua true tung y'Oy: toa do di^m ddi xiing etia A la: (3; 2; 1).

IA=IC

(1)

IA=ID



Suy ra

2=1-X

X

-3 = - 1 - y

y =2

-l = 2-z

z=3

IC=

=-1
A ( - l ; 2; 3)

(4-x; l - y ; l l - z )

IC = V ( 4 - ; c ) ' + ( ! - > ' ) ' + ( 1 1 - ^ ) '

A1


ID = ( - 8 - x ; - 2 - y ; 2-z)
ID



ngoai tiep t i i dien A B C D c6 tarn I(-2;4;5) va c6 ban kinh r = l A = 9

Tinh ^AO (O la trong tam ciia mat BCD ciia hinh t i i dien).
G i a i : a. Neu bon d i ^ m A , B, C, D la b6n dinh cua m6t hinh tir diSn thi

ba vector A B , AC, AD khong ddng phang.

Vay phuong trinh mat cau la:

A B = (-4;4;1), AC = (2;2;2) = 2 ( 1 ; 1 ; 1 ) , A D = ( - 3 ; - 5 ; 3 )

(x+2)' + ( y - 4 ) ' + ( y - 5 ) '

=81

Cdch I: Xet bieu thirc

Vidu8
Cho a = (3,;2;2)va 6 = (18;-22;-5). T i m cbie't

= 14,c 1 a,c

c tao vdfi true tung goc t u .

±bva

-4

1


Giai

Cdch 2. Ta khong tim dugc cap so x, y thoa man A B = xAC

Goi c = (x; y; z)
V i : fl 1 c

< 0 < = > y < 0 nen chon y = - 6

Dap so: c = (-4; -6;12)

- 2 y + 10 = 0

3x-z + ll = 0
y = 4

z = -2y
y = ±6

V i c tao vdi true tung goc til nen c.j

zf



+ 3y

he nay v6 nghiem

+ yAD,


67. Toa d6 trung diem cac canh cua tam giac ABC la (1;3;2), (0;2;0).
(2; -2; 4). Tim toa do cua cac dinh tam giac ABC.

b. Goi 0(x, y, z) ta c6:
AO = AB + BO

68. Tim tren true hoanh mot diem each deu hai diem A(l; -3;7) va B(5;7;- -5).
AO = AC + CO
69. AABC CO A(l;2; -1), B(2; -1;3), C(-4;7;5). Tim d6 dai ducmg phan
AO=AD+DO

giac trong BD.

3AO = AB + AC + AD-(OB + OC + OD).
Vi O la irong tarn tarn giac BCD ntn OB + OD + OC = 0.
1
1
Suyra: AO =-{AB + AC + AD) =-(-5;l;5)

70. AABC CO A(-4; -1;2), B(3;5; -10). Tim toa d6 dinh C bie't trung di^m
canh AC thu6c true tung,^ trung diem canh BC thu6c mpOxz.
71. AABC CO A(6;2;3), goc toa d6 la trung diem canh AC. Trong tam G ciia

64. Cho tii dien ABCD. Goi A', B', C , D' la cac di^m theo thir tu chia cac
doan thang AB, BC, CD, DA theo ty s6 k:

75. Cho lang tru diing ABCA,B,C, c6 day ABC la tam giac vudng
AB = A C = a, AA, =. aV2 . Goi M, N 1^ lugt la trung diem ciia doan

A'A

BB

CC

DD

AB

BC

CD

DA

AA, va BC,. Chiing minh MN la dirdng vu6ng goc chung cua cac ducmg

=k

1. CMR vdfi moi di^m O bat ky trong kh6ng gian, ta lu6n c6:
OA + OB + OC + OD = OA' +OB+OC'

+OD'


J

Kh6ng CO mat x va y (A = 0, B = 0) thi mat phing do song song hoac
Itrung voi mat phang Oxy. Tuong tu mat phang Ax +D = 0 song song hoac
ttrung vdi mat phang Oyz, mat phang By + D = 0 song song hoac triing v6i
lat phang Oxz
f

