VINH I hiet ke bai giang H)NHHQC|2 -...

T R A N VINH

I hiet ke bai giang

H)NHHQC|2 TAP HAI

N H A X U A T B A N H A N0I

TRAN VINH

THIET KE BAI GIANG

H INH HOC 12 TAP 2

NHA XUAT B A N HA NOI

Chi/dNq 2

M A T NON, M A T T R U , M A T C A U

P h a n 1

Gidl THlfu CHL/CJNG

I . CAU TAG CHUONG

§ 1. Khai niem ve mat trdn xoay

§ 2. Mat eau

On tap chuang II

I. Muc dich ciia chuong

• Chuang II nham cung ca'p cho hpc sinh nhiing kien thdc co ban ve khai niem cac

khd'i trdn xoay trong khdng gian ma chii yeu la mat ndn, mat tru va mat cau.

Mat ndn trdn xoay : Day, dudng sinh va dudng trdn day.

Dien tich xung quanh va dien tich toan phan ciia mat ndn.

The tfch ciia mat ndn.

• Mat tru trdn xoay la gi ?

Dien tich xung quanh va dien tich toan phSn cua mat tru.

The tfch ciia mat tru.

• Mat eSu la gi ?

Dien tfch ciia mat cau.

Thi tich ciia mat cau.

II. MUC TIEU

1. Kien thiirc

Nam dupe toan bp kien thiic ca ban trong chuong da neu tren.

Hieu cac khai niem cac mat trdn xoay: Mat ndn, mat tru va mat eau.

Nam dupe eac cdng thiic tfnh dien tfch, the tfch cua eac mat trdn xoay.

2. KI nang

Tinh dupe dien tfch xung quanh, dien tfch toan phan ciia cac hinh trdn xoay.

Tfnh dupe the tfch cua hinh lang tru, hinh ndn.

3. Thai do

Hpc xong chuong nay hpc sinh se lien he dupe vdi nhieu van de thuc te sinh ddng,

lien he dupe vdi nhiing van de hinh hpc da hpc d Idp dudi, md ra mdt each nhin

mdi ve hinh hpc. Tii dd, cac em cd the tu minh sang tao ra nhiing bai toan hoac

nhiing dang toan mdi.

Ket ludn:

Khi hpc xong chuong nay hpc sinh can lam tdt eac bai tap trong sach giao khoa va

lam dupe cac bai kiem tra trong chuong.

P h a n 2

cAc &Al SOAN

{Ti^p theo)

§2. Mat cau

(tiet 6, 7, 8, 9, 10)

I. MUC TIEU

1. Kien thurc

HS nam duoc:

1. Khai niem chung vd mat cau.

2. Diem thudc mat cau , diem d trong va diem d ngoai mat cSu.

3. Giao cua mat cau va mat phang.

4. Giao cua mat cdu va dudng thang.

5. Tie'p tuye'n ciia mat cau.

6. The tfch va dien tich cua mat c^u.

2. KI nang

• Ve thanh thao cac mat cau.

• Xac dinh dupe mdt mat phang la tie'p dien cua mat cau, mot mat phang la

tie'p tuye'n cua mat cau.

• Xac dinh dupe vi trf tuong ddi cua mat phang va mat eau.

• Tfnh dupe the tfch va dien tfch cua mat cau.

3. Thai do

• Lien he dupe vdi nhieu van de thuc te' trong khdng gian.

• Cd nhieu sang tao trong hinh hpc.

• Hiing thii trong hpc tap, tfch cue phat huy tfnh dpc lap trong hpc tap.

II. CHUAN BI CUA GV VA HS

1. Chuan bi ciia GV:

• Hinh ve 2.15 de'n 2.26.

• Thudc ke, phan mau,...

2. Chuan bi cua HS :

Dpc bai trudc d nha, cd the lien he cac phep bien hinh da hpc d Idp dudi

in. DHAN PHOI TH6I LUONG

Bai dupe chia thanh 5 tiet:

Tie't 1: Tir dau den het phan I

Tie't 2: Tie'p theo de'n het phan II

Tie't 3: Tie'p theo den het phan III

Tie't 4: Tie'p theo den het phan IV

Tie't 5: Hudng din bai tap

IV. TIEN TDlNH DAY HOC

n. DAT VAN D€

Cau hdi 1.

Hinh trdn xoay la gi ? Hay ke mdt vai hinh trdn xoay da hpc.

Cau hdi 2.

Hau ndu each tao ra hinh ndn va hinh tru.

C^u hdi 3.

Mat tru trdn xoay va hinh tru gidng va khac nhau d nhihig diem nao?

R. Bni 1V161

HOATDONCl

I. MAT CAU VA KHAI NIEM L I £ N QUAN DEN MAT CAU

1. Mat cau

GV neu cau hdi :

HI. Em hay neu khai niem hinh cau theo each nghi ciia minh.

H2. Qua dia ciu cd phai la hinh ciu hay khdng?

• GV su dung hinh 2.14 trong SGK va dat van de:

H3. Cho hai diem M^ va M2, hay so sanh OM^ va OMj.

H4. Khi cat mdt mat eau va mdt dudng thing di qua O, ta dupe nhiing doan thang.

Hay so sanh cac doan thang dd.

H5. Mdt doan thang di qua O eat mat cau tai A va B. Hay neu vi trf ciia O ddi vdi

AvaB.

H6. Mdt doan thang di qua O cat mat cau tai A va B. Mdt diem M bat ki tren mat

cau dd. Hay do gdc AMB .

H7. Neu dinh nghia mat cau theo y ciia em.

• GV neu dinh nghia :

Tap hgp nhirng diem cdch deu mgt diem O cho trUdc mgt khodng khdng

ddi la mgt mat cdu.

H8. Hay neu kf hieu dudng trdn tam O ban kfnh R.

H9. Day cung cua dudng trdn la gi ?

HIO. Hay ndu tfnh chat ciia day cung Idn nhat.

• GV neu dinh nghia dudng kfnh cua dudng trdn:

Dudng kinh cua dudng trdn la ddy cung di qua O.

H I 1. Hay so sanh dp dai eiia dudng kfnh vdi ban kfnh, vdi day cung bat ki?

H12. Khi bie't dudng kfnh ciia mat cau thi cd xac dinh dupe tam va ban kinh cua

mat cau hay khdng?

• GV neu su xac dinh cua mat cau:

Mgt mat cdu xdc dinh khi bie't tdm vd bdn kinh hoac dudng kinh ciia nd.

2. Diem nam trong va nam ngoai mat cau. Khd'i cau.

HI3. M thudc mat cau (O ; R). Hay so sanh OM va R.

H14. M d ben trong mat cau (O ; R). Hay so sanh OM va R.

H15. M d ben ngoai mat cau (O ; R). Hay so sanh OM va R.

• GV neu cac khai niem diem nam trong, nam tren va nam ngoai mat cau va cho

HS dien vao bang sau:

OM

R

Vi trf

3

5

3

Tren

7

6

9

8

• GV neu dinh nghia khd'i eau

Tap hgp tdt cd cdc diem thudc mat cdu S(0; R) vd tdt cd cdc diem d trong

mat cdu ggi la khdi cdu.

HI6. Danh dau x vao 6 ma M thuoc khd'i cau

OM

R

Thudc khd'i

cau

3

3

3

4

7

6

9

9

3. Bieu di^n mat cau

• GV neu eac bieu dien khd'i cau theo y sau

Ve mdt dudng trdn.

Tam mat cau la tam dudng trdn.

8

- Cac mat phang cdn lai cat mat cau la mdt hinh dupe bieu dien la hinh Elip.

Tham khao hinh 2.16

• GV neu qua ve each bieu dien mat cau nhd phep chie'u .

HI7. Hay ve mdt mat cau di qua ba diem A, B va C. Mat cau dd cd duy nhat

khdng?

4. Dudng kinh tuye'n va vl tuye'n cua mat cau

HIS. Hay neu khai niem dudng kinh tuye'n va vT tuye'n trong dia If.

• GV sii dung hinh 2.17 va dat cac cau hdi:

HI9. Kinh tuyen la ca dudng trdn Idn hay niia dudng trdn Idn?

H20. Hay ve kinh tuye'n va vi tuye'n.

• GV neu dinh nghia kinh tuye'n va vl tuye'n:

Giao CIM mat cdu vd nira mat phdng cd bd la true cua mat cdu ggi la kinh tuye'n.

Giao ciia mat cdu vd mat phdng vudng gdc vdi true ciia nd ggi la vT tuye'n.

• Thuc hien ^ 1 trong 4 phiit.

Hoat ddng cua GV

Cdu hdi 1

Tam giac ABC cd dac diem gi?

Cdu hdi 2

Hoat ddng ciia HS

Ggi y trd Idi cdu hdi 1

Tam giac can


Gpi H la trung diem AB ; OH cd dac diem gi ?

Cdu hoi 3

0 thudc mat phang nao ?

OH 1 AB.

Goi y trd Idi cdu hoi 3

Mat phang trung true ciia AB.

HOATDONC 2

II. GIAO CUA MAT CAU VA MAT PHANG

GV sii dung hinh 2.18.

Dat OH = h ;

1. Trudng hpp h > r

H21.SosanhOMvar.

H22. M nam trong hay ngoai mat cau?

• GV ket luan M nam ngoai mat ciu va giao ciia mat ciu va mat phang la khdng cd.

2. Trudng hpfp h = r

H23. H thuoc S. Diing hay sai ?

H24. M 7 H thi M khdng thudc S. Diing hay sai.

• GV ket luan :

Giao cua mat cau S va mat phang P: Mat phang (P) tiep xiic vdi mat cau.

• Dua vao hinh 2.19 GV dua ra cac cau hdi sau:

H25. Mpi dudng thing thudc (O) khong di qua H cd giao vdi mat ciu hay khdng?

• GV neu dinh If:

Dieu kien cdn vd du de mat phang (P) tie'p xuc vdi mat cdu S(0 ; r) tgi H

la (P) vudng gdc vdi bdn kinh OH tgi H.

H26. Mpi mat phing vudng gdc vdi OH deu tiep xiic vdi S. Diing hay sai ?

H27. Chiing minh khi (P) tie'p xiic vdi S tai H thi OH < OM.

10

2. Trudng hop h < r

Trong hinh 2.20

H28. H thudc mien trong cua S. Diing hay sai ?

H29. M thudc (P), M thuoc S thi OM cd dac diem gi ?

H30. Cho OH = h tfnh HM.

• GV ket luan :

Giao cua mat ciu S va mat phang P la dudng trdn tam H ban kfnh r' - yr -h

H31. Khi h = 0, tim tam dudng trdn la giao cua (P) va (S).

• GV neu ket luan

Mat phdng di qua tdm ggi la mat phdng kinh. Giao ciia (P) vd (S) Id

dudng trdn vd ggi la dudng trdn Idn.

• Thuc hien A2 trong 5'

cau a

Hoat ddng cua GV

Cdu hdi 1

Cho bie't dp dai OH'.

Cdu hdi 2

Cho bie't OM

Hoat ddng cua HS

Ggi y trd Idi cdu hoi 1

O H ' = -2


OM = r

11

Cdu hdi 3

Tfnh MH'


V 4 2

cau b

Hoat ddng cua GV

Cdu hdi 1

Ta cd OH' = a, OK = b so sanh OH va OK.

Cdu hdi 2

Tfnh H'M va KN.

Cdu hdi 3

So sanh H'M va KN.

Hoat ddng cua HS


OH < OK.


HS tu tfnh

Ggi y trd loi cdu hoi 3

H'M > KN.

• GV ket luan :

Mat phang cang gan tam thi dudng trdn giao cd ban kinh cang Idn

12

HOATDONC3

III. GIAO CUA MAT CAU VA D U O N G T H A N G . TIEP TUYEN CUA

MAT CAU

1. Dinh nghia

• GV su dung hinh 2.22 va dat ra cac cau hdi:

GV cho HS chi ra doan thing d, r.

1. Trudng hop d > r

H32. So sanh OM va OH.

H33. M d trong hay ngoai mat cau?

H34. A cd cat mat cau hay khdng?

• GV neu dinh nghia:

Khi d > r thi A vd (S) khdng cdt nhau

2. Trudng hgp d = r

• GV sii dung hinh 2.23 va dat ra eac cau hdi:

13

H35.SosanhOMvaOH.

H33. M ? H thi M d trong hay ngoai mat cau?

H34. A cd cat mat cau hay khdng?

• GV neu dinh nghia:

Khi d = r thi A vd (S) tie'p xuc nhau

• GV neu dinh If:

Dieu kien cdn vd du de A tie'p xdc vdi mat cdu S(0 ; r) tgi H la A vudng

gdc vdi bdn kinh OH tgi H.

3, Trudng hop d < r

• GV sii dung hinh 2.24 va dat ra eac cau hdi:

H35. So sanh OH va OM.

H36. Mpi diem thudc doan MN cd thudc khd'i cau khdng.

• Ket luan :

Khi d < r thi dudng thdng A cdt (S) tgi hai diem phdn biet.

• GV dat bai toan : Cho biet d, r hay tfnh MN.

K L : M N = 2Vi^

GV cho HS dien vao bang sau

d

r

MN

3

4 4

7

5

3

0

14

• De tdng ket GV nen cho HS dien bang sau :

d

r

Giao cua A va (S)

3

4

5

4

4

4

• GV tdng ket chii y bdi bang sau :

Qua mdt diem d tren

mat cau

Qua mdt diem d ngoai

mat cau

S6 TIEP TUYEN v6l(S)

Vd sd

Vd sd

TJNH CHAT CUA CAC TIEP TUYEN

Cac tiep tuyen cimg vudng gdc

vdi ban kfnh tai tiep diem

Dp dai eac doan thing ndi diem

dd va tiep diem bang nhau

• GV neu chii y :

Mat cdu ggi Id ndi tie'p da dien ne'u nd tie'p xuc vdi tdl cd cdc mat ciia da

dien dd vd khi dd la ciing ndi da dien ngoai tie'p mat cdu.

• Thuc hien ^ 3 trong 5'

cau a

Hoat ddng cua GV

Cdu hdi 1

Gpi O la tam ciia hinh vudng canh a. Chiing minh 0 la giao

diem cua cac dudng cheo.

Hoat ddng ciia HS

Gffi y trd Idi cdu hoi 1

HS tu chiing minh.

15

Cdu hdi 2

Tfnh OA.

Cdu hdi 3

Xac dinh tam mat cau.

va ban kfnh cua

Gffi y trd Idi cdu hdi 2

aS 0 A = ^ ^

2


HS tu ket luan.

caub Hoat ddng ciia GV

Cdu hdi 1

Gpi 0 la tam cua hinh vudng canh a. Chiing minh 0 la giao

diem ciia cac dudng cheo.

Cdu hdi 2

Tfnh khoang each tii O den cac canh.

Cdu hdi 3

Xac dinh tam va ban kfnh cua mat cau.

Hoat ddng ciia HS


HS tu chiing minh.


a d = -

2 Gffi y trd Idi cdu hoi 3

HS tu ket luan.

cau c Hoat ddng cua GV

Cdu hdi 1

Gpi O la tam cua hinh vudng

canh a. Chiing minh O la giao

diem cua cac dudng cheo.

Cdu hdi 2

Tfnh khoang each tii 0 de'n cac mat

Cdu hdi 3

Xac dinh tam va ban kfnh cua

mat eau.

Hoat ddng cua HS


HS tu chiing minh.


2 Ggi y trd Idi cdu hdi 3

HS tu ket luan.

16

HOATDONC 4

IV. CONG THtrC DI$N TICH MAT CAU VA T H ^ TICH MAT CAU

• GV neu cdng thuc dien tfch mat cau:

S-4m^

• Vdi 71 = 3,14. Hay dien vao bang sau vdi 2 chir sd thap phan

r

S

1 2 3 4

• GV neu cdng thiic the tfch mat ciu:

V = — w 3

• Vdi 71 a: 3,14. Hay dien vao bang sau vdi 2 chvr sd thap phan

r

V

1 2 3 4

I GV neu chii y

a) Dien tich ciia mat cdu bdn kinh r bdng 4 Idn dien tich dudng trdn Idn

cua mat cdu dd.

b) The tich V ciia mat cdu bdng the tich ciia khdi chdp cd dien tich ddy

bdng dien tich mat cdu vd chieu cao bdng bdn kinh ciia mat cdu.

GV cho HS dien vao bang sau:

r

Dien tfch

dudng trdn

Idn

1 2 3 4

H.hoc 12/2 17

Thuc hien ^ 4 trong 5'

Hoat ddng cua GV

Cdu hdi 1

Xac dinh tam va ban kfnh mat

cau ngoai tiep hinh lap phuong.

Cdu hdi 2

Tfnh the tfch V.

Hoat ddng cua HS


Tam O la tam hinh lap phuong, ban

aV3 kfnh r = .

2


HS tu tfnh.

HOATDQNG 5

TOM T^T Bfil HQC

1. Tap hgp nhirng diem cdch deu mgt diem O cho trudc mgt khodng khdng ddi la

mgt mat cdu.

2. Mgt mat cdu xdc dinh khi bie't tdm vd bdn kinh hoac dudng kinh cda nd.

3. Tap hgp tdt cd cdc diem thudc mat cdu S(0; R) vd tdt cd cdc diem a trong mat

cdu ggi Id khdi cdu.

4. Dieu kien cdn vd du de mat phdng (P) tie'p xdc vdi mat cdu S(0 ; r) tgi H Id (P)

vudng gdc vdi bdn kinh OH tgi H.

5. Mat phdng di qua tdm ggi la mat phdng kinh. Giao ciia (P) vd (S) la dudng trdn

vd ggi la dudng trdn Idn.

6.

Giao ciia (P) va (S)

d > r

Khdng cd

d = r

(P) tie'p xiic (S)

d < r

(P) cat (S) theo

mdt dudng trdn

18

Giao cua Ava (S)

d > r

Khdng cd

d = r

A tie'p xiic (S)

d < r

A cat (S) tai 2 diem

HOATDONC 6

MQT SO Cfia HOI T R 6 C NGHIEM

Hay dien dung (D) sai (S) vao cac kh^ng djnh sau :

Cdu 1.

(a) Mdt mat phing cit hinh cSu thi cat theo mdt dudng trdn.

(b) Dudng trdn Idn ed ban kfnh bang ban kfnh mat ciu.

(c) Dudng trdn Idn di qua tam O.

(d) Mdt hinh ciu cd vd sd dudng trdn Idn.

Trd Idi.

a

D

b

D

c

D

d

D

n D D •

Cdu 2.

(a) Mdt dudng trdn xoay quanh mdt dudng thing ta dupe mdt liinh cdu [ ]

(b) Mdt mat ciu quay quanh mdt dudng kfnh cua nd ta dupe mdt mat eSu. [ ]

(c) Niia dudng trdn quay quanh mdt dudng kinh ciia nd ta dupe mdt mat ciu. [ ]

(d) Ca ba khang dinh tren deu sai. [1

Trd Idi.

a

S

b

D

c

D

d

S

19

Cdu 3. Gpi d la khoang each tii O cua mat cau S(0 : r) den mat phing (P)

(a) d > r thi (P) cat (S) D

(b) d < r thi (P) cit (S) [ ]

(c) d = r thi (P) cit (S) D

(d) Ca ba khang dinh tren deu sai U

Trd Idi.

a

S

h

D

c

S

d

S

Cdu 4. Gpi d la khoang each tii O ciia mat cau S(0 ; r) den mat phang (P)

(a) d > r thi (P) tiep xiic vdi (S) •

(b) d < r thi (P) tiep xiic vdi (S) [ ]

(c) d = r thi (P) tie'p xiic vdi t (S) [ ]

(d) Ca ba khing dinh tren deu sai |_|

Trd Idi.

a

S

b

S

e

D

d

S

Chon khang dinh dung trong cac cau sau:

Cdu 5. Gpi d la khoang each tilt O cua mat ciu S(0 ; r) de'n mat phing (P)

Dien vao chd trdng sau :

d

r

Vi tri tuang ddi

cua (P) vd (S)

3

5

4

4

5

4

5

8

20

Cdu 6. Gpi d la khoang each tir O ciia mat cau S(0 ; r) den dudng thing A


d

r

Vi tri tuang ddi cua A vd (S)

3

5

4

4

5

4

5

8

Cdu 7. Cho hinh lap phuang ABCD. A'B'C'D' tam O. Tam cua mat cau ngoai tiep

hinh lap phuong la:

(a) O; (b) A;

(c) B; (d) C.