* Ne'u A,B,C,D khac 0, bang each dat a = - — , b = - — , c = - — ta c6
A.
B
C
\ dua (1) \i dang: - + ^ + - = 1(2)

a b e


* Neu D = 0 thi (or) di qua gd'c toa do va ngugfc lai.
* Neu trong phuomg trinh (1) khong c6 mat x(A = 0) thi mat phang
AXQ + Byo +Czo+D
tuong ung se song song hoac chiia true Ox.
d(Mo,a) =
^A'+B'+C
Tuomg tu vdri y va z.
* Khoang each gifia hai mat phang song song la khoang each tiir m6t.
* N6'u plijong trinh mat phang c6 dang Cz + D = 0
diim bat ky cua mat phang nay den mat phang kia.
49


'

B. Vf DU

Mat phdng (R) vu6ng goc vdi hai mat phang (?) va (Q) nen nhan hai
veeto phap tuye'n ciia hai mat phang nay lam cap veeto chi phuomg. Vay

V i du 1: Viet phuong trinh mat phang di qua diem ( 2 ; - l ; - 1 ) va vu6ng

;vecto phap cua mp (R) la:

goc vdi true eao.

r

0



-1

0

= (0; 1; 1)
Mat phang (R) di qua A(-l;2;3) nen phuong trinh mat phang (R) la:

V i du 2:

0(x + l ) + ( y - 2 ) + ( z - 3 ) = 0

Viet phuong trinh mat phang qua ba diem
c:>y + z - 5 = 0
M(3;-1;2);N(4;-1;-1);Q(2;0;2)
Vi du 4:
Giai:

Trong kh6ng gian vdi he true toa dd Oxyz, cho tii dien ABCD vdi

Veeto phap tuyen ciia mp (MNQ) la:
—

0

-3

(2; -1; 6); B(-3; -1; -4); C(5; -1; 0), va D ( l ; 2; 1).

-3

1.

Trong khdng gian Oxyz cho di^m A ( - l ; 2; 3) va cac mat phang

=>C5.C4 = (-3).8 + 0.0 + 6.4 =0

(P):x-2 = 0 , ( Q ) : y - z - l =0

C4 1 Cfi nen A ABC vudng tai C.

Viet phuong trinh mat phang (R) di qua di^m A va vuong goc vdi hai
mat phang (?) va (Q).
(Trich de thi vao D H LuSt Ha Noi, 1999)
Giai: Veeto phap tuye'n ciia (P):
Veeto phap tuye'n cua (Q):

= (-5; 0; -10); CA = (-3; 0; 6);Cfl = (8; 0; 4)

= (1; 0; 0)

CA = 7(-3)'+0'+6' =3V5
CB = V 8 ' + 0 ' + 4 ' = 4V5, AB = 5V5
vay

SABG = ^ .

3V5 . 4V5

= 30


Khoang each tir D den mp (ABC) la:
2 +1
d(D, (ABC)) = Vo'+i'+o'

Dap so:
Vi du 6:

z„ = - 2
z„ = - 4 4

M,(0;0;-2) va M2 (0;0;-44)

Tim tap hop nhung di^m M(x,y,z) each mp (P):
4x - 4y - 2z + 3 = 0 m6t khoang bang 2.
Giai:

= 3

The tfch tir didn DABC la:
j3.30 = 30(dvtt)

Vr du 5:

Tim mot diem \xtn true cao each d6u diem A(l; -2;0) va mp
(P): 3x - 2y + 6z - 9 = 0.
Giai:

Goi M € Oz CO toa dp M(0; 0; z^). Ta c6:
MA = (1; -2; -Zo)


(Q):5x-3y + z-18

=0

(Q): 3x + 2 y - z + 3 = 0

2. (P): 2x + 3 y - 5 z - 1 5

A'x + B'y + C'z + D ' = 0
A'+B'+C^^

ditn (P; Q) hay nam trong hai goc ke nhau cua nhi dien nay?
=0

'Ax + By + Cz + D = 0
Do do (d):

89. Xet xem dia'm A(2; - 1 ; 3) va goc toa do O ciing n^m trong goc ciia nhi
1 (P): 2 x - y + 3 z - 5

NH6

Ta da biet giao tuyen cua hai mat phang phan biet cat nhau la mot dudng

86. Viet pt mp each d^^u hai mp: (P) 3x + 2y - z + 3 = 0 va (Q)
3x + 2 y - z - l

A. L t

1. Phirong trinh tong quat cua^ircmg thang

=0

92. Trong khong gian Oxyz cho hinh lang tru diing ABC.A,B|C, v6i
A(0;-3;0), B(4;0;0); C(0;3;0); B,(4;0;4).
a) Tim toa do cac dinh A, va C,. Viet phuofng trinh mat c^u c6 tarn la A
va tiep xiic vdi mat phang (BCC,B,).
b) Goi M la trung diem ciia A,B, .Viet phuong trinh mat phang (P) di
qua hai diem A, M va song song vdi BC,.

x =

x^+at

y = yo+bt
Z =

trong do a^ + b^ + c^

0

(2)

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
Tit (2) va neu a, b, c deu khac 0, khii t d cac phuong trinh tren ta c6:


XQ+at

y =

yo+bt

Z =

ZQ+Ct

^

a,a

.= 0

a,MM

^0

a,a

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


(Mo € ( A ) , a

vecto chi phuomg ciia ( A ) .

Icliido A / / ( a )

2) Khoang cdch giita dudng thing vd mat phang song song

3) Neu Aa + Bb + Cc = 0 va
nghiem Idii do A c

A' <=>

AXQ

+ Byo +

CZ^ + D

= 0 thi (*) v6 so

(a).

sau:

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:



• 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'
Cdch 2: S\x dung cong thiic: d( A ;A') = h =
a,a
(M e A, a la vecta chi phuofng ciia A. M ' e A', a'la vecto chi phuong
ciia A'.

Giai: Ta c6:
+

x + y-z + 3 = 0

(1)

2x - y + 5z - 4 = 0

(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



z = - l + 2t
x-2 = -t
Giai: Ta c6:

x-2

^ =t
2
z+1
=t

_y

2
z+1_ y

»

2x +

y-A-Q

[y-z-\ 0

I 5 x - 7 y + 2 z - 3 = 0vampOxy
^ Giai: Mat phang Oxy c6 vecto phap tuyen k = (0; 0; 1) va di qua goc
toa d6 nen c6 phuorng trinh: z = 0.

V i du 2.


.

VfduS.
Viet phuong trinh ducmg thang song song
voi hai mp:

V i du 3.
Viet phuong trinh tham s6' (tiir do suy ra PT chmh iic) ciia ducmg thang
' ;c + ; ; - z + 3 = 0
biet PT t6ng quat:
l2x-;; + 5z-4 = 0

3 x + 1 2 y - 3 z - 5 = 0 , 3 x - 4 y + 9z + 7 = 0,
va cat hai ducmg thang:
x + 5 _y-3
-4

Hinh 40

_z + \' + l
3

-2

z-2


G i a i : Ducmg tiiang (A) piiai t i m la giao tuye'n cua liai mp (P) ^'a mp(Q)
M p (P) cluia (d,):

tim la: 9y + 3z + 5 = 0.

^

= (3; 12; - 3 )

= 3(1; 4; - 1 ) , va (Q) c6 cung

vecto phap tuye'n vdfi ( Q ) : 3x - 4y + 9z + 7 = 0.

V i d u 7.

.^=(3;-4;9)

Viet PT mp qua dudng thang:

rx-3jv + 7z + 36 = 0

Vecto chi phuong u^ ciia (A) la tich c6 hudng ciia n, va n ^ .

^

=^ Vecto chi phuong
4
"A

. «2 ] =

=[«,



3

3

2

2

-4

-3

-4

2

8

8

-3

= (25; 32; 26)

15

0

'^^'^^ S^'^


G i a i : M P phai t i m thuoc chCim mp:

PT mp (P) la: 25(x + 5) + 32(y - 3) + 26(z + 1) = 0
o

"^^^

85

m = n
%5m = \9n

Chon m = n = 1, ta C O mp phai t i m la: 3x - 2y + 6z + 21 = 0
Chpn m = 19, n =85, ta c6 mp phai t i m la: 189x + 28y + 48z - 591 = 0.
Vf d u 8. T i m toa do giao didm cua ducmg thang:
.x-1

y-A

z-5

va mp 3 x - y + 2 z - 5 = 0


Giai: PT ducmg thartg c6 thi vie't dudri dang tham s6:
h^7

=2
yQ=-3=^Q(2;-3;2)

2x + y - 2 z + 3 = 0

3 x - 6z+15 = 0 o

phap

tuyen

n = (3; 1; -2) ciia (P) lam vecto chi

+3=0

phucng, suy ra phuong trinh ducmg thang
x = \ 3t
AB la: • y = 3 + t . Toa do giao di^m I

z = -4-2/
(1)
(2)

ciia AB va (P) la nghiem ciia he PT:

x = -5 + 2z.