Trd Idi. (a).

Cdu 8. Cho hmh lap phuong ABCD. A'B'C'D' tam O. Tam ciia mat ciu tiep xiic

vdi tat ca cac mat ciia hinh vudng hinh lap phuong la :

(a) O; (b) A;

(c) B; (d) C.

Trd Idi. (a).

Cdu 9. Cho hinh lap phuong ABCD. A'B'C'D' canh a. Ban kfnh cua mat cau tiep

xiic vdi tat ca cac mat cua hinh lap phuong la :

(a) a;

, ,aV3 (c)--r-

(b)f.

(d)a^/^

Trdldi. (b).

Cdu 10. Cho hinh lap phuong ABCD. A'B'C'D' canh a. Ban kfnh cua mat cau

ngoai tie'p hinh lap phuong la :

(a) a;

. ,aV3 ( O — ;

Trdldi. (c).

( b ) ^ 2

(d)aV2

21

Cdu 11. Cho hinh lap phuong ABCD. A'B'C'D' canh a. Dien tfch xung quanh ciia

mat cau tie'p xiic vdi tat ca cac mat ciia hinh vudng hinh lap phuong la :

(a) 47ta"; (b)7ia^

(c) 87ra ; (d) 127ia

Trdldi. (b).

Cdu 12. Cho hinh lap phuang ABCD. A'B'C'D' tam O. The tfch ciia mat cau tiep xiic vdi tat ca cac mat cua hinh vudng hinh lap phuong la :

(a)-7:a ;

(c) — 7ia ; ,3

(b) - 7 i a ^ 6

(d) - 7 : a ^ 3

Trdldi. (b).

Cdu 13. Cho hinh chdp S.ABC ndi tie'p hinh ciu tam O ban kinh r (hinh ve)

Biet tam giac ABC la tam giac deu, SO 1 (ABC). Canh AB bing

(a)rV3; (b)2rV3

(c)4rV3; (d)3rV3

Trd Idi. (a).

22

Cdu 14. Cho hinh chdp S.ABC ndi tiep hinh cau tam O ban kfnh r (hinh ve) s

Biet tam giac ABC la tam giac deu, SO 1 (ABC). The tfch hinh chdp bing :

(a) 471 r V3 ;

(c)4r'V3;

Trd Idi. (a).

Cdu 15. Cho hinh ciu S(0 ; 4) nhu hinh ve

(b) -71 r V3; 3

(d)3r'V3

AB bing :

(a)2V3;

(c)3;

Trdldi. (a).

(b)3V3

(d)6.

Cdu 16. Cho hinh cau S(0 ; 4) nhu hinh ve

Ban kfnh dudng trdn tam O bing :

(a) V7;

( c ) 3 ;

Trdldi. (a).

(b)3V7

(d)6.

HOATDQNC 7

naOTNG D^N Bfil TOP SGK

Bai 1. Hudng ddn. Sii dung dinh nghia hinh eau:

Ddp sd. Mat ciu tam O (trung diem AB) ban kinh AB

24

Bai 2. Hudng ddn. Dua vao tfnh chat cua mat phang tmng true va dinh nghia mat eau:

A

B - - C

Ddp sd

- Dung tam E dudng trdn ngoai tiep tam giac CBD.

Ke Ex L (BCD).

Dung mat phing trung true ciia AB.

I la tam.

HS tu chiing minh.

Bai 3. Hudng ddn. Dura vao tfnh chit ciia giao diem mat cau va mat phang

Tap hop la dudng thing di qua tam O' cua dudng trdn va A 1 mp(O')

Bai 4. Hudng ddn. Dua vao dinh nghia mat cau.

A

Goih (O') la dudng trdn ndi tiep tam giac ABC. Sii dung bai 3.

Bai 5. Hudng ddn. Dua vao vf trf tuong ddi cua mat phang va mat cau:

cau a.

Chiing minh 4 diem A, B, C va D ciing thudc mdt dudng trdn.

cau b. Hudng ddn.

Ta cd MA. MB = MO' - r ' l Trong dd r' la ban kfnh dudng trdn tam O'

MO'2 = MO^ - 0 0 ' ^ = d - 0 0 ' 2

r '2=r^-00 '2

26

Bai 6. Hudng ddn. Dua vao tfnh chat tie'p tuyd'n ciia mat ciu.

- Chiing minh Al va BI la hai tie'p tuyen cua mat ciu.

- Chiing minh AMAB = ALAB.

Bai 7 Hudng ddn. Dua vao dinh nghia hinh tru, tfnh chat ciia hinh tru, dien tfch

xung quanh va the tfch hinh tru. B

\ O

cau a. Chiing minh O la giao diem eac dudng cheo la tam ciia mat ciu.

^ a^+b^+c^

cau b. Tam O' la tam hmh chu nhat ABCD.

Va 2+b2

Bai 8 Hudng ddn. Dua vao tfnh chat ciia tie'p tuye'n mat ciu. HS tu giai. Bai 9 Hudng ddn. HS tu giai. Bai 10 Hudng ddn. HS tu giai.

27

On tap churdng II

(tiet 11,12)

1. MUC TIEU

1. Kien thurc

HS n im duac:

1. Khai niem chung ve mat trdn xoay.

2. Mat tru va cac tfnh chat ciia mat tru.

3. Mat cau va cac tfnh chat cua mat cau.

4. Giao ciia mat cau va dudng thing.

5. Tiep tuyen ciia mat cau.

6. The tfch va dien tfch ciia mat cSu.

2. KI nang

• Giai thanh thao cac bai toan lien quan den mat ciu, mat tru.

• Xac dinh dupe mdt mat phing la tie'p dien cua mat ciu, mdt mat phing la

tie'p tuyen ciia mat cau.

3. Thai dp

• Lien he dupe vdi nhieu vin de thuc te' trong khdng gian.



II. CHUAN DI CUA GV VA HS

1. Chuan bi cua GV:

• Thudc ke, phan mau, ...

2. Chuan bj cua HS :

Dpc bai trudc d nha, cd the lien he cac phep bie'n hinh da hpc d Idp dudi

28

ra. PHAN PHOI THOI LUONG

Bai dupe chia thanh 2 tie't :

Tie't 1: Chiia bai tap va on tap.

Tie't 2: Kiem tra 1 tie't.

IV. TIEN TDINH DAY HOC

HOATDONC 1

I. M O T s d CAU HOI ON TAP

Cdu hdi 1. Neu dinh nghia mat trdn xoay.

Cdu hdi 2. Neu mdt sd hinh trdn xoay trong thuc te ma em biet.

Cdu hdi 3. Mdt tam giac thudng quay quanh mdt canh cd dupe mdt hinh ndn khdng?

Cdu hdi 4. Mdt hinh ndn dupe tao thanh nhu the nao?

Cdu hdi 5. Mdt hinh binh hanh quay quanh mdt canh cd dupe mot hinh tru khdng?

Cdu hdi 6. Mdt hinh tru dupe tao thanh nhu the' nao?

Cdu hdi 7. Neu khai niem true, dudng sinh va dudng cao cua hinh ndn.

Cdu hdi 8. Day ciia hinh ndn la hinh gi?

Cdu hdi 9. Neu khai niem true, dudng sinh va dudng cao cua hinh tru.

Cdu hdi 10. Neu dinh nghia mat ciu bing cac each khac nhau.

Cdu hdi 11. Hinh cau va mat ciu khac nhau nhu the' nao?

Cdu hdi 12. TCr mdt diem ke hai tie'p tuyen de'n mat cau.

a) Khi nao thi ke dupe? b) Neu cac tfnh chat cua hai tiep tuyen dd.

Cdu hdi 13. Neu cac vi trf tuong ddi ciia mat phing va mat cau..

29

Cdu hdi 14. Neu cac vi trf tuong ddi eiia dudng thing va mat ciu.

Cdu hdi 15. Neu sU xac dinh mat cau.

Cdu hdi 16. Neu cac cdng thiic tinh dien tfch mat ciu va the tfch khd'i ciu.

HOATDONC 2

n. MOT s o CAU HOI TRAC NGHIEM ON TAP CHUONG II

Hay chon cau tra Idi dung.

Cdu 1. Cho hinh chdp ndi tiep mdt hinh ndn

(a) Hai hinh chdp va hinh ndn cd dudng cao trdng nhau;

(b) The tfch hinh chdp va the tfch hinh ndn bing nhau;

(c) The tfch hinh chdp Idn hon the tfch hinh ndn;

(d) Ca ba y tren deu diing.

Trd Idi. (a).

Cdu 2. Cho hinh chdp luc giac ddu canh day la 2V3 ndi tiep mdt hinh ndn cd dudng cao la 3

S

30

Dudng cao ke tii S ciia mdi mat bdn ciia hinh chdp la :

(a)2VlO; (b)VlO

To (c) (d)10

Trdldi. (b).

Cdu 3. Cho hinh chdp luc giac deu canh day la 2v3 ndi tiep mdt hinh ndn cd dudng cao la 3

s

Ban kfnh dudng trdn day la

(a) 2V3;

, , 2V3

(b) 2V6

(d) V3.

Trd Idi. (a).

Cdu 4. Cho hinh chdp luc giac deu canh day la 2v3 ndi tiep mdt hinh ndn cd

dudng cao la 3 S

Dudng sinh la

(a) 2 ^ ;

, , 2V3. ( 0 — ,

Trdldi. (b).

(b) 2V6

(d) V3.

31

Cdu 5. Cho hinh lang tru luc giac deu canh day la 2\/3 ndi tiep mdt hinh tru cd

dudng cao la 3 B

" C

A' , ^-

B'

Dien tfch xung quanh cua hinh lang tru la la :

(a)36V3; (b) 16V3

(c)46V3; (d)26V3.

Trd Idi. (a).

Cdu 6. Cho hinh lang tru luc giac deu canh day la 2V3 ndi tiep mot hinh tru cd dudng cao la 3

A B

A' B'

D' E'

32

Dudng sinh ciia hinh tru la :

(a)2V3;

( c ) 3 ;

Trd Idi. (c).

(b)3V3

(d)6.

Cdu 7. Mdt hinh ciu cd dudng trdn Idn ngoai tiep mdt tam giac deu canh 1 cd ban

kfnh la

(a) 27i; (b)37t

(d)6.

Trd Idi. (e).

Cdu 8. Mdt hinh cau cd dudng trdn Idn ngoai tiep mdt tam giac deu canh 1 cd dien

tfch toan phan la :

H hoc 12/2 33

(b)3V3

(d) 671.

(a)2V3;

(c)-i=; V3

Trd Idi. (c).

Cdu 9. Gpi d la khoang each til O ciia mat ciu S(0 ; r) den mat phing (P) Dien vao chd trdng sau :

d

r

Vi tri tuang ddi

cda (P) vd (S)

6

5

5

4

4

4

8

8

Cdu 10. Gpi d la khoang each tii O ciia mat cau S(0 ; r) den dudng thing A


d

r

Vi tri tuang ddi

ciia A vd (S)

4

5

4

5

5

7

9

8

Cdu II. Cho hinh lap phuong ABCD. A'B'C'D' tam O. Tam ciia mat cau ngoai

tie'p hinh lap phuong la

(a) O; (b) A

(c) B; (d) C.

Trd Idi. (a).

Cdu 12. Cho hinh lap phuang ABCD. A'B'C'D' tam O. Tam ciia mat ciu tiep xuc

vdi tat ca cac mat cua hinh vudng hinh lap phuong la :

(a) O; (b) A

(c) B; (d) C.

Trd Idi. (a).

34

HOATDONC 3

m . HUONG DAN BAI TAP SGK

Bai 1. Hudng ddn.

cau a diing vi : Qua ba diem ABC ta cd mdt mat phing. Giao ciia mat ciu va mat phing la dudng trdn.

Bai 2. Hudng ddn. D

Dudng sinh ciia hinh ndn : BD - av2

Dudng cao ciia hinh ndn AB = a

35

Bai 3. Hudng ddn.

Til tam E cua day hinh ndn ke dudng cao SE.

- Mat phing trung true ciia mdt dudng cao bat ki cat SE tai O.

Chiing minh O la tam mat cau ngoai tiep hinh chdp.

Bai 4. Hudng ddn.

- Chiing minh tam I dudng trdn ndi tie'p tam giac ABC thudc mat cau dd.

- Chiing minh ABC la tam giac deu.

- Chiing minh SI 1 (ABC).

Bai 5. Hudng ddn.

A

36

a) HS tu giai.

Tfnh HB = r = 1V3

- Dudng sinh khdi tru a. Bai 6. Hudng ddn.

Ke mat phing trung true cua SA cit SO tai I.

Chiing minh I la tam mat ciu.

De tfnh r = SI ta su dung tam giac ddng dang

SI SE „ SE.SA — = — ^ SI = SA SO SO Bai 7. HS tu giai.

Cau hdi trac nghiem :

HS tu giai.

37

HOATDONC 4

MOT SO D^ KI^M TRA CHUONG H

De sd 1

Cdu i . ( 3 d) Cho hinh hop chii nhat ABCDA'B'CD' canh a, b, c.

a) Xac dinh tam mat cau ngoai tie'p hinh hop.

b) Tfnh dien tfch mat cau dd.

Cdu 2. (4 d) Cho hinh ndn cd ban kfnh day la 4, dudng cao la 6.

a) Tfnh dien tich xung quanh va the tfch hinh ndn.

b) Tfnh ban kfnh hinh cau ngoai tiep hinh ndn dd.

Cdu 3. (3d) Cho hinh tru cd ban kfnh day la 4, chieu cao 3. Cit hinh tru bdi mot mat phing song song vdi true.

a) Hinh thiet dien la hinh gi?

b) Neu cac dung hinh dd biet hinh dd cd dien tfch la 18.

De sd 2

Cdu 1. i 6 d).

Cho hinh cau cd ban kfnh la 3.

a) Tfnh the tfch khd'i cau va dien tfch mat ciu.

b) Tfnh the tfch hinh tru ngoai tiep hinh cau.

Cdu 2. (4 d).

Cho hinh ndn cd ban kfnh day la 6, dudng cao la 8.

a) Tfnh dd dai dudng sinh.

b) Mdt mat phing vudng gdc vdi true cit hinh ndn theo mdt dudng trdn ban kinhd la 2. Tfnh the tfch hinh ndn cut tao thanh.

38

Chi/ofNq 7

PHlTOlVG P H A P I W A 0 6 TROafG HHOIVG GIAIV

P h a n 1

Gidl THlfu CHUOHG

I. CAU TAO CHUONG

§ 1. He tpa dp trong khdng gian

§ 2. Phuong trinh mat phing

§3. Phuang trinh dudng thing trong khdng gian

On tap chuang III

On tap cud'i nam

1. Muc dich ciia chuong

• Chuong III nhim cung cap cho hpc sinh nhiing kien thiic co ban ve khai niem ve tpa dp trong khdng gian va nhiing ling dung ciia nd.

Tpa dp vecto va tpa dp diem.

- Bieu thiie tpa dp cua cac phep toan vecto .

Tfch vd hudng cua hai vecto.

Phuang trinh mat cau.

• Gidi thieu vd phuang trinh mat phing trong khong gian.

Vecto phap tuyen cua mat phing.

- Phuong trinh tdng quat ciia mat phing.

- Dieu kien de hai mat phang song song, vudng gdc.

Khoang each tir mot diem den mdt mat phing.

39

• Phuang trinh dudng thing trong khdng gian:

Phuang trinh tham sd cua dudng thing.

- Dieu kien de hai dudng thing song song.

- Dieu kien de hai dudng thing cheo nhau.

- Dieu kien de hai dudng thing cit nhau.

II. MUC TIEU

1. Kien thurc

Nam dupe toan bd kien thiic co ban trong chuong da neu tren.

Hieu cac khai niem va tfnh chat vecto trong khdng gian.

Hieu va bie't dupe mdi quan he giiia vecto phap tuye'n va cap vecto chi phuang

ciia mat phing.

Hieu va biet dupe mdi quan he giiia vecto phap tuye'n va vecto chi phuong ciia

dudng thing.

2. Kl nang

Xac dinh dupe cac vectp trong khdng gian.

Van dung dupe eac tfnh chit de giai bai tap.

- Chiing minh dupe hai mat phing, song song, vudng gdc.

Lap dupe cac phuang trinh dudng thing va phuang trinh mat phing.

Xac dinh dupe vi tri tuong ddi cua dudng thing va dudng thing, dudng thing va

mat phing, giiia hai mat phing.

3. Thai dp

Hpc xong chuang nay hpc sinh se lien he dupe vdi nhieu van de thuc te' sinh ddng,

lien he dupe vdi nhiing van de hinh hpc da hpc d Idp dudi, md ra mot each nhin

mdi ve hinh hpc. Til dd, cac em cd the tu minh sang tao ra nhCrng bai toan hoac

nhiing dang toan mdi.

Ket ludn:

Khi hpc xong chuong nay hpc sinh cin lam tdt cac bai tap trong sach giao khoa va

lam dupe cac bai kiem tra trong chuong.

40

P h a n 2

cAc BAI SOAN

§1. He toa do trong khong gian

(tiet 1, 2, 3, 4)

1. MUC TIEU

1. Kien thurc

HS n i m duoc:

1. Khai niem tao do vecta trong khdng gian, tpa do diem va dp dai vecto.

2. Bieu thiic tpa dp ciia cac phep toan : cdng, trii vecto; nhan vecta vdi mdt so thirc.

3. Bieu thiic tpa dp ciia tfch vd hudng ciia hai vecta.

4. Riuong trinh mat cau.

2. KT nang

• Thuc hien thanh thao cac phep toan ve vecto, tfnh dp dai vecto,

• Vie't dupe phuang trinh mat ciu.

3. Thai do

• Lien he dupe vdi nhieu van de thue te' trong khdng gian.


• Humg thii trong hpc tap, tich cue phat huy tfnh dpc lap trong hpc tap.

II. CHUAN DI CUA GV VA H&

1, C h u a n bi cua GV:

• Hinh ve 3.1 den 3.3.



Dpc bai trudc d nha, cd the lien he cac phep bie'n hinh da hpc d Idp dudi

III. PHAN PHOI THOI LUONG

Bai duoc chia thanh 4 tie't:

Tie't 1

Tie't 2

Tiet 3

Tie't 4

Tii dau de'n het phan I.

Tie'p theo den he't phan II.

Tie'p theo de'n het phan III.

Phan IV va hudng dan bai tap.


a. DRT VA'N Di

Cau hdi 1.

Nhic lai khai niem hinh hop, hinh chdp.

Cau hdi 2.

Cho hinh lap phuong ABCDA'B'CD'

a) Chiing minh cac canh ciia hinh lap phuong xuit phat tii mdt dinh vudng gdc vdi nhau.

b) Cho canh ciia hinh lap phuong la a, tfnh dp dai dudng cheo ciia hinh lap phuong.

n. isni MOI

HOATDONCl

I. TOA DO CUA DIEM VA CUA VECTO

1. He toa do

GV mo ta he true toa do trong khong gian va neu cau hoi :

HI. Hai vecto i, j cd vudng gdc vdi nhau hay khdng?

H2. Vecto k cd vudng gdc vdi tat ca cac vecta thudc mat phing (Oxy) khdng?

42

• GV sir dung hinh 3.1 trong SGK va dat van de:

H3. Hay dpc ten cac mat phing tpa dp.

H4. Hay ke' ten cac vecto dan vi.

H5. Cd the ed them mdt gdc tpa dp niia khac O hay khdng?

H6. Hay neu cac tfnh chit ciia mat phing tpa dp, vecto don vi?

-2 — -2 — - 2 H7. Tfnh i = i.i, j = j . j , k = k.k.

H8. Tfnh i.j,j.k, k.i .

• Thue hien ^ 1 trong 4 phiit.

Su dung hinh ve 3.2. GV cho HS len bang ve lai hinh va hudng din HS thuc hien z

N

Hoat ddng cua GV

Cdu hoi 1

Bieu dien OM theo OE va ON.

Cdu hdi 2

Bieu dien OE theo OH va OK.

Hoat ddng cua HS

Gffi y trd loi cdu hoi 1

OM = OE + ON

Ggi y trd loi cdu hdi 2

OE=OH+OK.