Hinh41

Dat z = t va thay vao m6t trong hai PT ciia (d), ta c6 PT tham s6:
x = -5 + 2t
(d) \ = 7 - 2t


x = -2

u = (2; - 2 ; 1). Viet PT mp (a) qua P(4;l;6) va vu6ng goc v6i (d) nSn

y=2

nhan u ciia (d) lam vecta phap tuydn.

z = -2

V a y P T m p ( a ) l a : 2 ( x - 4 ) - 2 ( y - l ) + z - 6 = 0 o 2 x - 2 y + z - 1 2 = 0.
Tim toa d6 giao diem A cua (d) va ( « ) bang each thay PT tham s6' ciia
(d) vao PT ( a ): 2(-5 + 2t) - 2(7 - 2t) + 1 - 12 = 0
t = 4. Thay lai vao PT tham s6' ciia (d) dugc toa d6 A(3; - 1 ; 4).
Theo tinh chat doi xiing thi A la trung die'm PQ, de dang c6:

t = -\

I(-2; 2; -2).

Theo tinh chat doi xiing thi I la trung di^m AB nfin de dang tinh dugc
toa do diem I .
-2 =|(1+^.)

3 = ^(4 + ^ , )
y,

=\^z,)


Vi du 11.

-1 -1

2
-1

M,M^

3

\ + 2z-2

3

2
[M1,W2]=

=0

1 1

2

2 2

- J

= 5(1;-1;-1)


= 1.1 +4.(-3) + (-3).(-3) = -.2

trSn m p 2 x - y + z - l = 0
U„U2

Giai: Ducmg thang da cho thupc chiim mp:
m(5x - 4y - 2z - 5) + n(x + 2z - 2) = 0
«(5m

+ n)x - 4my + 2(-m + n)z - 5m - 2n = 0 (P)

= V l ' + 4 ' + ( - 3 ) ' =:,V26

f A p dung c6ng thiic tinh khoang each ciia hai dufimg thang cheo n hau vao
bJli toan ta c6:

V i P 1 Q (mp Q: 2x - y + z - 1 = 0) ndn phai c6:
Hp

±

Hp

HQ

[u,,

d(D,A) =


=0
=0

V i du 12. (Dai hoc su pham thanh pho Ho Chi Minh - A, B - 20(X))
Trong khSng gian vdi he true toa d6 Oxyz cho cae ducmg thing:

x - z s i n a + cosaj = 0

( a l a t h a m so)
y-zcosa -sina = 0 •

1) Xac dinh vecta chi phuong ciia (d).
2) Chiing minh (d) tao vdi true Oz m6t goe kh6pg phu thu6e a.
3) Viet PT hinh ehi^i (d') ciia (d) trdn mp Oxyj
4) Chiing minh vdd moi gia tri a ducmg thing (d') lu6n tiS'p xuc vdi mdt

(D,):

•

(D^):

1 "

2

"

x + 2y-3
2x-y



95. Viet PT mat phang chiia ducmg thang

x-2z^0
3x-2y

va vuong goc

+

z-3^0

v6i mat phang: x - 2y | f z + 5 = 0.
x^7
x-1
2

96. Chimg to rang hai ditcmg thang:

_y + 2 _
~ -3 ~

+ 3t

z-5
va < y = 2 + 2t
4
z = \-2t



1) Chung to rang hai ducmg thang do cheo nhau.
2) Tinh khoang each gifla chiing.

+ 2y + 3z + 4 = 0.

3) Vie't PT ducmg thang qua M(2; 3; 1) va cat (d,), (dz).

98. Chumg minh rang hai iducmg thang sau day cat nhau:
(d,): X = 2t - 3, y = 3l[ - 2, z = 4t + 6.

;c = 2^ +1
5. Cho hai ducmg thing : (d,)

(d^): X = t + 5, y = -4rtj - 1, z = t + 20.

=0

- X - V +i 4

=0

(da)

vaCdj)

z = 3r-3

99. Tinh khoang each giiJj^ hai ducmg thang:
2x~z-\l


z

- Y

-

•>

ducmg thang •

'x + y-z

+2=0

x+l =0

3 '~ -2

_z-2
~

2

^

•

^


2) Tinh khoang each tiir goc toa do den mp chiia hinh binh hanh.
(Dai hoc dan lap Dong D6 Ha Noi, khdi A, 1997).


112. Trong khong gian v d i h6 tea d6 Oxyz, cho mat cSu

108. A A B C CO A ( l ; 2; 5) va PT hai trung tuye'n 1^:
x-3

_ y-6

-2

2

_ z-1

. x-4

^ y-2

1

-4

1

^

z-2

z = \ 2t

a) Vie't phuong trinh mat phang (P) chiia ducmg thang A, va song song
vdi ducfng thang A2.
b) Cho diem M (2; 1; 4). T i m toa d6 diem H thu6c ducmg thang A2 sao
cho doan thang M H c6 do dai nho nha't.
(Trich de thi vao dai hoc khoi A - 2002).
110. Trong khong gian vdi he toa do Oxyz cho hinh chop S.ABCD c6 day
ABCD la hinh thoi, A C cat BD tai goc toa do O. Biet A(2; 0; 0), B(0; 1; 0),
S(0; 0; 2 \/2 ). Goi M la trung diem ciia canh SC.
a) Tinh goc va khoang each gifla hai ducmg thang SA, B M b) Gia sir mat phang ( A B M ) cat ducmg thang SD tai d i ^ m N . Tinh the'
tich khoi chop S.ABMN.
(Trich de thi vao dai hoc khoi A - 2004).
111. Trong khong gian Oxyz cho 2 du5ng t h i n g :
d,
'-2

+ y^ +

z + 2
-1

1

'x = -l + 2t
vad.,: \ = \ t
z = 3

1) Chung minh d , va 02 cheo nhau
2) Viet phucmg trinh dudng thang d vu6ng goc vdi mat phang (P):


C. a = ( 3 ; l ; 2 )

D. a = ( l ; 3 ; 2 )

A.

x-2 _y__ z + \

B.

x-2_
y _ z+1
~4~~ f 6
I

114. Cho vecto a {2; 3; -1), b(0; 1; 4), c ( l ; 0; -3). Xac dinh toa d6 cua
vecto 2 a -b - 2 c.
A.(2;5;l),

B. (2; 5; 0),

C.(l;4;l),

x+2
C.
4

A.


x-3
4

x-Z_y-6
x-3
4

+

+ 4x - 2y - 20 = 0

A.(1;-2;0);R = 5,

B. (-2; 0; 1); R = 5,

C.(-2; 1;0);R = 5,

D. (-2; 1; 0); R = 4
1..

A. Cat nhau;

>

117. Viet phuong trinh mat phang qua goc toa do 0(0; 0; 0) vk hai diem
P(4;-2; l),Q(2;4;-3)
A.

X



B. Cheo nhau;

C. Song song;

D. Triing nhau.

II. BAI TAP.
5. Cho tii dien ABCD. E, F, I theo thu tu la trung diem ciia AB, CD, EF.

C. X + 7y + lOz = 0
D. 3x + 7y + lOz = 0
118. Tinh Idioang each tir diem M ( l ; - 1 ; 2) den mat phing (P) c6 phuong
trinh lOx + lOy + 5z + 2 = 0
A.l;

y-6
2

It. Xac dinh vi tri tuong ddi cua hai ducmg thang

D. Hinh chu nhat

116. Tim tam va ban kinh hinh ciu c6 phuong trinh la:
+

x + 2 _ [y _ z + 1
D.
4
6

badiemA,B,C.
c) Thiet lap PT mat ciu ingoai tiep tii didn
(Dai hoc Bach khoa Ha ]Noi, nam 1996)
126. Cho mat ciu (S) c6 PT: (x - 1)^ + (y - 1)^ + z^ = 6 va hai duofng thing:
(di): X = 1 + 2t, y = 3 - 2t, z = 1 + 2t
(dj): X = 1 - t, y = 2 + 2t, z = 1 - 3t
Viet PT mp tidp xuc mat c^u (S) dong thdi song song vdi (dj) va (d2).
127. Trong kh6ng gian vcti l?^ toa do Oxyz cho ba diim A(l; 0; 0),
B(0; 2; 0) va C(0; 0; 3).
1) Viet 0iuong tnnh tong quat cua cac mp (OAB), (OBQ, (OCA) va (ABQ
2) Xac dinh toa d6 tam I ciia mat ciu n6i tiep tii dien OABC.
3) Tim toa d6 diem J d6'i xiing vdi I qua mat phang ABC.
(Dai hoc Hue'-2000)
128. Trong khdng gian vdi he toa d6 Oxyz, cho diem A(l; 2; 1) va ducmg
thang (d) CO PT: - = — = z + 3.
3 4
1) Vict PT mp di qua A va chiJa ducmg thang (d).
2) Tinh khoang each tilt die'm A da'n ducmg thang (d).
(Dai hoc Kien true Ha Noi, nam 1997).