43

Cdu hoi 3

Tim cac mdi quan he giiia cac

vecto ON OH va OK

Cdu hdi 4

Bieu dien OM theo i, j va k.


OK = x.i OH = y.j ,ON =


OM = x.i + y.j + z.k

= z.k

2. Toa dp cua diem

GV sir dung hinh 3.2 va dat cau hdi:

H9. Cho ba sd thuc x, y va z. Cd bao nhieu diem M thda man OM = x.i + y.j + z.k.

HIO. Cho OM = x.i + y.j + z.k. Cd bao nhieu bd sd sd thuc x, y va z thda man he

thiie tren.

• GV tra Idi va neu dinh nghia :

Bd ba sd thuc (x; y z) thda mdn OM = x.i + y.j + z.k ggi la tga do diem

M vd ki hieu M (x ; y ; z) hoac M = (x ; y ; z).

HI 1. Cho M (0 ; 0 ; 0) Hay chi ra M.

HI2. Cho M(0 ; 1 ; 2). Hdi M thudc true nao ?

HI3. Cho M(l ; 0 ; 2). Hdi M thudc true nao ?

H14. Cho M(l ; 2 ; 0). Hdi M thudc true nao ?

3. Toa dp vecto


Trong khdng gian cho vecta a. Bg ba sd (x ; y ; z) thda mdn

a = x.i + y.j + z.k ggi la tga do cua vecta a. Ki hieu a(x;y;z) hoac

a = (x;y;z).

HI5. Vecta OM va diem M cd ciing tpa dp khdng?

44

• GV neu nhan xet trong SGK:

Tga do cua OM chinh Id tga do cua M.

• Thue hien A2 trong 4 phiit.

Su dung hinh ve 3.2. GV cho HS len bang ve lai hinh va hudng din HS thuc hien

Hoat ddng cua GV

Cdu hoi 1

Tim tpa dp cua AB.

Cdu hoi 2

Tim tpa dp ciia AC.

Cdu hoi 3

Tim tpa dp ciia A C

Cdu hdi 4

Tim tpa dp ciia AM.

Hoat ddng cua HS


AB(a;0;0)


AB(a;b;0).


AB(a;b;c)


AM(- ;b ;c ) 2

HOATDQNC 2

II. BIEU THtrC TOA D O CUA CAC PHEP TOAN VECTO

• GV neu dinh If:

Trong khdng gian Oxyz cho ba vecta a(ai;a2;a2) vab(bi ;b2;b3) fa cd :

a + b = (aj +bi;a2 + b2;a3+b3)

a - b = (a, - b p a 2 - b 2 ; a 3 - b 3 )

ka = (kai;ka2;ka3), trong dd k la mgt sdthuc.

• GV hudng din HS chiing minh dinh If tren.

H16. Hay so sanh cac tpa dp ciia a va b khi a = b .

• GV neu he qua 1:

Hai vecta bdng nhau thi cdc tga do tuang img bdng nhau.

HI7. Hay viet cac bieu thiic tpa dp cua he qua 1.


Vecta 0 cd cdc tga do bdng 0.

HI8. Hay viet cac bieu thiic tpa dp ciia he qua 1.


Hai vec ta cdng phuang thi mdi tga do cua vec ta ndy bdng k Idn tga do

tuang Umg cda vec ta kia..

HI9. Hay vie't cac bieu thiic tpa dp cua he qua 3.


Khi bie't tga do ciia AvdB ta cd tga do ciia AB bdng cdch lay mdi tga do

tuang img cda B trit di tga do tuang img ciia A..

H20. Hay viet cac bieu thiic tpa dp cua he qua 4.

46

• Trong sach GK khdng cd vf du nhung GV nen lay vf du minh hpa cho dinh If va he qua nay. Vi day la kien thiic rat quan trpng..

Vfdu.ChoA(l ;1 ; 1),B(-1 ; 2; 3) va C (0 ; 4 ;-2).

a) Hay tim cac tpa dp ciia AB va AC .

b) Tim tpa dp ciia vec to 3AB .

c) Tfnh AC + 3AB.

• GV gpi mdt HS giai cau a.

Hoat ddng ciia GV

Cdu hoi 1

Tim AB.

Cdu hoi 2

Tim AC

Hoat ddng ciia HS

Ggi y trd Idi cdu hoi. 1

HS tu giai.


HS tu giai.

• GV gpi tie'p HS thii 2 tra Idi cau b va sau dd gpi HS thii ba lam cau c.

HOATDONC 3

III. TICH VO HUONG

1. Bieu thurc toa do cua tich vd hudng

• GV ndu dinh If

Trong khdng gian Oxyz cho ba vecta aia^;32;a2) vab(b, ;h2;h-^) ta cd:

a.b = ajb2 + a2b2 + a3b3.


2. tTng dung

a) Do ddi cua vectff

• GV neu cau hdi sau :

H21. Trong boat ddng 2, hay tfnh dp dai AC

47

GV neu dinh nghia :

Cho a(a| ;a2 ;a3) khi do do ddi ciia vecta aki hieu va:

3.1 ~r 3,T ~r 3.-5

b) Khodng cdch gida hai diem

H22. Cho A (XA ; yA ;ZA) va B (Xg ; y^ ;z^).

Xac dinh AB.

Tfnh AB.

GV neu ket qua 2:

Khodng cdch gida hai diem AB Id

AB = AB = 7(XB-XA)^+(yB-yA)^+(ZB-ZA)^

c) Gdc giita hai vectff

• GV neu cdng thiic tfnh gdc giiia hai vecto :

Cho cdc vecta iij = (x,; yj ; Zj), M2 = (- 2 ! }'2 > ^2) ^^ ^^ ^ ^^y >*'

^1-^2+yi>'2+^1^2 ta cd COS(M] , M2) =

^x\+y1 + z\ yjxj+yj+zl

H23. Khi nao hai vecto vudng gdc vdi nhau ?

• GV neu he qua :

M] J. U2 o iii.ii2 = 0 <^ x^X2 + yxy2 + 21Z2 = 0.

H24. Vecto to 0 vudng gdc vdi mpi vecto.


Hoat ddng cua GV

Cdu hoi 1

Tfnh b + c.

Hoat ddng ciia HS


b + c = (3;0;-3)

48

Cdu hdi 2

Tfnh a(b + c l .

Cdu hoi 3

Tfnh a + b .

Cdu hoi 4

Tfnh a + b


a(b + c) = 3.3 + 0.0 + (-3). l = 6.


a + b = ( 4 ; - l ; - l ) .


a + b -Vl8

HOATDONC 4

IV. PHUONG TRINH MAT CAU

• GV neu each chia mdt sd khd'i da dien va dat cau hdi:

H25. Tfnh khoang each giiia hai diem M(x ; y ; z) va I (a ; b ; e).

H26. Bie't khoang each dd la r, hay lap bieu thiie mdi quan he dd.

• GV neu dinh If

Mat cdu tdm I(a ; b ; c), bdn kinh r cd phuang trinh

(x-a)^ +iy-b)'^ + iz-c)^ =r^



Hoat ddng ciia GV

Cdu hoi 1

Hay xac dinh a, b va c

Cdu hdi 2

Xac dinh r.

Cdu hoi 3

Viet phuong trinh mat ciu.

Hoat ddng cua HS

Ggi y trd Idi cdu hoi I

a = 1, b = -2 vac = 3.


r = 5.


( x - l ) 2 + ( y + 2 ) 2 + ( z - 3 ) ^ = 2 5 .

H.hoc 12/2 49

H27. Hay neu mdt dang khac ciia phuong trinh mat ciu.

• GV neu nhan xet:

Phuang trinh x + y^ + z + 2ax + 2b\ + 2cz + d = 0 la phuang

trinh ciia mat cdu khi vd chi khi d^ + b^ + c' > d. Khi dd tdm mat

cdu la diem I(-a ; -b ; -c )vd bdn kinh mat cdu la

r = yja^ +b' +c^ -d.

H28. d phai thoa man deu kien gi de x^ + y" + z + 2ax + 2by + 2cz + d = 0 la

phuang trinh ciia mat ciu ?

• GV cho HS thuc hien vi du trong SGK.

HOATDONCL

TOM T ^ Bfit HOC

1. Cho cac vecto u^ = (x , ; y i ; z , ) , ^2 = (A^ ; y2 : '-2) va sd k tuy y, ta cd :

1) »i = fh <=> xi = X2, yi = y2. zi = ^2

2) », - U2 = (AI + x , : y, + y , ; ?i + -2)

3) ;ii - i?2 = (- 'i - ^'2; >'i - >''2; i - ^2)

4) kn^ = (A-A'i; ky^ ; .fe,)

5) N|.U2 = .Vj.vo + 3'iy2 + z,Z2

6) K | = V"i" = v- T +>'i" +^\

7) cos((7, iin) = "•"'"- —'•'- ''^ vdi ii, ?t 0; U2 ^ 0

V-T +.^T + - r \-^'2 +-^': + - :

8) M, -L ih <=> i<i .NT = 0 <=> A', AT + y,y2 + -1-2 = ^ •

2. M =(A- ; y ; z) <=> 0/if = xi + yj + zk

50

3. Cho hai diem Aix^ ; y^ ; z^) va Bixg; VB ' ^fi)-

1) AB = ixg-X/^;yg-y^;zB-z^)

i 1 1 9

2) A 5 - ^ ( A : g - x ^ ) +{yB-yA) +{^B-^A)

4. Mat ciu tam I(a ; b ; c), ban kfnh r cd phuang trinh

(x- f l )2+(y- fc )^+(z-c )2=r2

2 2 2 •) -

Phuong trinh x + y + z + 2ax + 2by + 2cz + d = 0 la phuofng trinh cua mat cau 2 2 2 •'

khi va chi khi a + b + c > d. Khi dd tam mat cau la diem I(- a ; - b ; - c) va

ban kfnh mat cau la

•I? + b^+c-d.

HOATDONC 6

MOT SO C^U HOI TR^C NQHim

Hay dien dung (D) sai (S) vao cac khing dinh sau :

Cdul. Cho a = ( l ;2 ;3) , b = ( - 2 ; 3 ; - l ) . Khi dd a + b cd toa dp la

(a) a + b cd toa dp la (-1 : 5 ; 2) Lj

(b) a - b c d t o a d d l a ( 3 ; - l ;4) •

(c) b - a cd toadd la (3 ; - l :4) [ ]

(d) Ca ba khing dinh tren dSv; sai \_\

Trd Idi.

a

D

b

D

c

S

d

S

51

CAM 2. Cho a = ( l ;2 ;3) , b = ( - 2 ; 3 ; - l ) . Khi dd a + b cd toaddia

(a)3a + b cd toaddIa ( l ; 9 ; 8 ) •

(b) a - 2 b c d t o a d d l a ( 5 ; - 4 ; 5 ) •

( c ) 2 b - a c d t o a d d I a ( 5 ; - 4 ; 5 ) [ ]

(d) Ca ba khing dinh tren deu sai [ j

Trd Idi.

a

D

b

D

c

s

d

s Cdu5. Cho a = ( l ;2 ;3) , b = ( - 2 ; 3 ; - l ) . Khi dd a + b cdtoaddia

(a) a.b = 1 •

( b ) a . b = - l •

( c ) 2 b . a = 2 [ ]

(d) Ca ba khing dinh tren deu sai Q

Trd Idi.

a

D

b

S

e

D

d

S

Cdu 4. Cho hinh cau cd phuong trinh : (x -1)^ + (y + 2f + (z + 3)^ = 2

(a) Tam ciia hinh cau la 1(1 ; -2 ; -3)

(b) Tam cua hinh cau la I(-l ; 2 ; 3)

(c) Ban kfnh ciia hinh cau la 2

(d) Ban kfnh ciia hinh cau la yf2

Trd Idi.

U U D D

a

D

b

S

c

S

d

D

52

Chon khang djnh diing trong cac cau sau:

Cdu 5. Trong cac cap vecto sau, cap vecto ddi nhau la

(a)a = ( l ; 2 ; - l ) , b = ( - l ; - 2 ; l ) ;

(b)a = ( l ; 2 ; - l ) , b = ^ ( l ; 2 ; - l ) ;

(c)a = ( - l ; - 2 ; l ) , b = ( - l ; - 2 ; l ) ;

(d)a = ( l ; 2 ; - l ) , b = ( - l ; - 2 ; 0 ) ;

Trd Idi. (a).

Cdu 6. Cho hinh ve :

z

. . - ; • " "

.---i"" i ,/f""" .. . j - ' " i _,J--''''

i ,.<-'' i „.••!'••' i ...]••''•'

i ,i ' ' ' 1 ,.k--i---,-j--'-'-

_....f i ..i--^

.i---i ]i:r(

L

-TE^I

^ 1 i l J . - - ' ' ..•••'•' . . - • •

Ic 1 i

'! - 4 ^

:::::... :': ^-"B,--'

Diem D cd toa dp la

(a) (5 ; 1 ; 0 ) ;

(c) (1 ; 5 ; 0 ) ;

Trd Idi. (d).

(b) (0 ; 1 ; 5);

( d ) ( l ; 0 ; 5 ) .

53

Cdu 7. Cho hinh ve :

Diem C cd toa dp la

(a) (4 ; 4 ; 0 ) ; (b) (4 ; 0 ; 4)

Trdldi. (b). Cdu 8. Cho hinh ve :

(c) (0 ; 4 ; 4) (d) (0 ; 0 ; 4)

Dl-

•"" \ y"\ r

.--1" \L^o -••*" --'"'' .--•'' -••"' ---•"' -•'''

^ ;

Diem A ed toa dd la

(a) (0 ; 2 ; 0 ) ; (b) (2 ; 0 ; 2)

Trd Idi. (c).

(c) (2 ; 0 ; 0) (d) (0 ; 0 ; 2)

54

Cdu 9. Cho hinh ve

Diem B cd toa dd la

.-••'• I O f - -

-r - - . -3-^--

.y"'\ k i f : ;

ic

..y...y.B^:::i...-"'

^ :^ y' • .y'\ ji^o.-' •••••[ .-•-• . - - - . . - - - . - - - " V

(a) (4 ; 4 ; 0 ) ;

Trd Idi. (a).

Cdu 10. Cho hinh ve

X

Diem E cd toa dp la

(a) (3 ; 4 ; 3 ) ;

Trd Idi. (a).

(b) (4 ; 0 ; 4)

A

z

..-••<"] A""' i .-'!'' i .-•i''"'

i .i---"i > i -

L

--

.

').. :.--': A--^ ...--'

j^..^::l..^:l..^:;:

(b) (4 ; 3 ; 4)

f-

--f _: ./ i

...^

(c) (0 ; 4 ; 4 ) ;

i i i c i i

£T[ M 1 1 1 ; ; ; ; ;

1 i i 1 i i

• ; . . - • • . . . . - • ' . . . - • • . , . - • y

-:::;£L::::1..--""

(c) (3 ; 4 ; 4 ) ;

(d) (0 ; 0 ; 4)

(d) (3 ; 0 ; 4)

55

HOATDONC 7

HOG NG D^N Bfil T6P SGK

Bai 1, Hudng ddn. Dua vao tfnh chat cua eac phep toan vecto

Caua.Tacd 4a = {8;-20;12) ; — b ^ ' 3

0;- 2 I 3 ' 3

; 3c-(3;21;6).

Tii dd ta cd ke't qua.

caub. Tacd ; 4b = (0;8;-4) ; -2c = ( -2;-14;-4) .

Tilr dd ta cd ket qua.

Bai 2. Hirdng ddn. Dua vao tfnh chat chit XQ - - ( x ^ + Xg + x^ );

y G = ^ ( y A + y B + y c ) ; ZG= gl^A+ZB+zc)

Bai 3. Hudng ddn. Dua vao tfnh chat cua phep toan toa dp. Hai vecto bing nhau. A(1 ;0;1)

D(1 ; -1 ; 1

B(0 ; 1; 2)

C'(4 ; 5 ; -5

Tfnh toa dp diem C bang each gpi C(x ; y ; z) va D C - ( x - l ; y + l ;z- l)

AB = (-1;2;1) va DC = AB Ta cd C(2; 0; 2)

Tfnh toa dd A' bang each : AA' = DD' ta ed A'(3 ; 5 ; 6)

Tuong tu ta cd B' (4 ; 6 ; -5), D'(3 ; 4 ; -6).

56

Bai 4. Hudng ddn. Dua vao tfnh chat ciia tfch vo hudng hai vecto

U\.U2 =x\X2+yiy2+h^2

a) a.b = 6 .

b) c.d = - 2 1 .

Bai 5. Hirdng ddn. Dua vao phuong trinh mat cau.

a) Phuang trinh mat cau dupe vie't dudi dang :

( x - 4 f + ( y - l ) 2 + z2=16.

Tir dd ta ed tam va ban kfnh mat cau.

b) Phuang trinh mat ciu dupe vie't dudi dang :

(x-1)^ + 4

y + -3

^2 :^2

z + -19

Tii dd ta cd tam va ban kfnh mat cau.

Bai 6. Hudng ddn. Dua vao phuong trinh mat cau.

a) Xac dinh tam mat cau : I = (3;-1 ;5), ban kfnh mat ciu r = 3.

Tii dd ta cd tam va ban kfnh mat ciu

( x - 3 f + ( y + l ) ' + ( z - 5 f = 9 .

b) Xac dinh tarfi mat cau : C - (3 ; -3; l ) , ban kfnh mat ciu r = v5

Tii dd ta cd tam va ban kfnh mat ciu

( x - 3 f + ( y + 3 ) ^ ( z - l f = 5 .

§2. phtiTdng trinh mat phang

(tiet 5, 6, 7, 8, 9)

I. MUC TIEU

1. Kien thiirc

HS n im ducfc:

1. Vecto phap tuye'n ciia mdt mat phing, cap vecto chi prfiuong ciia mat phing.

2. Sir xac dinh mdt mat phing.

3. Biet duoc phuong trinh tdng quat va phuong trinh tham so cua mat phang.

4. Xac dinh dupe didu kien de hai mat piling song scHig va hai mat piling vudng gdc.

2. Kl nang

• Lap dupe phuong trinh mat phing khi biet mdt diem va vecto phap tuyen, khi

bie't mot diem va cap vecto chi phuong.

• Xac dinh dupe vi trf tuong ddi eiia hai mat phing, hai mat phing song song, hai

mat phing vuong gdc.

• Tim dupe khoang each tii mdt diem den mdt mat phing.

3. Thai dp

• Lien he dupe vdi nhieu van de cd trong thuc te' ve mat phing trong khdng gian.



n. CHUAN BI CUA GV VA H6

1. C h u a n bi cua GV:

• Hinh ve 3.4 den 3.8 trong SGK.

58


• Chuan bi sin mdt vai hinh anh thuc te' trong trudng ve hai mat phang vudng gdc ,

hai mat phing song song.


• Dpc bai trudc d nha, dn tap lai mdt sd kien thiic da hpc.

• Chuan bi thudc ke, biit chi, biit mau de ve hinh.

m. DHAN PHOI T H 6 I LUONG

Bai nay chia thanh 5 tiet:

Tie't 1: tii diu den het dinh nghia phan I.

Tie't 2 : tie'p theo den het muc 1 phin II.

Tie't 3: tie'p theo den het phan II.

Tie't 4 : tie'p theo den het muc 1 phan III.

Tie't 5: tie'p theo den het phin IV.


n. DRT VAN ff>€

Cau hdi 1.

Cho hinh lap phuong ABCD.A'B'CD' cd A trung vdi gdc toa dp. AB

triing vdi Ox, AD triing vdi Oy, AA' triing vdi Oz

a) Tim toa dp tat ca cac dinh cua hinh vuong.

b) Tim toa dp vecto AM vdi M la trung diem C C

Cau hdi 2.

Neu mdt so tfnh chit co ban cua phep toan ve vecto.

59

B. Bni MOI

HOATDONCl

I. VECTO PHAP TUYEN CUA MAT PHANG

GV neu mot so eau hdi sau day:

HI. cd bao nhieu dudng thing vudng gdc vdi mat phang

H2. Mot mat phing xac dinh khi nao ?


Cho mat phdng (a). Ne'u vecta n khdc vecta 0 cd gid vudng gdc vdi mat

phdng (a) ggi la vecta phdp tuyen cua mat phdng (a)

H3. Cho n la vecto phap tuyen ciia (a), hdi k n cd la vecto phap tuyen cua (a) khdng?