.4

19. Trong khdng gian v6i ht toa d6 Oxyz cho ba diem H -;0;0
K 0;1;0 ,1

a) Viet PT giao tuya'n ciia mp (KHI) va mp x + z = 0 of dang chinh tac.
b) Tinh cosin cia goc phang tao bdi mp (KHI) va mp Oxy.
(Dai hoc Giao thdng Van tai Ha N6i, nam 1997).
T30. Cho hai ducmg thkg c6 PT: (d) j
(A): x-5.

(ViSn Dai hoc Md Ha Ndi, khdi A, nam 1997).
133. Cho hai didm A(0; 0; -3), B(2; 0; -1) va mat phang (P) cd phuong trinh
la: 3 x - 8 y + 7 z - l =0.
1) Tim toa dd giao di^m ciia dudng thang di qua hai di^m A, B vdi mat
phang (P).
2) Tim toa dd diem C nam trfen mp(P) sao cho tam giac ABC deu.
(Dai hoc Qudc gia Ha Ndi - A - 2000).
4Z

7^


4) Chiing minh rang nS'u di^u kien tren duac nghiSm diing t h i mot trong
hai so b, c nho hem a, s6' con lai 16n ban 2a.

ON TAP CUOl N A M

j39. Trong mp (?) cho du6ng thang d co dinh va mot diem c6' dinh O g d, mot
goc vuPng Oxy quay quanh O, Ox va Oy cat d tai A va B. Cho d' ± P

134. Mot hinh h6p chu nhat c6 do dai dudng cheo d, no tao vdri day goc a
va

mat ben

\dn goc

p . Chiing minh the tich hinh hop bario

d^'sin a sin /? -yjcosia + P) cos(a -


tie'p xiic

Cho tam giac din ABC canh a. Tren dudng thdng d vuong goc vdi m;ii
phang (ABC) tai A lay diem M . Goi H la true tam tam giac ABC, K I t
true tam tam giac BCM.

. Cho hinh c^u (O, R)

v6i mat phang (P). Cho hinh non (nam

1) Chiing minh rang M C 1 (BHK) va H K 1 (BMC).

bang X .

2) K h i M thay doi tren d, tim gia t r i Idn nhat cua the tich t i i dien KABC.

a) Cho X < 2R va X < h. Tinh t6ng dien tich S cua hai thiet dien. Bieu

137. Cho hinh chop t i i giac deu S.ABCD vdfi day la hinh vuong A B C D co

thiic t i m dupe co con thich hop kh6ng n€\x h < x < 2R (keo dai cac

cdng phia vdi hinh cin doi vdfi (P), day thupc (P), dudng cao h, ban kinh
day bang R. Cat hai hinh bang mp (Q) // (P), each nhau mot khoang

canh bang a. Mat ben tao vdi day mot goc 60".

i duofng sinh ciia hinh non dd chiing cat (Q).


kien can va du de B A C = I v la be + 2a^ = a (b + c).

b) T i m tren dudng thang d diem M , sao cho khogng each tir M den mat

3) Cho biet a va b + c = d, B A C = Iv. Tinh the

phang (a) bang 2 V3 .

tich cua hinh chop the(

a va d. Lap phuong trinh de tinh b, c trong truomg hop nay. T i m diei

c) Viet phuong trinh mat cin (S) co dudng kinh AB. Xet vi t r i tuong doi

kien de tinh dupe b, c.

giiia mat c^u (S) va mat phang (a).