• GV ndu chii y :

Neu n la vecta phdp tuyen cua mp{a) thi kn {k^Q) cdng la vecta

phdp tuyen cda mp(^a).

• GV neu va hudng din HS giai bai toan 1. ( Su dung hinh 3.4).

60

Hoat ddng ciia GV

Cdu hdi 1

De chiing minh n la vec to phap

tuyen ciia (a) ta can chiing minh

va'n de gi ?

Cdu hoi 2

Hay chiing minh nhan dinh tren.

Hoat ddng cua HS


Ta chiing minh n.a = n.b = 0 .


HS tu chiing minh.

GV neu nhan xet:

Vecta n thod mdn n.a = n.b - 0 ggi Id tich cd hudng cua hai vecta

a va b , ki hieu n = a A b hoac n =

Thuc hien A l trong 5 phiit.

a,b

Hoat ddng cua GV

Cdu hoi 1

Tim toa dp vecto AB

Cdu hoi 2

Tim toa dd vecto AC

Cdu hoi 3

Tim vec to phap tuyen ciia mat

phing (ABC)

Hoat ddng cua HS


AB = ( 2 ; l ; - 2 )


AC = (-12;6;0)


n = ( l ;2 ;2) .

61

HOATDONC 2

II. PHUONG TRINH TONG QUAT CUA MAT PHANG

• GV neu va cho HS giai bai toan 1. (Sii dung hinh 3.5)

M(x ;y ; x)

Hoat ddng cua GV

Cdu hdi I

Tim toa dp vecto M Q M

Cdu hdi 2

M Q M va n cd quan he vdi nhau

nnirthe'nao ?

Cdu hoi 3

Hay chiing minh nhan dinh" ciia

bai toan.

Hoat ddng cua HS


HS tu tfnh.


MoM.n=0.


HS tu chiing minh.

• GV neu va cho HS giai bai toan 2. (Sir d ling hinh 3.5)

Hoat ddng cua GV r Hoat ddng cua HS

Cdu hdi I Ggi y trd Idi cdu hdi I

Hay chpn diem MQ thoa man

phuong trinh da cho.

Cdu hdi 2

Gpi (a) la mat phang qua M„ va

HS tu chon.


HS tu chiing minh.

nhan n(A;B;C) lam vecto p\rd~i

tuye'n. Chiing min- M ihup'? (a)

khi Ax + By + Cz +D = 0. 1

62

1. Djnh nghia

• GV neu dinh nghia sau :

Ax + By + Cz + D = 0 trong dd A^ + B^ + C^ > 0 ggi la phuang trinh

tdng qudi ciia mat phdng (or).

H4. Tim vecto phap tuyen eiia mat phing : Ax +By + Cz + D-0

• GV neu nhan xet a) :

(a) cd phuang trinh Ax + By + Cz + D = 0 thi vecta phdp tuye'n cua (a)

la n = (A;B;C)

H5. Lap phuang trinh mat phing di qua Mo(x,); yo; Z;,) va nhan n = ( A ; B ; C ) la

vecto phap tuyen.

• GV neu nhan xet b) :

Mat phdng (a) di qua diem MQ (XO,>'O,ZO) '' ' " ^ vecta phdp tuye'n n (A ;

B Old A(x-Xo) +B(y-yo) + C(z-Zo) = 0.

• Tliuc hien ^ 2 trong 5 phut.

GV gpi ba HS len bang dien vecta phap tuye'n vao d trdng sau:

HSl:

(a)

VTPT:n

4x - 2y - 6z + 7 = 0 4x + 2 y - 6 z + 7 = 0

HS2:

(a)

VTPTin

4x + 2y -+6z + 7 = 0 4x + 2 y - 6 z - 7 = 0

63

HS3:

(a)

VTPT:n

4x - 2y - 6z - 7 = 0 4x +2y + 6z + 7 = 0


Hoat ddng ciia GV

Cdu hdi 1

Xac dinh MN

Cdu hdi 2

Xac dinh MP

Cdu hdi 3

Xac dinh VTPT cua (MNP)

Cdu hdi 5

Lap PTTQ ciia mat phing.

Hoat ddng cua HS


MN = (3 ; 2 ; I)


MP = (4 ; 1 ; 0).


n = (-1 ; 4 ; -5)


X - 4y + 5z -2 = 0.

2. Cac trudng hop rieng

a) Mat phdng di qua gdc tog dg:Sit dung hinh 3.6.

H6. Diem O (0 ; 0 ; 0) thudc mat phing (a). Tim D.

• GV ke't luan :

Mat phdng (a) di qua gdc tog do O khi vd chi khi D = 0.

b) Mol Irong cdc he sd: A, B hgc C bang 0 ; Sii dung hinh 3. 7

H7. A = 0, mat phing (a) va Ox cd quan he nhu the' nao ?

• GV ke't luan :

Mat phdng (a) song song (hoac chira) true tog do Ox khi vd chi khi

A = 0.

H8. Phat bieu trong trudng hpp : B = 0 hac C= 0.

64

GV neu tdng quat:

Mat phdng (a) song song vdi true tog do ndo dd khi vd chi khi he sd

tuang img cda bie'n sd'bdng 0.

• Thuc hien A4 trong 5 phiit.

Hoat ddng ciia GV

Cdu hoi 1

B = 0 thi-(a) cd dac diem gi ?

Cdu hdi 2

C = 0 thi (a) cd dac diem gi ?

Hoat ddng cua HS


(a)//Oy.


(a) // Oz.

GV neu tdng quat:

Mat phdng (a) song song vdi true tog do ndo dd khi vd chi khi he sd

tuang img cua bie'n sd'bdng 0.

c) Mat phdng (a) triing vdi mot trong cdc mat phdng tog do: Su dung hinh 3. 8

H9. A = 0, B = 0 mat phing (a) va mp(Oxy) cd quan he nhu the' nao ?

• GV ke't luan :

Mat phdng (a) song song hoac triing vdi mat phdng (Oxy) khi vd chi khi

A = B = 0.

HIO. Phat bieu trong trudng hop : B = 0, C= 0 hoac C = 0, A = 0.


Hoat ddng cua GV

Cdu hdi 1

A = C = 0 thi (a) cd dac diem gi?

Cdu hdi 2

B = C = 0 thi (a) cd dac diem gi?

Hoat ddng ciia HS


(a) // Oxz.


(a) // Oyz.

H.hoc 12/2 65

GV neu tdng quat:

Mat phdng (a) song song vdi mat phdng tog do ndo dd khi vd chi khi he

sd tuang img cda cdc bie'n sd'bdng 0.

• GV neu cac cau hdi

Hll . Tii phucmg trinh : Ax + By + Cz +D = 0 <=>- + - + - = 1 bing each nao? a b c

GV nhan xet: Ax + By + Cz + D = 0 vdi cac he sd A, B, C, D deu khac 0.

D , D £> ^ u V .. . . Khi dd bang each dat a = ; b = ; c = . ta dua phuong tnnh tren ve

dang : — + - + - = 1 a b c

• GV neu nhan xet:

X y z ' ., ' Phuang trinh — + — + - = 1 goi la phuang trinh dogn chdn cua mat phdng.

a b c '

HI2. Phuong trinh doan chin ciia (a) cit cac true theo diem nao?

GV nhan xet:

Rd rdng mat phdng cd phuang trinh (2) cdt cdc true Ox, Oy, Oz Idn lU0

tgi cdc diem M(a ; 0 ; 0), N(0 ; b ; 0) vd P(0 ; 0 ; c).

• GV neu vf du trong SGK va giai.

HOATDONC 3

m. Difiu KlfeN DE HAI MAT P H A N G SONG SONG, HAI MAT PHANG

VUONG GOC.


1 Hoat ddng ciia GV

Cdu hoi 1

Xac dinh vecto phap tuyen ciia (a)?

Hoat ddng cua HS


n = (l;-2;3)

66

Cdu hoi 2

Xac dinh vecto phap tuye'n ciia (P)?

Cdu hoi 3

Hai vecto tren quan he nhu the' nao?


ir-(2;-4;6)

Gffi y trd loi cdu hoi 3

Hai vecto tren cdng tuyd'n.

• GV dat vin de :

Trong khdng gian toa dp Oxyz, cho hai mat phing (a) va (a') lin luprt cd

phuong trinh :

ia):Ax + By + Cz + D = 0

ia') :A'x + B'y + C'z + D' = Q;

chiing lan lupt cd eac vecto phap tuyen la «(A ; 5 ; C) va n\A ;B'; C).

Khi nao (a) va (a') song song ?

Khi nao (a) va (a') vudng gdc ?

1. Dieu kien de hai mat phang song song

H13. Khinao(a)//(a')

67

• GV ke't luan : (a) // (a') c^ hai vecto phap mye'n ciia hai mat phing dd cdng tuye'n.

H14. Hay vie't bieu thiic toan hpc de hai mat phing (a) va (a ' ) song song.

• GV ke't luan :

, ^ „ , ,^ fn = ki? f(A;B;C) = k ( A ; B ' ; C ) a// a' o \ o ^

[ D ^ k D ' [ D ^ k D '

, , , „ fn = kir' [(A;B;C) = k ( A ; B ' ; C ) a)= a c^ { <^<

[D = kD' l D = kD'

• GV cd the neu each khac cua ke't luan tren cho de hieu hon:

Cho hai mat phdng (a) vd (a') Idn lu0 cd phuang trinh :

(a) :Ax + By + Cz + D = 0

(a') :A'x + B'y + C'z + D' = 0

a) Hai mat phdng dd song song khi vd chi khi

A__B__C_ D_

A'~ B'" C'^ D'

b) Hai mat phdng dd trdng nhau khi vd chi khi

A^_B__^_D_

A'~ B'~ C'~ D'

• GV neu chii y :

Hai mat phdng (a) vd (a') cdt nhau o(A ; B ; C) i^ k(A ; B'; C)

GV cd the neu each khac cho di nhd:

Cho hai mat phdng (a) vd (a') Idn lu0 cd phuang trinh :

(a) :Ax + By + Cz + D^O

(a') A'x + B'y + C'z + D' = 0

Hai mat phdng dd cdt nhau khi vd chi khi A . B • C ^A': B': C

68

• GV neu vf du trong SGK va giai. GV cd the neu vf du khac.

• Sau day la vf du khac :

Cho hai mat phing (a) : x -my + 4z + m = 0

i^):x-2y + (m + 2)z-4 = 0.

Hay tim gia tri ciia m de :

a) Hai mat phing dd song song.

b) Hai mat phing dd trdng nhau.

c) Hai mat phing dd cit nhau.

Cau a.

Hoat ddng cua GV

Cdu hoi 1

Neu dieu kien de (a) // (P).

Cdu hdi 2

Xac dinh m de (a) // (P).

Hoat ddng eiia HS


A _ B _C D

A'' B'" C'^ D'


1 -m 4 m

I - 2 m + 2 -4

m-2

Caub.

Hoat ddng ciia GV

Cdu hdi 1

Neu dieu kien de (a) = (P).

Cdu hoi 2

Xac dinh m de (a) = (P).

Hoat ddng cua HS


A B C D

A'~ B'~ C'~ D'


1 -m 4 m

1 - 2 m + 2 -4

Khdng cd m.

69

cau c. Hoat ddng cua GV

Cdu hdi 1

Neu dieu kien de (a) cit (P).

Cdu hdi 2

Xac dinh m de (a) cit (P).

Hoat ddng cua HS


A:B :C ^A': B': C


m ^ 2.

2. Dieu kien de hai mat phang vudng gdc

GV sii dung hinh 3.12 va dat cac cau hdi:

H15. Nhan xet ve hai vecto Uj va n2 .

• GV neu dieu kien :

(a,) ± ( 0 2 ) 0 n,'.n^ = O o A A ' + B B ' + C C = 0

• GV neu vf du trong SGK va giai :

Hoat ddng cua GV Hoat ddng ciia HS

Cdu hoi 1

Mat phing da cho cd cap vecto chi phuong nao ?

Cdu hdi 2

Xac dinh vecto phap tuye'n ciia mat phing cin lap.

Cdu hdi 3

Xac dinh mat phing can lap.


AB va n.


Mat phing cd vecto phap tuyen la

n = AB, n = (-I;13;5)


X - 13y-5z+5 = 0.

70

HOATDONC 4

IV. KHOANG CACH TtTMOT D I £ M DEN M O T MAT P H A N G

• GV neu dinh If:

Trong khdng gian Oxyz cho mat phdng (a) cd phuang trinh :

Ax + By + Cz + D = 0 vd diem M fx ; y^ ,• ZQ).

Khdng cdch tirM„ de'n (a) ki hieu d^M , (o.)) vd dugc tinh theo cdng thitc:

' AJCQ + ByQ + CZQ + D\ dfMo,fa))=- I lA^ + B^ +C^

• De chiing minh dinh If tren, GV can dua ra cac budc sau

Gpi M,(x, ; y, ; z,) la hinh chieu ciia M,, tren (a).

Tfnh dd dai MJMQ.U

Tfnh dp dai : M,Mo .

» Thuc hien vf du 1 trong 4'

Hoat ddng cua GV Hoat ddng cua HS

Cdu hdi 1

Tfnh diO, ia))

Cdu hoi 2

TmhdiM,ia))


|2.0 - 2 . 0 - 0 + 31 diO, ia))

^2'+{-2f+{-lf


diO, ia))

_ |2 .1 -2 . ( -2 ) -13 + 3| 4

^ 2 2 + ( _ 2 f + ( - 1 ) 2 3

71

• Thuc hien vf du 2 trong 4'

Hoat ddng ciia GV

Cdu hdi 1

Chpn mdt diem M bat ki thudc (a).

Cdu hdi 2

Tfnh diM, ia)).

Hoat ddng ciia HS


GV cho HS chpn diem bat ki.


diM, ia))

diM, ia)) = 3.

• Thuc hien A 7 trong 5 phiit.

Hoat ddng cua GV

Cdu hdi 1

Hai mat phing nay cdng song

song vdi mat phing nao?

Cdu hdi 2

Tfnh khoang each giiia hai mat phing dd.

Hoat ddng cua HS


MP(Oyz)


d((a), (p)) =1 -8 -(-2)1 = 6.

TOM TfiT B^l HQC

1. Vecto n^O gpi la vecta phdp tuye'n ciia mat phing ( a ) neu gia eiia vecta n

vudng gdc vdi mat phing (or).

2. Trong khdng gian Oxyz, cho mat phing id) di qua diem A/Q(XQ,>^0'^O) ^^ ^^

vecto phap tuyen niA; B ;C):

ia) : Aix - XQ) + Biy- yo) + Ciz - Zg) = 0.

Dat D = -iAxQ + ByQ + CZQ) thi phuong trinh ciia mat phing (or) dupe viet dudi

dang :Ax + fiy + Cz + D = 0 trong dd A^ + B^ + C^ > 0.

3. Cac trudng hop rieng:

72

Phuang trinh cua (a)

By + Cz +D = 0

Ax + By +D = 0

Ax + Cz +D = 0

Cz +D = 0

By +D = 0

Ax+D = 0

Dgc diem cua {a)

(a) song song hoac chira Ox.

(a) song song hoac chira Oz.

(a) song song hoac chira Oy.

(a) song song hoac triing Oxy.

(a) song song hoac trimg Oxz.

(a) song song hoac Irimg Oyz.

X y z 4. Phuang trinh doan chan : —I- — + — = 1.

a b c

5. Cho hai mat phing (a) va (a') lin lupt cd phuang trinh :

( a ) : Ax + By + Cz + D = 0

(a ' ) : A'x + B'y + Cz + D' = 0

a) Hai mat phing dd cit nhau khi va chi khi A : B : C T A' : B' : C. b) Hai mat phing dd song song khi va chi khi

A__B__C_ D_

A ' ~ f i ' " C ' D' c) Hai mat phing dd triing nhau khi va chi khi

A__B__£^_D_

A'~ B'~ C'~ D'

6. Khoang each tii mdt diem den mat phing :

\AXQ + ByQ + CZQ + D\ diMQ,ia)) = ^- i A^ + B^ + C

73

HOATDONC 5

MQT SO C6U HOI TRAC NGHI|M

Cdu 1. Hay dien diing, sai vao cac d trdng sau day:

(a) Mat phing x + 3 y - z + 2 = 0cd vecto phap tuyen la (1 ; 3 ; -1)

(b) Mat phing x + 3 y - z + 2 = 0cd vecto phap tuyen la (1 ; 3 ; 2)

(c) Mat phing x 3 y - z + 2 = 0cd vecto phap tuyen la (1 ; -3 ; -1)

(d) Mat phing -x + 3y - z + 2 == 0 cd vecto phap tuye'n la (-1 ; 3 ; -1)

Trd Idi.

a

D

b

S

c

D

d

D

Cdu 2. Hay dien diing, sai vao cac d trdng sau day:

(a) Mat phing 3y - z + 2 = 0 cd vecto phap tuyen la (0 ; 3 ; -1)

(b) Mat phing x + 3y + 2 = 0 cd vecto phap tuyd'n la (1 ; 3 ; 2)

(c) Mat phing x - z + 2 = 0cd vecto phap tuyen la (1 ; 0 ; -1)

(d) Mat phang -x + 3 y - z =:0cd vecto phap tuyen la (-1 ; 3 ; -1)

Trd Idi.

a

D

b

S

c

D

d

D

D D • D

D 0 D D

Cdu 3. Cho hinh lap phuong canh 1 nhu hinh ve.

74

Hay dien dung, sai vao cac d trdng sau day:

(a) Mat phing ABCD cd phuang trinh la z = 0

(b) Mat phing BB'CC cd phuong trinh la x = 1

(c) Mat phing A'B'C'D' cd phuong trmh la z = 1

(d) Mat phing CCD'D cd phuang trinh la y = 1

Trd Idi.

a

D

b

D

c

D

d

D

Cdu 4. Dien vao d trdng sau

D D D D

Phuong trinh ciia (a) Dgc diem ciia (a)

(a) song song hoac chiia Ox.

(a) song song hoac chira Oz.

(a) song song hoac chda Oy.

(a) song song hoac triing Oxy.

(a) song \i.ng hoac triing Oxz.

(a) song song hoac triing Oyz.

Chgn cdu trd Idi dung trong cdc bdi tap sau:

Cdu 5 Cho hinh ve. Hinh lap phuang ABCD.A'B'CD' cd canh 1.

Mat phing (A'BD cd phuang trinh nao sau:

(a)x + y + l = 0 ; (b)x + y + z = l

(c) x + z = 1 ; (d) y + z + 1 = 0

Trdldi (b).

Cdu 6. Cho hinh ve. Hinh lap phuang ABCD.A'B'CD' cd canh 1. , z

Mat phing (CBD) cd phuang trinh nao sau:

(a) -X + y +z + 1 = 0 ; (b) -x + y + z = 1

(c) X + z = 1 ; (d) y + z + 1 = 0

Trd Idi . (b).

76

Cdu 7. Cho mat phing cd phuang trinh (P) : x + 2y + 3z - 1 = 0.

Mat phing nao sau day song song vdi (P)

(a) 2x +4y + 6x 1 = 0 ; (b) 2x +4y - 6z -2 = 0;

(c) 2x - 4y + 6z -2 = 0; (d) - 2x +4y + 6z -2 = 0.

Trd Idi (a).

Cdu 8. Cho mat phing cd phuong trinh (P): x + 2y + 3z - 1 = 0.

Mat phing nao sau day triing vdi (P)

(a) 2x +4y + 6z -2 = 0; (b) 2x +4y - 6z -2 = 0;

(c) 2x - 4y + 6z -2 = 0; (d) - 2x +4y + 6z -2 = 0.

Trd Idi (a).

Cdu 9. Cho mat phing cd phuong trinh (P): x + 2y + 3z - 1 = 0.

Mat phang nao sau day vudng gdc vdi (P)

(a) 2x +4y + 6x -2 = 0; (b) 2x +4y - 6x -2 = 0;

(c) 2x - 4y + 6x -2 = 0; (d) -3x + z -2 = 0.

Trd Idi. (d).

Cdu 10. Cho mat phing cd phuong trinh (P) : x + 2y + 3z - 1 = 0.

Khoang each tilr M(l, 2, -1) den (P) la :

( a ) ^ ; ( b ) i - ; (c) ^ ; (d) ^ Vl4 14 6 7

Trd Idi (a).

HOATDONC 6

HaQNG DfiN GIfil Bfil TfiP SfiCH GIfiO KHOfi

Bai 1. Sii dung phuong trinh ciia mat phing.

a) Hudng ddn. Six dung cong thiic : Aix -XQ) + Biy -yo) + Ciz - ZQ) = 0.

Ddp sd. 2x + 3y + 5z -16 = 0.

77

b) Hudng ddn. Vecta phap tuye'n cua mat phang dd la : [u, vl = (2; 6; 6)

Su dung cdng thiie : Aix -XQ) + Biy -yo) + Ciz - ZQ) = 0..\

Ddp sd. X - 3y + 3z - 9 = 0.

- - X V 7 c) Hudng ddn. Sit dung phuang trinh doan chin : h -2— -\ = 1

- 3 - 2 - 1 Ddp so. 2x + 3y + 6z + 6 - 0.

Bai 2. Sii dung tfnh chit cua trung diem va mat phang trung true.

•Trung diem eiia AB la M = (3 ; 2 ; -5).

. AB = ( 2 ; - 2 ; - 4 )

Ddp so. (a) : X - y - 2z + 9 = 0.

Bai 3. Sii dung cac trudng hpp rieng cua mat phang.

a) Hudng ddn. Six dung hoat ddng 4.

Ddp sd. mp(Oxy): z = 0, mp(Oxz): zy= 0, mp(Oyz): x = 0.

b) mp(a) //(Oxy) nhan k(0;0;l) lam vecto phap tuye'n.

Ddp sd. (a) : z + 3 = 0.

Tuong tu : (p) //(Oyz): x -2 = 0 ; (y) //(Oxz): x - 6 =0 .

Bai 4. Su dung phuong trinh tdng quat cua mat phing.

a) Hudng ddn. mp(a) chiia true Ox va di qua P se nhan i va OP lira cap vecto

chi phuang.

i ^ = [i,OP] = (0 ; -2 ; l ) .

(a) : 2y + z = 0.

b) Hudng ddn. mp(P) chiia true Oy va di qua P se nhan j va OQ lam cap vecto

chi phuang.

i^ = p,OQ] = (-3;0;-l).

(P) : 3x + z = 0.

78

c) Hudng ddn. mpiy) chiia true Oz va di qua R se nhan k va OR lam cap vecto chi phuang.

" Y = k,ORj = (4;3;0).

(y): 4x + 3y = 0.

Bai 5. Sii dung phuang trinh tdng quat ciia mat phing.

Nen cd hinh ve de HS trung binli de tudng tupng.

A(5; 1;3)

D(4 ; 0 ; 6)

B(1 ; 6 ; 2)

C(5 ; 0 ; 4)

a) Hudng ddn. mp(ACD) se nhan AC va AD lam cap vecto chi phuang.

r ^ = [AC,AD] = ( -2 ; - l ; - l ) .

(ACD):2x + y + z-14 = 0

Tuong tu : (BCD): 6x + 5y + 3z -42 = 0.

b) Hudng ddn. mp(a) nhan AB va CD lam cap vecto chi phuong.

n^ = [AB,CD] = (lO;9;5).

(p): 10z + 9y + 5z-74 = 0.

Bai 6. Sii dung phuong trinh tdng quat ciia mat phing.

Nen cd hinh ve de HS trung binh de tudng tupng.

79

Hudng ddn. mp(a) se nhan xxn = (2;-l;3) lam vecto phap tuye'n.

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

Bai 7. Su dung phuang trinh tdng quat ciia mat phing.

Mat phing (a) can lap nhan: AB va nn n lam cap vecto chi phuong . Do dd

% = ( l ; 0 ; - 2 ) .

(a) : X - 2z + 1 = 0.

Bai 8. Sii dung tfnh chat: Hai mat phing song song khi va chi khi:

A^_B__C_ D_

A'~ B'~ C D'

. 2 m 3 -5 ^, ., a) — = — = — yt. — Tu do ta CO n = -4, m = 4.

/? -8 -6 2

. 3 -5 m - 3 ^ . ^ . . 10 9 b) — = — = — ^— I u d o t a c o n = , rn= —

2 « - 3 1 3 2 Bai 9. Sii dung true tiep cdng thiic tfnh khoang each

IAJTQ + ByQ + CZQ + D diMQ,ia)) = ^-

V ^ B^ +C'

a) diMQ, (a)) = 5

80

b) diMQ, ia)) = — .

13

c) diMQ, (a)) = 2.

Bai 10. Sir dung phuong phap tpa dp. Nen ve hinh de ddi tucmg HS trung binh de tudng tupng.

Tit hinh ve HS hay vie't tpa dp cac dinh cua hinh lap phuong.

a) Chiing minh hai mat phing (AB'D') va (CBD) cd chung mdt vecto phap tuyen.

b) Lap phuong trinh mat phing (CBD) va tfnh khoang each tur A den mat

phang dd.

Ddp sd. d = —;=. V3

H.hoc 12/2

§3. Phtfctog trinh dtfofng thang

trong k h o n g g i a n

(tie1t 10,11, 12,13,14,15)

I. MUC TIEU

1. Kien thiirc

HS n im dugc:

1. Hiuong tiinh tham so ciia dudng thing: Vecto chi phuong, cap vecto phap tuyen

ciia dudng thing.

2. Dieu kien de hai dudng thing song song, cat nhau va cheo nhau.

2. Kl nang

• Lap dupe phuong trinh dudng thing.

• Tfnh dupe khoang each ttr mdt diem den mdt dudng thing.

3. Thai do

• Lien he dupe vdi nhieu van de cd trong thuc te' ve dudng thing trong khdng gian.





• Hlnh ve 3.14 de'n 3.17 trong SGK.


• Chuan bi san mdt vai hinh anh thuc te' ve dudng thing.

2. Chuan bj cua HS

Dpc bai trudc d nha, dn tap lai mdt sd kie'n thiic da hpc ve dudng thing.

82

m. DHAN PHOI THOI LUQNG


Tie't 1 : Tii dau den het dinh If 1.

Tie't 2 : Tiep theo de'n het phan I

Tie't 3 : Tie'p theo den het boat dong 4.

Tie't 4 : Tie'p theo den het vf du 2.

Tiet 5 : Tie'p theo de'n het vf du 4.

Tie't 6 : Phin cdn lai va bai tap.

IV. TIEN TDlNH DAY HOC

n. DRT VAN DC •

Cau hdi 1.

Neu khai niem ve vecto phap tuyen ciia mat phing.

Cau hdi 2.

Neu dieu kien de hai mat phing song song.

Cau hdi 3.

Neu cdng thiic tfnh khoang each tii mdt diem den mdt mat phing.

Cau hdi 4.

Neu each lap phuong trinh mat phing di qua ba diem.

B. BRI MOI

HOATDONC 1

DAT VAN DE

• GV dat ra mdt so tinh hudng:

HI. Hay neu phuong trinh tham sd ciia dudng thing trong mat phing.?

H2. Hai dudng thing trong khdng gian cd nhiing vi trf tuong ddi nao?

83

H3. Trong khdng gian phuang trinh dudng thing dupe lap nhu the nao?

H4. Hai mat phing cat nhau theo mdt giao tuyen. Phai chang mot dudng thang dupe xac dinh bdi hai mat phing cit nhau.

HOATDQNC 2

I. PHUONG TRINH THAM SO CUA DUCJNG T H A N G


Hoat ddng cua GV

Cdu hdi 1

Neu dieu kien de ba diem thing hang.

Cdu hoi 2

Hay chiing td ba diem dd thing

hang.

Hoat ddng ciia HS


Ba diem MQ, M , , M J thing hang khi

va chi khi MoMj = kMiM2


Ta cd

M o M i = ( t ; t ; t ) ; MjMj = ( t ; t ; t ) .

Tii dd ta cd dieu phai chiing minh.

• GV neu dinh If:

Trong khdng gian Oxyz cho dudng thdng A di qua Mo(xo;yo;zo) va

nhdn a = (a,;a2;a3) la vec ta chi phuang. Dieu kien cdn vd du de M(z ;

y ; z) ndm tren A Id cd mat sdthUc t sao cho

X = XQ + t a j

y = yo+ta2

z = ZQ + ta3

GV hudng din HS chiing minh dinh If tren.

GV neu dinh nghia :

84

Phuang trinh tham sd cua dudng thdng A di qua MQ (xo;yo;Zo) ^^ "^'^'^

a = (a, ;a2 ; a3) la vec ta chi phuang cd dgng :

x = Xo + ta,

y = yo + ta2

z = Zo + ta3

H5. Viet phuang trinh tham sd dudng thing A di qua gdc tpa dp va nhan a ( l ; l ; 2 )

lam vecto chi phuong.

H6. Mpi dudng thing deu cd the viet dudi dang tham sd. Diing hay sai ?

• GV neu chii y :

Neu cac tpa dp cua vecto a deu khac 0 thi A cd the viet dudi dang :

X - X Q ^ y - y p ^ z - z o

aj a2 a3

• GV cho HS thue hien vf du 1. (xem SGK).

• GV hudng din HS thuc hien vf du 2 kl ludng:

Hoat ddng ciia GV

Cdu hoi 1

Xac dinh vecta AB.

Cdu hoi 2

Viet phuong trinh tham sd cua dudng thing AB.

Hoat ddng cua HS

Ggi y trd Idi cdu hdi I

AB = ( l ; - 2 ; 3 ) ,


Chii y ring ta cd the sii dung hai diem

A hoac B.

«

x = l + 2t

y = - 2 + 2t hoac •

z = 3 -3 t

'x = 3 + 2t

y = 2t

z = 3t

85

• GV hudng din HS thuc hien vf du 3 ki ludng:

Hoat ddng cua GV

Cdu hoi 1

Dieu kien de mdt dudng thing

vudng gdc vdi mdt mat phing la gi?

Cdu hdi 2

Hay chiing minh bai toan.

Hoat ddng cua HS


Vecta chi phuong ciia dudng thing va vectc

piiap tuyen ciia mat p^ing cdng tuyeii.

Ggi y trd Idi cdu hdi 2 Ta ed vecto chi phuang cua d la :

a = ( l ;2;3) .

Vecto phap tuyen ciia (a) la :

^ = (2;4;6)


Hoat ddng cua GV

Cdu hdi 1

Hay xac dinh diem M.

Cdu hoi 2

Xac dinh vecta chi phuong ciia A.

Hoat ddng ciia HS


M (-1 ; 3 ; 5).


i = ( 2 ; - 3 ; 4 ) .

HOATDONC 3

n. Dl£u KIEN DE HAI DUONG THANG SONG SONG, CAT NHAU, CHEO NHAU


cau a

Hoat ddng cua GV

Cdu hdi 1

Chiing minh M thudc d.

Hoat ddng cua HS


HS tu chiing minh.

86

Cdu hoi 2

Chiing minh M thuoc d'


HS tu chiing minh.

caub Hoat ddng cua GV

Cdu hoi 1

Tim vecto ehi phuong cua d.

Cdu hdi 2

Tim vecto chi phuong cua d'

Hoat ddng cua HS


HS tu tim


HS tu tim.

• GV dat vin de :

H7. Trong khdng gian d va d' cd nhirng vi trf tuong ddi nao ?

H8. Chiing ta cd the tim dupe didu kien cua mdi trudng hop cu the hay khdng ?

1. Dieu kien de hai dudng thang song song


d song song vdi d' khi vd chi khi chiing khdng cd diem chung vd hai vecta

chi phuang a vd a' cdng phucmg.

H9. Hay viet bieu thufc toan hpc de d // d'

HIO. Khi nao d trung d'

• GV ke't luan tdm tit.


Hoat ddng cua GV

Cdu hoi 1

Hay tim vecto chi phuong cua d.

Cdu hdi 2

Hay tim vecto chi phuong cua d'

Hoat ddng cua HS

Gffi y trd Idi cdu hdi I

GV gpi HS tra Idi. i


GV gpi HS tra Idi.

87

Cdu hdi 3

Chiing minh hai vecto chi phuong

ciia d va d' eiing phuang.

Cdu hdi 4

Chiing minh d va d' khdng cd

diem chung.


HS tu chiing minh.


HS tu chiing minh.

• Thuc hien &.4 trong 3 phiit.

Cach 1

Hoat ddng cua GV

Cdu hdi I

Hay chpn diem M e d .

Cdu hoi 2

Chiing minh M thudc d'

Hoat ddng ciia HS


HS tu chpn diem M nao dd.


HS tu chiing minh.

vay d va d' triing nhau.

De chiing minh hai dudng thdng trdng nhau ta chgn diem M bdt ki thuoc

dudng thdng ndy vd chimg minh nd thudc dudng thdng kia.

Cach 2

Hoat ddng cua GV

Cdu hdi 1

Chiing minh hai vecto chi phuang eiia hai dudng thing ciing

phuang.

Cdu hdi 2

Chiing minh d va d' cd diem chung.

Hoat ddng ciia HS


HS tu chiing minh.


HS tu chiing minh.

2. Dieu kien de hai dudng thang cat nhau

Xet cac dudng thing cd phuong trinh :

88

x = Xn +a,t

y = yo+a2t

z = Zo +a3t

d' :

x = x'o + a'i t '

y = y o + a 2 t '

z = z'o + a'3 t'

• GV neu cac eau hdi sau

H l l . Hai dudng thing cit nhau cd may diem chung ?

HI2. Khi hai dudng thing cit nhau thi he phuang trinh :

x'o+a'i t' = Xo +ait

yo + a'2t' = yo + a2t

Zo+a '3t ' = Z o + M

cd bao nhieu nghiem.


Hai dudng thdng cdt nhau khia vd chi khi he phuang trinh

x'o + a'i t' = Xo +a]t

• y o + a 2 t ' = yo+a2t

z 'o+a'3t ' = Zo+a3t

cd dung mdt nghiem.

• GV neu chii y trong SGK


Hoat ddng ciia GV

Cdu hoi 1

De tim giao diem cua hai dudng thing ta cin xet he nao?

Cdu hoi 2

Hay tim t va t'

Cdu hoi 3

Chiing minh he cd nghiem duy nha't.

Hoat ddng cua HS


GV gpi HS tra Idi.


GV gpi HS tra Idi.


HS tu chiing minh.

89

3. Dieu kien de hai dudng thang cheo nhau

HI3. Hai dudng thing cheo nhau cd diem chung hay khdng ?

HI4. Khi hai dudng cheo nhau thi he phuang trinh :

x'o + a'i t' = Xo +ait

y'o + a-2t' = yo+a2t

zo + a ' j t :Zo+a3t

cd nghiem hay khong ?

HIS. Hai dudng thing cheo nhau thi hai vecta chi phuang cd ciing phuong hay

khdng ?


Hai dudng thdng cheo nhau khi vd chi khi a vd a' khdng cimg phuang vd

he phuang trinh :


y'o + a'2t ' = yo + a2t

z'o+a 31' = Zo + a3t

vd nghiem.

• Thuc hien vf du 3 trong 5'.

Hoat ddng cua GV

Cdu hdi 1

Xac dinh hai vecto chi phuang cua hai dudng thing?

Cdu hoi 2

Chiing minh hai vecto ehi phuang dd khdng cimg phuang.

Cdu hdi 3

Chung minh he

Hoat ddng cua HS


GV gpi HS tra Idi.


GV gpi HS tra Idi.


HS tu chiing minh.

90

l + 2t = l + 3t'

- l + 3t = - 2 + 2t'

5 + t = - l + 2t'

cd nghiem duy nhat.

Thuc hien vf du 4 trong 5'.

Hoat ddng ciia GV

Cdu hoi 1

Xac dinh hai vecto chi phuong

cua hai dudng thing?

Cdu hoi 2

Hai dudng thing vudng gdc khi

nao ?

Cdu hdi 3

Chiing minh a.a' = 0 .

Hoat ddng ciia HS


GV gpi HS tra Idi.


Khi hai vecto chi phuong ciia chiing

vudng gdc.


HS tu chiing minh.

• GV neu cac vi tri tuong ddi cua dudng thang va mat phang

Xet mat phing : (a) : Ax + By + Cz + D = 0

Dudng thing d

x = Xo + ait

y = yo + a2t

z = Zo +a3t

H15. Giao diem cua (a) va d la nghiem cua phuang trinh nao ?

• GV cho HS tra Idi va ket luan :

Giao diem eiia (a) va d la nghiem ciia phuong trinh :

A(Xo + a,t) + B(yo + ajt) + C(Z(, + ajt) + D = 0. (1)

He (1) vd nghiem thi d // (a).

He (1) cd vd sd nghiem thi d thudc (a).

He (1) cd 1 nghiem thi d cit (a).

HI6. Khi nao d vudng gdc vdi (a) ?

• Thuc hien A s trong 5 phiit.

cau a.

Hoat ddng cua GV

Cdu hdi 1

De tim sd giao diem ta

phuang trinh nao ?

Cdu hdi 2

Phuang trinh dd cd nghiem

khdng ?

Cdu hdi 3

Hay ke't luan.

xet

hay

Hoat ddng cua HS


(2 + t) + (3 -1) + 1 - 3 = 0

» 4 = 0


Phuang trinh vd nghiem.


d // (a)

caub. Hoat ddng ciia GV

Cdu hdi 1

De tim sd giao diem ta phuong trinh nao ?

Cdu hdi 2

Phuang trinh dd cd nghiem

khdng ?

Cdu hdi 3

Hay ke't luan.

xet

hay

Hoat ddng cua HS


(l + 2t) + ( l - t ) + l - t - 3 = 0

o 0 = 0


Phuong trinh cd vd so nghiem.

Gffi y trd lai cdu hoi 3

d thudc(a)

cauc. Hoat ddng cua GV

Cdu hdi 1

De tim sd giao diem ta xet phuong trinh nao ?

Hoat ddng ciia HS

Gffi y trd Idi cdu hoi I

( l + 5 t ) + ( l - 4 t ) + l + 3 t - 3 = 0

<» 4t = 0

92

Cdu hdi 2

Phuang trinh dd cd khong ?

Cdu hoi 3

Hay ke't luan.

nghiem hay


Phuang trinh cd 1 nghiem.


d c i t ( a )

HOATDONC 4

TOM TfiT Bfir HQC

1. Phuong trinh tham sd cua dudng thing A di qua Mo(xo;yo;zo) va nhan

a = (ai;a2 ;a3) la vec to chi phuong cd dang :

X = XQ + taj

y = yo+ta2

z = ZQ + ta3

2. d song song vdi d' khi va chi khi chiing khdng cd diem chung va hai vecto chi

phuong a va a'ciing phuong.

3. Hai dudng thing cit nhau khi va chi khi he phuang trinh :


y 'o+a '2t ' = yo + a2t

z'o + a'31' = Zo +a3t

cd diing mdt nghiem.

4. Hai dudng thing cheo nhau khi va chi khi a va a' khdng ciing phuang va he

phuong trinh :

x'o + a'i t' = Xo + ait

y 'o+a'2t ' = yo+a2t

z'o+a'31' = Zo + a3t

vo nghiem.

93

HOATDONC 5

MQT SO C^U HOI TRBC NGHIEM

Cdu 1. Hay dien dung (D) sai (S) vao cac cau sau

(a) Dudng thing

x = l + 2t y = 2 + 3t cd vecto chi phuong la a (2 ;3 ; - l ) Ll

z = - l - t

(b) Dudng thing

(e) Dudng thing

(d) Dudng thing

Trd Idi.

x = l + 2t

y = 2 + 3t cd vecto chi phuang la a ( l ; 2 ; - l ) |_J

z = - l - t

x = l + 2t

y = 2 + 3t cd vecto phap tuyd'n la a

z = - l - t

i;0;-.) D

x = l + 2t

y = 2 + 3t cd vecta phap tuyen la a ( l ; 0 ; - l )

z = - l - t

a

D

b

S

c

D

d

S

Cdu 2. Hay dien diing sai vao cac eau sau :

'x = l + 2t

y = 2 + 3t di qua diem M ( 1 ; 2 ; - 1 ) (a) Dudng thang

(b)) Dudng thing

z = - l - t

x = l + 2t

y = 2 + 3t di qua diem M ( 2 ; 3 ; - 1 )

z = - l - t

D

•

D

94

(c) ) Dudng thing

x = l + 2t

y = 2 + 3t di qua diem M(0;2;l)

z = - l - t

(d) Ca ba eau tren deu sai

Trd Idi.