143. Trong khdng gian vori he tea d6 Oxyz cho 4 di^m: S(2; 2; 6), A(4; 0; Oj
x=t
x + 3>'-l = 0
B(4; 4; 0), C(0; 4; 0).
A,: y = -t va A j : • [y^z-2=Q
a) Chung minh rSng hinh chop SABCO la hinh chop Hi giac d^u.
z = -A-2t
b) Tmh th^ tich cua khoi chop SABCO.
la) Vie't phucmg trinh mat phang (P) chiia A, va song song v6i A2.
c) Vie't phuang trinh mat ciu ngoai tie'p hinh chop S.ABCO.
l b ) Tinh khoang each giiia A, va Aj.

a) Viet phuang trinh mat phang (Q) di qua A, B va vuong goc v6i m.'
phang (P).
b) Goi ducmg thang A la giao tuye'n ciia hai mat phang (P) va (Q). Hay
vie't phuang trinh chinh tac cua dudng thang A.
c) Goi H la hinh chieu vu6ng goc cua A tren mat phang (P). Tim toa do
ciia diem H.
147. Trong khong gian vdi he toa 66 Oxyz cho hai du&ng thang


HLfdNG

1.

D A N GIAI - D A P

S O

a) Trong khoi da dien m6i
canh la canh chung ciia dung
2 mat
b) Cung sir dung tinh chat
tren

2.

Chia khoi lap phuofng thanh 6
khoi tii dien.

3.


(SAB)

BC

A B ' ± SC; A C X SC => SC 1

(AB'C).

(AB' 1 (SBC) nen AB' 1 B'C). Ta c6 AB' = ^
3

~

1

AB'

c) Cho S.AB'C CO chieu cao SC, day la A AB'C vuong b B' vi

^.

va A C . SC = a.AC

12
.a SC = a
ma

nen A C = ^

B'C = ^

day. Ta c6 A H = ^ . - ^
3
2

= ^ n e n

A H ' = A H tg60° =

1

A H , BC

1

AH

=>

A'AH

= 60°

=> A A ' H

= 30°

SA

3



AA'

=> BC 1 BB' => BCC'B' la hinh chG nhat.
c)

^S.ABC

= a

Vay the tich lang tru A B C . A ' B ' C =
b) BC

48

AA'

= 2AH

Vs.^sc _ SA
S.ABC

= 2a => SBCC-B' =

2a'

Tiir E ve EK 1 A A ' => BKC la thiet diSn thing cua lang tm. Ta tinh duoc

SC

B C la

ducmg xien c6 hinh chie'u tren, ( A A ' C C ) la A C => B C ' A = 30°.
=> A C i= A B . cotg30°' = AB> /3 ma A B = AC.cotg30° = b V3 => A C = 3b.

Vay

= a l Vl3 + 2a' = a'(2 + Vo

) don v i dien tich

Ve chop S.ABC. TiT B va B' ha B H 1 (SAC), B ' H ' ± (SAC). K h i do
BH
B'H'

A A , B D can ( do A , B =

BD X A , 0 . Mat khac BD 1 A C

BD 1 ( A , A O )

A,D)

That vay: ve A , K 1 A D => H K 1 A.K
a
AH
AK
coscp cos— =
. T2
AA, AH

i

1-

cos a
2 oc
cos —

AK

•
^ a sin« .
2


•

cos^a = 2a si

a
2

\

cos 2 a

H e (ABCD) va ke H M 1

_

AD (M e

A ' M H = 60°, A ' K H = 45°

A'K.

= xa,

8^.82

S2

—

— x y . s i n a .a


Vay the tich khdi h6p V = (AB.AD) sin, a . A , H =
^

I

SMEHK=

cosa

a
cos —
2

= cosa.

AA,

V = a.b.x = abc . ^ (don vi the tich).
G o i 0 0 , la giao tuyen ciia 2 mat
cheo ( A , C i C A ) va ( B , D i D B ) . Qua
l € 0 0 , ke lin lugt 2 ducfng thang
K E va M H deu vuong goc v 6 i
0 0 ] . K h i do a la goc gifia M H va
: E va M E H K la thiet dien thang
:iia h6p. Dat K E = x, M H = y t h i

a
cos a = coscp c o s y . (*)


=>

2x

A'M =

X

•

Goi O la tarn hinh vu6ng ABCD,
SO 1 (ABCD)
AD),

Goi E H la ducmg trung binh
cua ABCD, v i A D // (SBC)
=> khoang each ttr A den (SBC)
chinh la khoang each tir E
de'n (SBC). Ke E K
1
SH
EK 1 (SBC) => E K = 2a.

x