U

U

a

D

b

s

c

S

d

S

Cdu 3. Hay dien diing sai vao cac cau sau :

x = l + 2t

y = 2 + 3t cd phuang trinh chfnh tic la

z = - l - t

(a) Dudng thing x - l _ y - 2 _ z + l

2 " 3 ~ -1

D

(b)) Dudng thing

x = l + 2t ^ ^ , , V , , . , - ,v x - 2 y - 3 z + 1

y = 2 + 3t CO phuang trmh chinh tac la = =

D z = - l - t

(c) Dudng thing

(d) Dudng thing

Trd Idi.

x = l + 2t

y = 2 + 3t vudng gdc vdi mat phing 2x+3y-z+l= 0 |_|

z = - l - t

x = l + 2t

y = 2 + 3t vudng gdc vdi mat phing x+2y-z +3 = 0 |_J

z = - l - t

a

D

b

S

c

D

d

S

95

Chpn cau tra Idi diing trong cac cau sau:

'x = l + t

Cdu 4. Dudng thing A : y = 2 + 2t song song vdi dudng thang:

z = l - t

(a )d :

(c)

x - 3 _ y - 4 _ z + 5

x - 3 y - 4 z + 5

-1 2

Trd Idi (a).

-1

(b)

(d)

x - 3 _ y - 4 _ z + 5

x - 3 y - 4 z + 5

1 1

Cdu 5. Dudng thing A

(a) (1 ; 2 ; 1)

(c) (2 ; 3 ; 1)

Trd Idi. (a).

x = l + t

y = 2 + 2t di qua diem nao sau day:

z = l - t

( b ) ( l ; 2 ; - l )

( d ) ( l ; 3 ; - l )

Cdu 6. Dudng thang A :

x - 3 y - 4 z + 5

x = l + t

y = 2 + 2t va dudng thing nao sau day cat nhau

z = l - t

(a )d :

(c)

1 2 -1

x - 3 _ y - 4 _ z + 5

~ -1

(b)

(d)

x - l _ y - 2 _ z - l

^ ~ " 1 ~ 1

x - 3 _ y - 4 _ z + 5

1 " 2 ~ 1 - 1 2

Trd Idi (b).

Cdu 7. Cho A ( 1 ; 2 ; 3) va B(-l ; 1 ; 1). Phuong trinh dudng thing AB la

(a)

x = l - 2 t

y = 2 + lt

z = 3 + 2t

(b)

x = l - 2 t

y = - 2 - t

z = 3 -2 t

96

x = - l + 2t

(c) < y = 1 + It

z = l - 2 t

Trd Idi (d).

Cdu 8. Cho dudng thing A :

song vdi A la

'x = l - 2 t

(a) j y = 2 + It

z = 3 + 2t

(d)

x = l + 2t

y = 2 + lt

z = 3 + 2t

x = l + t

y = 2 + 2t, dudng thing di qua M(l ; -1 ; 2) va song

z = l - t

(b)

x = - l + 2t

(c) <{ y = 1 + It (d)

z = l - 2 t

Trd Idi. (b).

Cdu 9. Cap dudng thing nao sau day song song

x = l - t

y = - l - 2 t

z = 2 + t

x = l + 2t

y = 2 + lt

z = 3 + 2t

(a)

X

y z-

= l - 2 t

= 2 + lt

= 3 + 2t

va<

(c)

x = - l + 2t

y = 1 + It va

z = l - 2 t

x = l - t

y = - l - 2 t

z = 2 + t

x = l + 2t

y = 2 + lt

z = 3 + 2t

(b)

(d)

1-t

y = - l - 2 t va <y = 2 + lt

z = 2 + t

x = l + 2t

y = 2 + lt va

z = 3 + 2t

x = l + 2t

z = 3 + 2t

x = - l - 2 t

y = 2 - t

z = - 3 - 2 t

Trd Idi (d).

Cdu 10. Cap dudng thing nao sau day vudng gdc vdi (P) cd phuang trinh

2x+2y - z+2 = 0

(a)

x = l - 2 t

y = 2 + 2t

z = 3 - l t

(b)

x = l - t

y = - I - 2 t

z = 2 + t

H h^r 19/9 97

( c ) -

x = - I + 2t

y = l + lt

z = l - 2 t

Trd Idi (a).

(d ) .

'x = l + 2t

y = 2 + lt

z = 3 + 2t

HOATDQNC 6

naOFNG D^N Gl^l B^l T6P SffCH GI^O KHOfi

Bai 1. Sir dung phuong trinh tham sd hoac phuong trinh chfnh tic ciia dudng thing:

cau a. HS tu giai.

cau b. Chii y ring dudng thing A ±(P) <=> u^ = kup

Ta cd n(j = (l; 1;-1). Tii dd suy ra dudng thing cin tim.

cau c. d // d <=> Uj = kuj.

HS tu giai.

cau d. Phuang trinh dudng thang PQ cd Upg = PQ

HS tir giai.

Bai 2. GV nen ve hinh de de tudng tupng.

98

De tim hinh chieu ciia mdt dudng thing d tren mdt mat phang (a) ta lam nhu sau:

Budc 1. Lap phuog trinh mat phang (P) di qua d va vudng gdc vdi d

Mat phang nay di qua mdt diem cua d va cd vecto phap tuye'n n = u^, n^

Budc 2. Giao cua (a) va (P) la d' cd u^. = rnm,npl

cau a.

Xac dinh vecto phap tuye'n ciia (Oxy): k = (O; 0; 1).

Xac dinh vecto chi phuang cua d: Uj = (l;2;3) .

Xac dinh vecto phap tuye'n cua mat phing (a) di qua d va vudng gdc vdi Oxy:

n = [ud,nojJ = (-2; 1 ; 0).

Vecto chi phuang cua d' la : u = (1 ; 2 ; 0).

Ta cd M (2 ; -3 ; 0) thudc giao tuyen ciia (a) va (Oxy) nen nd thudc d'

'x = 2 + t

y = - 3 + 2t

z = 0

Phuang trinh dudng thang d' la

Cau b. Giai tuang tu cau a.

'x = 0

Ddpsd. j y = - 3 + 2t

z = l + 3t

Bai 3. Su dung true vi trf tuang ddi eiia hai dudng thing

cau a. Hudng ddn. Xet he phuang trinh :

- 3 + 2t = 5 + t'

- 2 + 3t = - l - 4 t '

^6 + 4t = 20 + t'

Giai ra ta ed hai dudng thing cit nhau.

99

Bai 4. Hudng ddn. Six dung true tiep tfnh chat ciia hai dudng thing cit nhau khi va

chi khi he :

x 'o+a ' i t ' = Xo + ait

y'o + a'2t' = yo+a2t

z'o + a'3t' = Zo + a3t

cd nghiem duy nhat:

Nhu vay hai dudng thing da cho cit nhau khi va chi khi he :

'l + at = l - t '

t = 2 + 2t'

^- l + 2t = 3 - t '

cd nghiem duy nhat.

Giai ra ta cd a = 0.

Bai 5. Hudng ddn. De tim giao diem ciia dudng thing d va mat phing (a) ta thay

cac gia tri cdn phu thudc vao tham sd t ciia x, y va z vao phuong trinh ciia (a). Giai

ra ta dupe t. Tit dd ta cd ket qua ;

cau a.

Ta ed 3(12 + 4t) +5(9 + 3t) -(1+ t) -2 = 0 o 26t + 78 = 0 « t = -3.

Tii dd ta cd : X = 0, y = 0, z = -2.

Dudng thing d cit mat phing (a) tai mdt diem duy nha't.

cau b. Giai tuong tu eau a ta cd d //(a).

cau c. d thudc mat phing (a).

Bai 6. Hudng ddn. Chiing minh A // (a) va tim mdt diem thudc A.

Tim khoang each tir diem dd de'n (a).

T a e d M ( - 3 ; - l ;-1) e A.

d(A, (a)) = d(M,(a)) = -3

100

Bai 7. Hudng ddn.

cau a. Dung mat phing (a) di qua A va vudng gdc vdi A. Giao diem cua (a) va

(A) la diem can tim.

Vecto phap tuyen eiia (a) la n^ = u^ = (l;2;l)

Mat phing (a ) : x +2y +z - 1 = 0.

''lo-i^ 2 2

Ttr dd giai nhu bai 6 ta cd giao diem la H

Cau b. Hudng ddn.

H chfnh la trung diem ciia AA'

^A' -2Xf^ - x ^

y A ' = 2 y H - y A

^A' = 2 Z j ^ - Z ^

Tit dd tacd A ' ( 2 ; 0 ; - l ) .

Bai 8. Hudng ddn.

cau a. Dung qua M mdt dudng thing A vudng gdc vdi (a). Dudng thing dd cd

vecto ehi phuong chfnh la vecto phap tuyen ciia (a). Giao ciia dudng thing A va

(a) chfnh la diem can tim.

'x = l + t

y = 4 + t

z = 2 + t

tiirddtacd H ( - l ; 2 ; 0 ) .

cau b. Giai tuong tu bai 7.

M (-3 ; 0 ; -2).

cau c. d(M, (a)) = MH = 2 ^ .

Bai 9. Hudng ddn.

De chiing minh hai dudng thing cheo nhau ta chiing minh

101

Hai dudng thing cd hai vecto chi phuang khdng cong tuye'n.

He phuang trinh

x'o+a , t = XQ +ait

y o + a'2t' = yo+a2t

z\+a\t' = Zr

vo nghiem

HS tu chung minh.

Bai 10. Hudng ddn.

Hinh ve :

Lo-ra3t

Dat he true tpa dp nhu hinh be.

Lap phuang trinh mat phang (A'BD) :x + y + z - l = 0 .

Lap phuang trinh mat phang (CB'D') :x + y + z - 2 = 0.

Ddp so.

d(A, (A'BD)) 1

v -d(A, (CB'D')) = - ^

V3

102

On tap chu'ofng 3

(tiet 16,17)

I. MUC TIEU

1. Kie'n thurc

HS n im duac:

1. Khai niem:

+ Dinh nghia he true tpa dp trong khdng gian.

+ Dinh nghia phuong trinh mat cau.

+ Dinh nghia phuong tiinh mat phang trong khong gian.

+ Dinh nghia phuong tiinh dudng thing trong khdng gian.

2. Mdt so dinh If va menh de quan trpng:

+ Cac tfnh chat ciia phep toan ve toa dp trong khdng giari.

+ Tich vd hudng ciia bed vecta.

+ Tich hdn hop ciia hai vecta.

+ Dieu kien de hai mat phang song song, hai mat phing cit nhau.

+ Vi trf tuang ddi ciia dudng thing va mat phing.

+ Vi trf tuong ddi cua hai dudng thing.

+ Khoang each tir mot diem den mot mat phing, den dudng thing.

2. Kl nang

• Lap dupe phuang trinh : mat phing, mat cau va dudng thing.

• Tfnh duoc khoang each tur mot diem den mot mat phing, den dudng thing.

3. Thai do

• Lien he dupe vdi nhieu van de cd trong thuc te' vdi mon hpc hinh hpc khdng gian.





• Chuin bi dn tap toan bo kie'n thiic trong chuang.

• Chuin bi mdt de'n hai bai kiem tra.

• Cho hpc sinh kiem tra va cham, tra bai.

2. Chuan bi cua HS :

On tap lai toan bd kien thiie trong chuong, giai va tra Idi cac cau hdi bai tap trong chuong.

in. PHAN PHOI THOI LUONG


Tie't 1 : On tap.

Tie't 2 : Kiem tra 1 tie't.

IV. TI!'N TPlNH DAY HOC

n. DRT VAN D€

Cau hdi 1.

Em hay nhic lai : Cac khai niem tpa dp trong khdng gian, phuong

trinh mat cau va phuang trinh mat phang.

Cau hdi 2.

Neu mdi quan he giiia dudng thing va mat phang.

Cau hdi 3.

Vie't cac cdng thiic tfnh khoang each.

B. Bni M 6 I

HOATDONCl

1. On tap kie'n thurc co ban trong chudng

a) Tdm tat h' thuyet co ban.

• Toa dp ciia diem va toa dp ciia vecto ddi vdi he toa dp Oxyz :

104

+ Diem M cd toa dp (x ; y ; z) Ci> OM = xi + yj + zk.

+ Vecto u cd toa dp (x ; y ; z) <=i> M = xi + yj + zk.

+ AB = [XQ - x^ ; yg - y^ ; Zg - z^).

• Tfch vd hudng va tfch vecto : Cho M = (x ; y ; z) va v = (x' ; y' ; z') thi:

+ Tfch vo hudng ciia M va v la sd : u.v = xx'+ yy'+ zz'

+ Mdt sd tfnh chat: u ± v c^ u.v = 0 ;

• Phuang trinh mat eau : phuong trinh cd dang

x^ + y^ + z^ + 2ax + 2by + 2cz + d = 0,

vdi dieu kien a + b + c > d, la phuong trinh mat cau cd tam (-a ; - b ; -c) va cd ban

kfnh R=V^ + b^+c'

• Phuong trinh mat phing : phuong trinh

Ax + By + Cz + D = 0,vdiA^ + B^ + C^>0

la phuong trinh ciia mat phing cd vecto phap tuye'n la HiA ; B ; C).

• Mat phing di qua diem (XQ ; yo ; ZQ) vdi vecto phap tuyen iA;B;C)c6 phuang tnnh:

A(x - XQ) + Biy- yo) + Ciz - ZQ) = 0.

• Phuang trinh dudng thing :

Phuang trinh tham sd

X = XQ + at

y = yQ+bt

Z - ZQ + ct

vdi a^ + b^ + c^> 0.

105

+ Phuang trinh chinh tdc :

X - x^ y - yn z - Zn

a b c

Dudng thing vdi phuang trinh dd di qua diem (XQ ; yo ; ZQ) va cd vecta chi phuang

u = ia; b; c).

• Vi trf tuong ddi giiia hai mat phing (a) va (a') :

Ne'u (a) cd phuang trinh : Ax + 5y + Cz + D = 0 va (a') cd phuang trinh : A'x + B'y

+ Cz + D' = 0, thi

+ (a) va ia') cit nhau khi va chi khi A : 5 : C ; A' : fi': C

+ id) va ia") song song khi va chi khi — = — = — ^ — V V V 7 5 6 A' B' C D'

+ ia) va (a') triing nhau khi va ehi khi — = — = — = — . ^ A' B' C D'

+ ia) va ia') vudng gdc vdi nhau khi va chi khi AA'+ BB'+ CC = 0

• Khoang each :

+ Khoang each giira hai diem A(x^ ; y^ ; z^) va fi(xg ; y^ ; Zg) :

+ Khoang each tir diem Mo(xo ; yo ; Zo) de'n mat phing (a) cd phuong trinh

Ac + fiy + Cz + D = 0 la :

VA^ + 5^ + C ^

b) Cau hdi trac nghiem nham dn tap kie'n thurc:

GV nen dua ra mot he thdng cau hdi trie nghiem nhim on tap toan bd kien thiic trong chuang.

Sau day xin gidi thieu mdt sd cau hdi:

106

/. HAY KHOANH TRON CAU DUNG, SAI TRONG CAC CAU SAU MA EM

CHO LA HOP LI.

Cdu 1. Hai dudng thing song song thi hai vecto chi phuong cdng truyen.

(a) Diing (b) Sai.

Cdu 2. Dudng thing A 1 (a) khi va chi khi vecto chi phuong cua A va vecto phap

tuye'n cua (a) cdng tuye'n.

(a) Diing (b) Sai.

Cdu 3. Cho hai dudng thing

X = x'o + a'i t'

y = y 'o+a '2t ' d :

X = Xo + ait y = yo + a2t vad '

z = Zo +a3t z- z'o + a'31'

He phuang trinh

X ' n + a ' i t ' = X n + a , t

(a) Diing


y 0 + a '2 t' = yo + a2t vd nghiem thi d // d'

it

(b) Sai.

z'o+a 3t zo + ast

X = Xo + ait

y = yo +a2t va d' <

z = ZQ +a3t

X = x'o + a'i t'

y = y o + a ^ t '

z = z'n + a' t'

He phuong trinh

x'o + a'i t' = Xo+ait

y '0 + a '2 t' = y0 + a2t vo nghiem thi d va d' cheo nhau

z'o + a'3t' = Zo+a3t

(a) Diing


(b) Sai.

d :

x = Xo +ait

y = yo + a2t vad '

z = Zo + a3t

X = x'o + a'i t'

y = y'o + a'2t '

z = z'o + a'3 t'

107

He phuang trinh


y o + a 2 t' = yo +a2t cd nghiem thi d vad ' cit nhau

z' o-ra '3 t ' -Zo+a3t

(a) Diing


X = x o + a 11'

y = yo +a2t va d'

(b) Sai.

d:

z = Zo+a3t

He phuong trinh

y = y'o + a'2t '

z- z'o + a'3 t'


y 0 + a '2 t' = yo + a2t cd nghiem duy nhit thi d va d' cit nhau

z 'o+a'3t ' = Zo+a3t

(a) Diing (b) Sai.

Cdu 7. Hai mat phing song song khi va chi khi hai vecto phap tuyen ciia chiing

cong tuyen.

(a) Diing (b) Sai.

Cdu 8. Hai mat phang vudng gdc vdi nhau khi va chi khi tfch vd hudng hai vecto

phap tuyen bing 0.

(a) Diing (b) Sai.

Cdu 9. Cho mat ciu (I, r) va mdt mat phang (a).

d(I, (a)) < r thi giao ciia mat ciu va (a) la dudng trdn.

(a) Diing (b) Sai.

Cdu 10. Cho mat ciu (I, r) va mdt mat phing (a).

d(I, (a)) = r thi (a) tie'p xiie vdi (I, r).

(a) Diing (b) Sai.

Cdu 11. Cho mat ciu (I, r) va mdt mat phang (a).

d(I, (a)) > r thi giao cua mat ciu va (a) la dudng trdn.

(a) Diing (b) Sai.

108

Cdu 12. Phuang trinh dudng thing AB nhan AB lam vecto chi phuang.

(a) Diing (b) Sai.

Cdu 13. Phuang trinh mat phing vudng gdc vdi AB nhan AB lam vecto phap tuyen

(a) Diing (b) Sai.

Cdu 14. Phuang trinh mat phing di qua ba diem A, B, C nhan AB,AC lam

vecto phap tuyen.

(a) Diing (b) Sai.

Cdu 15. Phuang trmh mat phing di qua ba diem A, B, C nhan AB,BC lam

vecto phap tuye'n.

(a) Diing (b) Sai.

Cdu 16. Phuang trinh mat phing di qua ba diem A, B, C nhan rCB,AC lam

vecto phap tuyen.

(a) Diing (b) Sai.

Cdu 17. Phuang trinh mat ciu tdng quat cd dang:

x^ + y^ + z + 2ax + 2by + 2cz + d = 0.

(a) Diing (b) Sai.

Cdu 18. Phuang trinh mat cau tdng quat cd dang:

2x^ + y^ + z + 2ax + 2by + 2cz + d = 0.

(a) Diing (b) Sai.

Cdu 19. Mat cau tam I(a, b, c) ban kfnh r la :

( x - a ) ' + ( y - b ) ' + ( z - c ) ' = r

(a) Diing (b) Sai.


( x - a ) ' + (y-b)2 + (z-c)^ = r'

(a) Diing (b) Sai.

109

Cdu 21. Phuang trinh mat ciu : phuong trinh cd dang

x^ + y + z + 2ax + 2by + 2cz + d = 0, 2 2 2

vdi dieu kien a + b + e < d, la phuong trinh mat eau cd tam (-a ; - b ; -c) va cd ban

kfnh r = yla^ + b^ + c^ - d

(a) Diing (b) Sai.

Cdu 22. Phuang trinh mat ciu : phuang trinh cd dang 2 2 2

X + y + z + 2ax + 2by + 2cz + d = 0, 2 2 2

vdi dieu kien a + b + c = d, la phuong trinh mat cau cd tam (-a ; - b ; -c) va cd ban

kfnh R = Va^ + b^ + c^ - d

(a) Diing (b) Sai.

Cdu 23. Phuang trinh mat ciu : phuang trinh cd dang 2 2 2

X + y + z + 2ax + 2by + 2cz + d = 0, 2 2 2 v

vdi dieu kien a + b + c > d, la phuong trinh mat cau cd tam (-a ; - b ; -c) va cd ban

kfnh R = Va^ + b^ + c^ - d

(a) Diing (b) Sai.

Cdu 24. M ± v <=>M.v = 0

(a) Diing (b) Sai.

Cdu 25. Cho M = (x ; y ; z) va v' = (x ; y' ; z'). Tfch vd hudng ciia M va v la sd

u.v = xx'+ yy'+ zz'

(a) Diing (b) Sai.

Cdu 26. Diem M cd toa dp (x ; y ; z) <=> OM = xi + yj + zk.

(a) Diing (b) Sai.

Cdu 27. Vecto u cd toa dp (x ; y ; z) <=> M = xi + yj + zk.

(a) Diing (b) Sai.

110

Cdu 28. Ne'u diem A = (x^ ; y^ ; z^) va diem fi = (x^ ; y^ ; Zg) thi

A5 = (x5 - x^ ; y^ - y^ ; Z5 - z^ ) .

(a) Diing (b) Sai.

Cdu 29. Phuang trinh chfnh tic :

= -—^^ = ^ voi abc ?i 0. a b c

Dudng thing vdi phuang trinh dd di qua didm (XQ ; yo ; zd) va cd vecto chi phuong

u = ia; b; c).

(a) Diing (b) Sai.

Cdu 30. Cho (a) cd phuong trinh : Ax + By + Cz + D = 0 va (a') cd phuang trinh :

.4'x + By + C'z + D' = 0

ia) va (ttO cit nhau khi va ehi khi /I : B : C ;i ^ ' : B ' : C

(a) Diing (b) Sai.

Cdu 31. Cho (a) cd phuong trinh : Ax + By + Cz + D = 0 va (a') cd phuang trinh :

A'x + B'y + Cz + D' = 0

(a) va (a') song song khi va chi khi —- = — = ~-; ^ —i A' B' C D

(a) Diing (b) Sai.

Cdu 32. Cho (a) cd phuang trinh : Ax + By + Cz + D = 0 va (a') ed phuong trinh :

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

ia) va ia') triing nhau khi va ehi khi —; = -— = — = —-.

ia) Diing (b) Sai.

Cdu 33. Cho (a) cd phuong trinh : Ax + By + Cz + D = 0 va (a') cd phuong trinh :

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

ia) va ia') vudng gdc vdi nhau khi va chi khi AA'+ BB'+CC = 0

(a) Diing (b) Sai.

I l l

Cdu 34. Khoang each giiia hai diem /l(x^ ; y^ ; z^) va B(xg ; yg ; Zg) :

/ '7 "J O

^B=^J{xg-x^) +{yB-yA) +{^B-^A)

(a) Diing (b) Sai.

Cdu 35. Khoang each tir diem Mo(xo ; yo ; Zo) ^ " ^^^ phing (a) cd phuang trinh

Ax + By + Cz + D = 0 la :

IAXQ + Byo+ CZQ + D\ diMQ, ia)) I lA^ + B^ +C-

(a) Diing (b) Sai.

/ / . DIEN DUNG, SAI VAO 6 THICH HOP

Hdy dien dung, sai vdo cdc d trdng sau ddy md em cho Id hgp li nhdt.

Cdu 36. Cho hai diem A(0 ; 1 ; 0), B (1, 0, 1)

(a) AB = ( 1 ; - 1 ; 1 )

(b) AB = 1

(c) AB = V3

(d) Ca ba cau tren deu sai.

Trd Idi.

a

D

b

S

c

D

d

S


(a) Phuang trinh dudng thing AB la x = t

y = l - t

z = t

D D D D

D

112

(b) Phuong trinh dudng thing AB la •

x = l + t

y = - t

z = l + t

(c) Phuong trinh dudng thing ABIa ^ ~ ^ = ^ =^~^

(d) Ca tja cau tren deu sai.

Trd Idi.

D

D

• a

D

b

D

c

D

d

S

Cdu 38. Cho phuong trinh x +y +z - 2x +4y - 2z = 6

(a) Day la mdt phuong trinh mat ciu

(b) Day la mdt phucmg trinh mat ciu tam I (1 ; -2 ; 1)

(c) Day la mdt phuang trinh mat cau cd ban kfnh r = 6

(d) Day la mdt phuong trinh mat ciu ban kfnh r = Vd

Trd Idi.

a

D

b

D

c

S

d

D

Cdu 39. Cho hinh vudng cd canh la 1 nhu hinh ve.

D D 0 D

A'

B'

B , - ' ' C

z

C

. A )

D'

D

H hoc 12/2 113

(a) Diem A (0 ; 0 ; 0)

(b)DiemD'(O; 1; 1)

(e)DiemC'(l ; 1 ; 1)


Trd Idi.

a

D

b

D

c

D

d

S


(a) Dudng thing A'D' cd phuang trinh

(b) Dudng thing CC'cd phuong trinh

(c) Dudng thing A'C'cd phuang trinh


x = 0

y = t

z = l

x = l

y = l

z = t

x = t

y = t

z = l

• D D

D

D

D

•

114

Trd Idi.

a

D

b

D

c

D

d

S

Cdu 41. Cho hinh vuong ed canh la 1 nhu hinh ve.

(a) Mat phing (CCD'D) cd phuong trinh y = 1

(b) Mat phing (CC'A'A) cd phuang trinh -x + y + 1 = 0

(c) Mat phing (BB'D'D) cd phuang trinh x + y = 0


Trd Idi.

a

D

b

D

c

D

d

S

n n D •

/ / / . CAU HOI DA Ll/A CHON

Chpn cau tra Idi diing trong cac bai tap sau:

Cdu 42. Cho ba diem M(l ;0 ;0) , A^(0;-2;0), P(0;0;l) . Ne'u MNPQ la mdt hinh

binh hanh thi toa dp cua diem Q la :

( a ) ( - l ; 2 ; l ) ;

(c) (-2 ; 1 ; 2 ) ;

Trd Idi (b).

(b) (1 ; 2 ; 1)

(d)(-2 ; 3 ; 4)

115

Cdu 43. Cho ba diem M(l;0;0) , A^(0;-2;0), P(0;0;l) . Neu MNPQ la mdt hinh

binh hanh thi PQ cd phuang trinh la

(a)

(c)

x = t

y = 2t;

z = l

x = l

y = 2t ;

z = t

(b)

(d)

x = 2t

y = t

z = l

x = t

y = 2t

z = - l

Trdldi (a).

Cdu 44. Cho ha diim ^ ( l ;2 ;0 ) , B( l ;0 ; - l ) , C (0 ; - l ; 2 ) .

Dd dai AB la :

(b)2

(d)V5.

(a) 2;

( c ) l ;

Trd Idi (d).

Cdu 45. Cho ba diem ^( l ;2 ;0 ) , B( l ;0 ; - l ) , C ( 0 ; - l ; 2 ) .

Dd dai BC la :

(a) 2; (b) VTT

( c ) l ; (d)V5

Trd Idi. (b).

Cdu 46. Cho tam giac ABC cd ^ = ( l ; l ; l ) , B = ( 0 ; - 2 ; 3 ) , C = ( 2 ; l ; 0 ) .

Phuang trinh mat phang (ABC) la

(a) 3x + y +3z +7 = 0 (b) 3x + y +3z -7 = 0

(c) 3x + y +3z +5 = 0; (d) 3x + y +3z - 5 = 0

Trd Idi (b).

116

Cdu 47. Cho tam giac ABC cd A = {\;\;\), B = ( 0 ; - 2 ; 3 ) , C = ( 2 ; l ; 0 ) .

Phuong trinh mat phang di qua M (1 ; 2 ; -7) va song song vdi mat phing (ABC) la

(a) 3x + y+3z+12 = 0 (b) 3x + y+3z-32 = 0

(c) 3x + y +3z +16 = 0; (d) 3x + y +3z - 22 = 0

Trd Idi (e). ^

Cdu 48. Cho tam giac ABC cd ^ = ( l ; l ; l ) , B = ( 0 ; - 2 ; 3 ) , C = (2 ; l ; 0 ) .

Phuang trinh dudng thing di qua M (1 ; 2 ; -7) va vudng gdc vdi mat phing (ABC) la

'x = l -3 t fx = l + 3t

(a)iy = 2 + t (b)]y = 2 - t

z = -7 + 2t [z = -7 + 2t

X = 1 + 3t X = 1 + 3t

y = 2 + t

z = -7 + 2t

(c)<{y = 2 + t (d)

z = 7 + 2t

Trd Idi. (d).

Cdu 49. Cho tam giac ABC cd ^ - ( l ; l ; l ) , B = ( 0 ; - 2 ; 3 ) , C = (2 ; l ; 0 ) . Mat

ciu tam /( l ; l ; - l ) tie'p xiic vdi mat phang toa dd [ABC) cd phuong trinh la :

( a ) ( x - l ) ^ ( y - l f + ( z + l ) ^ = |

(b)(x-l)^(y-lf+(z-lf=|

(c)(x + l ) 2 + ( y - l f + ( z + l ) ' = ^^ 19

( d ) ( x - l ) ' + ( y - l f + ( z + l ) ' =

Trd Idi (a).

( d ) ^ . 4

Trd Idi. (d).

36 19

117

HOATDQNC 2

2. Hirdng dan tra Idi cau hoi va bai tap on tap chifdng III

Bai 1. Hudng ddn.

cau a. De chiing minh 4 diem lap thanh mdt tii dien ta chiing minh cac diem do

khdng ddng phing.

Cach 1. Lap phuang trinh mat phang (ABC) va chiing minh D khdng thudc mat

phang dd.

Cach 2. Lap phuong trinh hai dudng thing AB va CD sau dd chiing minh hai

dudng thing dd cheo nhau.

HS tu giai.

cau b.

Ta sii dung cong thiic : eos( AB, CD) = AB,CD

AB.CD

Ta cd AB = (-1; 1; 0) , CD = (-2; 1; - 2). Tir dd ta cd gdc can tim la 45"

cau c. Tfnh khoang each tii A den mat phing (BCD)

(BCD) : X - 2y -2z +2 = 0.

d(A, (BCD)) = 1

Bai 2. Hudng ddn.

118

cau a. Tam I la trung diem ciia AB: 1(1 ; 1 ; !)•

2

cau b. Dua vao phuong trinh mat cau :

( S ) : ( x - l ) ' + ( y - l ) ' + ( z - l ) ' = 62.

cau e. (a) di qua A va vudng gdc vdi AB.

( a ) : 5x + y - 6 z - 6 2 = 0.

Bai 3. Hudng ddn.

cau a. Tim tpa dp cua cac vecto : Be va BD.

Vecto phap tuyen ciia (BCD) la : [ B C , B D ] .

Tii dd ta cd : (BCD) : 8x - 3y -2a + 4 = 0.

Chiing minh tpa dd A khong thda man phucmg trinh ciia (BCD).

36 cau b. AH = d(A, (BCD)) =

AB, CD cau c. Vecta phap tuyen cua (a) la

la : X - z + 5 = 0.

Bai 4. Hudng ddn. Dua true tiep vao phuang trinh dudng thing:

cau a. HS tu giai

. Tir dd ta cd phuang trinh ciia (a)

119

Ddp so. Phuong trinh AB :

X = 1 + 2t

y = - t

z = - 3 + 3t

cau b. Gpi d la dudng thing cin lap. Ta thiy ngay d va A cd ciing vecto chi

phucmg.

'x = 2 + 2t

Ddp sd. Dudng thing d : • y = 3 - 4t

z = - 5 - 5 t

Bai 5. Hudng ddn.

Tam I cua mat ciu (S): I (3 ; -2 ; 1). Ban kinh cua (S): r = 10.

Khoang each tii I de'n (a) bing EK = d(I, (a)) = 6.

Ban kfnh dudng trdn r' = Vr^-IK^ = VlOO-36 = 8.

• De tim tam K : Lap dudng thing A di qua I va vudng gdc vdi (a).

A cit (a) tai K.

Ddpsd K ( - l ; 2 ; 3 )

Bai 6. Hudng ddn.

cau a. Thay cac tpa dp ciia d vao phucmg trinh (a) ta duoc phuang trinh theo t.

Giai ra ta duac t. Tii dd suy ra giao diem.

Ddp so. Giao diem M(0 ; 0 ; -2).

cau b. Vecto phap tuyen ciia(j3) la vecta chi phucmg cua d.

120

Tir dd ta cd (p): 4x + 3y +z + 2 = 0.

Bai 7. Hudng ddn. Six dung tfnh chat hinh chieu trong khdng gian. Cdng thiic tfnh

the tfch. d

cau a. (a) nhan a lam vecta chi phuang.

(a) : 6x - 2y - 3z + 1 = 0.

cau b. Thay tpa dp cua d vao phuong trinh (a).

Ddp sd M (1 ; -1 ; 3)

'x = l + 2t

cau e. A chfnh la dudng thing AM : < y = -1 - 3t z = 3 + 6t

Bai 8. Hudng ddn. Phuong trinh mat phing (a) nhan [u^, u^.J lam vecta phap

tuyen, Khoang each t\x tam I ciia mat ciu (S) de'n (a) bing r.

Ta cd [u^, u7] = (4 ; 6 ; 5). Gpi (a) cd phucmg trinh :

4x + 6y + 5 z + D = 0.

Taml(5;-1 ;-13),r = 5.

Tii dd ta cd : (a) : 4x + 6y + 5 z + 51 ± V77 = 0.

Bai 9. Hudng ddn. Lap phuong trinh dudng thing A di qua M va vudng gdc vdi

(a) (nghia la nhan vecto phap tuyen cua (a) lam vecta chi phuang).

Giao diem ciia A va (a) la diem cin tim .

Ddp sd M(-3 ; 1 ; -2).

121

Bai 10. Hirdng ddn. Lap phuang trinh dudng thing A di qua M va vudng gdc vdi

(a). Giao diem cua A va (a) la hinh chie'u H cua M tren (a). H la trung diem ciia

MM' T u d d t a c d M '

Ddpsd. (6 ; 13 ;-4).

Bai 11. Hudng ddn.

Lap phuang trinh mat phing di qua d va vuong gdc vdi (Oxz):

Mat phing nay di qua E (0 ; -4 ; 3) thudc d va nhan Uj, j lam vecto phap tuye'n.

Lap phuang trinh mat phing di qua d' va vudng gdc vdi (Oxz) :

Mat phang nay di qua F (1 ; -3 ; 4) thudc d' va nhan

tuye'n.

- Giao cua hai mat phing da cho la dudng thing cin tim.

3

Ud''J lam vecto phap

Ddp sd. A

7 25

18 z =

Bai 12. Hudng ddn. Lap phuang trinh mat phing (a) di qua A va vudng gdc vdi A.

Giao diem H ciia (a) va A la hinh chie'u cua A tren A. Ta thay H la trung diem ciia

A A

Ddpsd. K' (-3 ; 2 ; 1).

122

HOATDQNC 3

Tra Idi cau hoi trac nghiem chUdng III

1

D

11

C

2

C

12

B

3

A

13

C

4

D

14

D

5

D

15

A

6

A

7

B

8

B

9

D

10

c

HOATDQNC 4

Gidi thieu mot so de kiem tra chUdng ill

De so 1

PIlAN 1. Cau hdi va bai tap trac nghiem, mdi cau 1 diem.

Cdu 1. Cho hinh lap phuong ABCDA'B'CD' canh 1, trong he true tpa dp Oxyz, O

triing A, AB thudc Ox, AD thudc Oy, AA' thudc Oz. Khi dd :

(a) Mat phing (ABCD) cd phuang trinh x = 0 ;

(b) Mat phang (A'B'C'D') cd phuang trinh z = 1;

(c) Mat phing (A'B'C'D') cd phuong trinh y = 1

(d) Ca ba cau tren deu sai

Hay chpn cau tra Idi diing.

Cdu 2. Cho A (1 ; 2 ; 3), B ( 0 ; -1 ; 2).

(a) Vecto AB = ( - l ; - 3 ; - l ) ;

(b)Vecta AB = (-1;3;-1);

(c) Vecto AB = (-1;-3;1);


123

X

y z-

X

y z

X

y z-

= l - t

= 2 - 3 t

= 3 - t

= l + t

= 2 + 3t

= 3 + t

= - t

= - l - 3 t

= 2 - t


Cdu 3. Cho A (1 ; 2 ; 3), B ( 0 ; -1 ; 2).

(a) Phuang trinh dudng thing AB la

(b) Phuang trinh dudng thing AB la

(c) Phuong trinh dudng thing AB la


Hay chpn cau tra Idi sai.

Cdu 4. Cho mat phing (a) cd phucmg trinh :

2x+ 3y - z+1 = 0 va diem M(l ; 0 ; 2)

Phuang trinh mat phing di qua M va song song vdi (P) la

(a) 2x+ 3y - z +2 = 0; (b) 2x+ 3y - z= 0

(c) 2x+ 3y - z - 1 = 0; (d) 2x+ 3y - z+1 = 0.

PhAN 2. Bai tap tu luan 6 diem

Cau 5. (6 diem) Cho hai mat phing (a) : 2x + 3y - z + 1 = 0 va

(P):x-y + z-2 = 0.

a) Lap phucmg trinh tham sd cua giao tuye'n A cua hai mat phang.

b) Tfnh khoang each tiir diem M (2 ; -3 ; 0) den dudng thing A.

124

De sd2

PhAN 1. Cau hdi va bai tap trac nghiem, mdi cau 1 diem.

Cdu 1. Khoang each tu M(0 ; 1 ; 0) de'n mat phing ( a ) : x + y - z -2 = 0 la

(a) 1 ; (b) 2

(c) 3 ; (d) 4


Cdu 2. Diem nao sau day thuoc mat phing (a) :x + y - z - 2 = 0

( a ) M ( l ; l ; l ) ; (b) N(l ;-1 ; 1);

( c ) P ( l ; l ; 0 ) ; ; ( d ) Q ( l ; l ; - l ) ;


Cdu 3. Cho A (0 ; 1 ; 2), B(l ; 0 ; 2) va C (0 ; -1 ; -1)

(a) AB = (1;-1;0) ; (b) AC = ( 0 ; - 2 ; - 3 )

(c) BC = ( - ! ; - ! ; - 3 ) (d) Ca ba eau tren deu sai

Hay chpn cau tra Idi sai.

Cdu 4. Dudng thing di qua M(l ; -1 ; 0) va vudng gdc vdi mat phing ( a ) :

X - y + 2z -7 = 0 la

( a ) x - y + 2 z - 4 = 0 ;

(c) X - y + 2z - 2 = 0

Chpn cau tra Idi diing

PhAN 2 . Bai tap tu luan 6 diem

Cau 5. (6 diem). Cho hai dudng thing A :

(b) X - y + 2z - 5 = 0

(d) X - y + 2z -3 = 0

x = l - 2 t

y = 3 - t vaA ' :

z = - 4 +1

x = - l + 2t

y = - 2 + t

z = 5 - t

a) Xac dinh vi trf tuong ddi ciia hai dudng thing.

b) Xac dinh dudng vuong gdc chung cua A va A'

125

De sd 3


Hdy dien dung sai vdo cdc khdng dinh sau:

Cdu 1. Cho hinh ciu cd phuang trinh : x + y +z^ -2x +2y -4z •

(a) Tam cua mat ciu la 1(1 ; 1 ; 2)

(b) Tam ciia mat ciu la I (1 ; -1 ; 2)

(c) Ban kfnh mat cau la r = 22

(d) Ban kfnh mat ciu la r = 4

Cdu 2. Cho hinh ciu cd phuong trinh : x + y +z^ -2x +2y -4z

(a) Mat phing (a) : x + y - z -1 tiep xiic vdi mat cau

(b) Mat phing ( a ) : x + y - z -1 cit vdi mat ciu

(e) Mat eau tren ludn ludn khdng cit (a)

(d) Ca ba khing dinh deu sai

Cdu 3. Cho dudng thing A :

x = l - t

y = 2 + 3t

z = 3 - t

(a) Mat phing x - 3y +z -3 vuong gdc vdi A

(b) Mat phang x - 3y +z -3 song song vdi A

(c) Phuang trinh cua A la : = = -1 3 -1

(d) Ca ba y tren deu sai

Cdu 4. Cho mat phing (a) : z -1 = 0

(a) Mat phing (a) // (Oxy)

22 = 0

n D D D

22 = 0

D D D D

D D

D

D

D

126

(b) (a) ± Oz D

(c)(a)//Ox D

(d) Ca ba y tren deu sai U


Cau 5. (6 diem) Cho mat phang (a) cd phuong trinh : 2x + 3y + z - I = 0 va diem

A ( l ; 2 ; - 5 )

a) Xac dinh A' ddi xiing vdi A qua (a).

b) Lap phuang trinh mat cau tam A va cit (a) theo dudng trdn cd ban kfnh la 2.

HI /ONG DAN - DAP AN

De sd 1


Hdy chgn cdu trd Idi dung trong cdc cdu sau

Cdul

b

Cdu 2

a

Cdu 2

d

Cdu 4

b

PhAN 2. Bai tap tu luan 6 diem (HS tu giai)

De sd 2


Cdu 1

a

Cdu 2

c

Cdu 2

a

Cdu 4

d


127

De sdS


Cdu 1.

a

S

b

D

c

S

d

D

Cdu 2.

a

S

b

D

c

S

d

S

Cdu 3.

a

D

b

S

c

D

d

S

Cdu 4.

a

D

b

D

c

D

d

S


Ban dpc tu giai.

128

On tap cuoi nam

(tiet 18,19)

1. MUC TtU

1. Kien thurc

HS n im duac:

1. Khai niem:

Chuang I

+ Dinh nghia khd'i da dien.

+ Dinh nghia hinh chdp, hinh lang tru

+ Dinh nghia khd'i da dien deu

+ The tfch va dien tfch ciia khd'i da dien.

Chuang2

+ Dinh nghia hinh trdn xoay, hinh ndn, hinh tru, va hinh ciu va cac khai niem co ban.

+ Khai niem ve the tfch, dien tfch eiia cac khd'i trdn xoay.

Chuang3

+ Dinh nghia he true tpa dp trong khong gian.

+ Dinh nghia phuang trinh mat cau.

+ Dinh nghia phucmg trinh mat phing trong khong gian.

+ Dinh nghia phucmg trinh dudng thing trong khong gian.

2. Mdt sd dinh If va menh de quan trpng:

Chuang 1

+ Cdng thiic tfnh dien tfch xung quanh va dien tfch toan phan ciia hinh ndn va hinh tru.

+ Phan chia va lip ghep khd'i da dien nhu the' nao?

Chuang2

+ Cdng thirc ve dien tfch xung quanh va dien tfch toan phin ciia hinh ndn, hinh tru

va hinh ciu.

+ Cdng thiic the tfch cua hinh ndn, hinh tru va hinh ciu.

Hhoc 12/2 129

Chuang 3

+ Cac tfnh chit ciia phep toan ve tpa dp trong khdng gian.

+ Tfch vd hudng ciia hai vecto.

+ Tfch hdn hop ciia hai vecto.

+ Dieu kien de hai mat phing song song, hai mat phing cit nhau.

+ Vi trf tuong ddi ciia dudng thing va mat phing.

+ Vi trf tuong ddi ciia hai dudng thing.

+ Khoang each til mdt diem den mdt mat phing, de'n dudng thing.

2. Ki nang

On tap toan bd cac kl nang ciia 3 chuong.

3. Thai do

• Lien he dupe vdi nhieu vin d6 cd trong thuc te vdi mdn hpc hinh hpc khdng

gian.


• Himg thii trong hpc tap, tfch cue phat huy tfnh ddc lap trong hpc tap.

11. CHUAN Bl Ci A GV VA HS


• Chuan bi dn tap toan bd kien thiic.

• Chuan bi mdt de'n hai bai ki^m tra.

• Oio hoc sinh kiem tra va chain, tra bai.

2. Chuan bi ciia HS :

On tap lai toan bd kie'n thiic, giai va tra Idi cac cau hdi bai tap trong chuong.

III. PHAN PHOI THC)I LUONG

Bai nay chia thanh 2 tie't:

Tie't 1 : On tap.

Tie't 2 : Kiem tra 1 tiet.

130

IV. TEEN TPiNH DAY HOC

n . OHT VAN » €

Cau hdi 1.

Hay thdng ke eac cong thiic tfnh dien tfch va the tfch.

Cau hdi 2.

Hai dudng thing ciing song song vdi mdt mat phing thi song song vdi

nhau. Diing hay sai ?

Cau hdi 3.

Viet cac cdng thiic tfnh khoang each.

B. Bni AAOI

s

HOATDQNCl

1. On tap kie'n thurc co ban

a) Tdm tat li thuyet co ban.

Chuffng 1

• Hinh da dien chia khdng gian lam hai phin (phan ben trong va phan ben ngoai),

nd gdm mdt sd hiru ban da giac phing thoa man hai dieu kien :

a) Hai da giac hoac khdng cd diem chung, hoac cd mdt dinh chung, hoac cd mdt

canh chung.

b) Mdi canh ciia mdt da giac la canh chung ciia diing hai da giac.

• Hinh da dien ciing vdi phan bdn trong ciia nd gpi la mdt khd'i da dien.

Mdi khd'i da dien cd the phan chia dupe thanh nhiing khd'i tir dien.

• Phep ddi hinh trong khdng gian la phep bie'n hinh bao toan khoang each giiia hai

diem bit ki.

• Hai hinh da dien bing nhauneu ed mdt phep ddi hinh bien hinh nay thanh hinh kia.

131

• Hai hinh tii dien bing nhau ne'u chiing cd cac canh tuang ling bing nhau.

• Cd 5 loai khd'i da dien deu : khd'i tii dien deu, khdi lap phucmg, klidi tam mat

deu, khd'i mudi hai mat deu, khd'i hai muoi mat deu.

The tfch cua khd'i hop chii nh$t bing tfch sd ba kfch thudc.

• The tfch ciia khd'i chdp bing mdt phan ba tfch sd ciia dien tich mat day va chieu

cao khd'i chdp.

• The tfch cua khd'i lang tru bing tfch sd cua dien tich mat day va chieu cao ciia

khd'i lang tru.

Chuang 2

• Mat ciu SiO ; R) la tap hap \M \ OM = R}. Khd'i ciu SiO ; R) la tap hop {M I

OM<R}.

Mat ciu la hinh trdn xoay sinh bdi mdt dudng trdn khi quay quanh mdt dudng

kfnh ciia dudng trdn dd.

Khd'i eau la hinh trdn xoay sinh bdi mdt hinh trdn khi quay quanh mdt dudng kfnh

cua hinh trdn dd.

• Giao ciia mat ciu 5(0 ; R) va mp(F) phu thuoc vao R va khoang each dtixO de'n

(F). Gia sii H la hinh chieu ciia O tren mp(B). Khi dd :

- Neu d < R thi giao la dudng trdn nim trdn (B) cd tam H, ban kfnh

r = V B 2 - r f 2

- Neu d = Rthi mp(B) tiep xiic vdi mat cau S(0 ; R) tai H.

- Ne'u d>R thi mp(B) khdng cit mat ciu SiO ; R).

• Giao ciia mat ciu SiO ; R) va dudng thing A phu thudc vao R va khoang each d

til O tdi A. Gia sit H la hinh chieu ciia O tren A. Khi dd :

- Neu d<R thi dudng thing A cit mat cau S(<9 ; R) tai hai diem phan biet.

- Neu d = R thi A tiep xiic vdi mat cau SiO ; R) tai H. Cac dudng thing tie'p xiic vdi mat cau tai H nim trdn tiep dien cua mat cau tai H.

- Ne'u d> R thi A khdng cit mat ciu 5(0 ; R).

• Ve cac tie'p tuyen cua mat ciu di qua mdt diem A nim ngoai mat ciu :

132

- Cac doan thing ndi A va cac tie'p diem bing nhau.

- Tap hop cac tie'p diem la mdt dudng trdn.

2 ' - 4 3 • Hinh eau ban kfnh R cd dien tfch bang 47tR cd the tfch bang —TTR

• Mat tru la hinh trdn xoay sinh bdi dudng thing / khi quay quanh dudng thing A

song song vdi /.

• Mat tru cd true A, ban kinh R la tap hpp cac diem each dudng thing A mot

khoang R.

• Hinh tru la phin mat tru nim giiia hai mat phing phan biet (B), iP') vudng gdc vdi

true ciia mat tru, ciing vdi hai hinh trdn gidi han bdi hai dudng trdn ( ^ ) va ( '^ ' ) la

giao tuye'n ciia mat tru vdi hai mat phing (B) va (B*).

Hinh tru la hinh trdn xoay sinh bdi bdn canh cua mdt hinh chir nhat khi quay

quanh mot dudng trung binh cua hinh chii nhat dd.

Dien tfch xung quanh cua hinh tru bing tich sd chu vi dudng trdn day va chieu cao.

Dien tfch toan phan cua hinh tru bing tdng dien tfch xung quanh va dien tfch hai day.

• Khd'i tru la hinh tru ciing vdi phin bdn trong hinh tru dd.

Khd'i tru la hinh trdn xoay sinh bdi mdt hinh chir nhat (ke ca cac diem nam trong

nd) khi quay quanh mdt dudng trung binh ciia hinh chii nhat dd.

The tfch khd'i tru bing tfch sd ciia dien tfch day va chieu cao.

C Mat ndn, hinh ndn, khd'i ndn

• Mat ndn la hinh trdn xoay sinh bdi dudng thing / khi quay quanh dudng thing A

cit / nhung khdng vudng gdc vdi /.

• Mat ndn dinh O, true A ((9 thudc A), gdc d dinh 2a la hinh tao bdi cac dudng

thing di qua O va tao vdi A mdt gdc bing a (0" < a < 90°).

» Hinh ndn la hinh trdn xoay sinh bdi ba canh cua mdt tam giac can khi quay

quanh true ddi xumg eiia tam giac dd.

• Dien tfch xung quanh cua hinh ndn bing mdt niia tfch sd chu vi day va dp dai

dudng sinh.

133

• Dien tfch toan phin cua hinh ndn bing tdng dien tfch xung quanh va dien tich day.

• Khdi ndn la hinh ndn ciing vdi phin ben trong eiia hinh ndn dd.

• Khd'i ndn la hinh trdn xoay sinh bdi mdt hinh tam giac vudng (ke ca phin trong)

khi quay quanh dudng thing chiia mdt canh gdc vudng.

• The tich cua khd'i ndn bing mdt phin ba tfch sd dien tfch day va chieu cao.

ChUffng 3

• Toa dp ciia diem va toa dp ciia vecto ddi vdi he toa dp Oxyz :

+ Diem M cd toa dp (x ; y ; z)<=> OM = xi + yj + zk.

+ Vecto u c6toadoix;y ; z) c^ u = xi + yj + zk.

+ AB = [xg - x^ ; yg - y^ ; ZB - ^A)•

• Tfch vd hudng va tfch vecto : Cho M = (x ; y ; z) va v = (x ; y' ; z') thi:

+ Tfch vd hudng eiia M va v la sd : M.V ^ XX'+ yy'+ zz'

+ Mdt sd tfnh chit: u ± v <=i>M.v = 0 ;

• Phuang trinh mat ciu : Phuong trinh cd dang

x^ + y^ + z + 2ax + 2by + 2cz + d = 0,

vdi dieu kien a + b^ + c^ > d, la phUOng trinh mat ciu cd tam (-a ; - b ; -c) va cd ban

kinh R = Va^ + b^ + c^ - d

• Phuang trinh mat phing : Phuong trinh

Ax + By + Cz + D = 0, vdi A^ + B^ + C^ > 0

la phuang trinh ciia mat phing cd vecto phap tuyen la HiA ; B ; C).

• Mat phing di qua diem (XQ ; yo ; ZQ) vdi vecto phap tuye'n (A ; B ; C) cd phuong

trinh :

Aix - XQ) + Biy- yo) + Ciz - ZQ) = 0.

• Phuang trinh dudng thing :

134

Phuang trinh tham sd:

X - XQ + at

y^ yQ + bt

Z - ZQ + ct

vdi a^ + b^ + c^> 0.

+ Phuang trinh chinh tdc :

x-x^

a ..y-y^

b z - Zo vdi abc ^ 0.

c

Dudng thing vdi phuong trinh dd di qua diem (XQ ; yo ; ZQ) va cd vecto chi phuang

u = ia;b;c).

• Vi trf tuong ddi giiia hai mat phang (a) va (aO :

Ne'u (a) ed phuong trinh : Ax + By + Cz + D = 0 va (aO cd phugng trinh : A'x + B'y

+ Cz + D' = 0, thi

+ ia) va ia') cit nhau khi va chi khi A : B : C ^ A': B ' : C

A B C D + ia) va ia') song song khi va chi khi — = — = — ? —

A ti Ly L)

A B C D + (or) va (a') trung nhau khi va chi khi — = — = — = — .

A B C D

+ (a) va (a') vudng gdc vdi nhau khi va chi khi AA'+ BB'+ C C = 0 .

• Khoang each :

+ Khoang each giua hai diem /i(x^ ; y^ ; z^) va B(xg ; y^ ; Zg) :

I T O "J

AB=yj{xg-x^) +{yB-yA) +{^B-^A)

+ Khoang each tii diem Mo(xo ; yo ; ZQ) den mat phing (a) ed phuong trinh

Ac + By + Cz + D = 0 la :

AXQ + Byo+ CZQ + D\ rf(Mo,(a))--

^A^ + B^ +C^

135

b) Cau hdi trac nghiem nham dn tap kie'n thurc:

GV nen dua ra mdt he thdng cau hdi trie nghiem nhim dn tap toan bd kien thiic

trong chuang.

Sau day xin gidi thieu mdt sd cau hdi:

/. HAY KHOANH TRON CAU DUNG, SAI TRONG CAC CAU SAU MA EM

CHO LA HOP LI.

Cdu I. Qua hai phep bien hinh : Phep tinh tien theo vecto v va phep ddi xiing qua

dudng thing dupe hai khd'i da dien bing nhau.

(a) Diing (b) Sai.

Cdu 2. Qua hai phep bie'n hinh : Phep ddi xiing tam O va phep ddi xiing qua dudng

thing duoc hai khd'i da dien bing nhau.

(a) Diing (b) Sai.

Cdu 3. Khd'i da dien ludn chiia trpn mpi doan thing cd hai diu thudc khd'i da dien

la khdi da dien Idi.

(a) Dung (b) Sai.

Cdu 4. Khd'i da dien ludn chiia trpn mpi dudng thing la khd'i da dien Idi.

(a) Diing (b) Sai.

Cdu 5. Khd'i tii dien cd 4 mat la tam giac deu la khd'i da dien deu

(a) Diing (b) Sai.

Cdu 6. Cd vd sd khd'i da dien deu

(a) Diing (b) Sai.

Cdu 7. Chi cd 5 khd'i da dien deu

(a) Diing (b) Sai.

Cdu 8. Khd'i da dien deu cd sd dinh va so mat bing nhau

(a) Diing (b) Sai.

Cdu 9. Da dien cd eac mat la tam giac thi tdng sd cac mat phai la sd chan.

(a) Diing (b) Sai.

136

Cdu 110. Trung diem cac canh ciia mdt tii dien deu la dinh cua mdt tii dien deu.

(a) Diing (b) Sai.

Cdu 118. Hinh lap phuong la mdt da dien deu.

(a) Diing (b) Sai.

Cdu 12. Hinh lap phucmg la luc dien deu.

(a) Diing (b) Sai.

Cdu 13. Hinh lap phuong la da dien deu dang {4,3}

(a) Diing (b) Sai.

Cdu 14. Hinh lap phuong la da dien deu dang {3, 4}

(a) Diing (b) Sai.

Cdu 15. Hinh bat dien deu la da dien deu dang {4, 3}

(a) Diing (b) Sai.

Cdu 16. Hinh 12 mat deu la da dien deu dang (3,5}

(a) Diing (b) Sai.

Cdu 17. Mpi mat phing deu cit mat ndn theo mdt dudng trdn

(a) Diing (b) Sai.

Cdu 18. Mpi mat phing vudng gdc vdi true deu cit mat ndn theo mdt dudng trdn

(a) Diing (b) Sai.

Cdu 19. Mpi mat phing di qua true deu cit mat ndn theo mdt tam giac can

(a) Diing (b) Sai.

Cdu 20. Mpi mat phing di qua dinh neu cit mat ndn thi cit theo mdt tam giac can

(a) Diing (b) Sai.

Cdu 21. Mpi mat phing cit mat tru theo mot dudng trdn

(a) Diing (b) Sai.

Cdu 22. Mpi mat phing vudng gdc vdi true ciia mat tru cit mat tru theo mdt dudng trdn

(a) Diing (b) Sai.

137

Cdu 23. Mpi mat phang di qua true cua mat tru cit mat tru theo mdt hinh chii nhat

(a) Diing (b) Sai.

Cdu 24. Mpi mat phing song song vdi true cua mat tru neu cit mat tru thi cit

theo mdt hinh chu: nhat

(a) Diing (b) Sai.

Cdu 25. Dudng thing A 1 (a) khi va ehi khi vecto chi phucmg cua A va vecto phap

tuyen ciia (a) cdng tuye'n.

(a) Diing (b) Sai.


X = Xo +ait

y = y'o + a'2t ' d : y = yo + a2t vad'

z = Zo+a3t z = z'o + a'31'

He phuang trinh


y 0 + a 2 t' = yo + a2t vd nghiem thi d // d'

z o+a'3 t' = Zo+a3t

(a) Diing


(b) Sai.

Xo + a,t

y = yo+a2t vad'

z = Zo +a3t

X = x'o+a'i t'

y = y 'o+a '2t ' z = z'o+a'31'

He phuang trinh


(a) Diing


y'o + a'2t ' = yo+a2t vd nghiem thi d va d' cheo nhau

|t

(b) Sai.

Zo + a'3t' = Zo+a3t

138

X = Xo + ait

y = yo + a2t vad ' •

z = ZQ +a3t

X = x'o + a'i t'

y = y o + a ^ t '

z = z'o + a'31'

He phuong trinh


(a) Diing


y'o + a'2 t' = yo +a2t cd nghiem thi d vad' cit nhau

it

(b) Sai.

z'o + a'3t' = Zo + a3t

d:

X = Xo +ait

y = yo + a2t vad'

z = ZQ +a3t

X = x'o + a'i t'

y = y'o+a'2 t'

z = z'o + a'3t'

He phuang trinh

(a) Diing


y '0 + a '2 t' = yo + a2t cd nghiem duy nhat thi d va d' cit nhau

z'o+a'3 t' = Zo + a3t

(b) Sai.

Cdu 30. Hai mat phing song song khi va chi khi hai vecto phap tuyen cua chiing

cdng tuyen.

(a) Diing (b) Sai.

Cdu 31. Hai mat phing vuong gdc vdi nhau khi va chi khi tfch vd hudng hai vecto

phap tuyen bang 0.

(a) Diing (b) Sai.


d(I, (a)) < r thi giao eiia mat cau va (a) la dudng trdn.

(a) Diing (b) Sai.


d(I, (a)) = r thi (a) tiep xiic vdi (I, r).

(a) Diing (b) Sai.

139

Cdu 34. Cho mat cau (I, r) va mdt mat phing (a).

d(I, (a)) > r thi giao ciia mat ciu va (a) la dudng trdn.

(a) Diing (b) Sai.

Cdu 35. Phuang trinh dudng thing AB nhan AB lam vecto ehi phuang.

(a) Diing (b) Sai.

Cdu 36. Phuang trinh mat phang vudng gdc vdi AB nhan AB lam vecta phap

tuyen

(a) Diing (b) Sai.

Cdu 37. Phucmg trinh mat phing di qua ba diem A, B, C nhan AB,AC lam

Aiecto phap tuyen.

(a) Diing (b) Sai.

AB,BC lam Cdu 38. Phucmg trinh mat phing di qua ba diem A, B, C nhan

vecto phap tuyen.

(a) Diing (b) Sai.

Cdu 39. Phuang trinh mat phing di qua ba diem A, B, C nhan CB,AC lam

vecto phap tuye'n.

(a) Diing (b) Sai.

Cdu 40. Phuang trinh mat ciu tdng quat cd dang:

x^ + y^ + z^ + 2ax + 2by + 2cz + d = 0.

(a) Diing (b) Sai.

Cdu 41. Phucmg trinh mat ciu tdng quat cd dang:

2x^ + y + z + 2ax + 2by + 2cz + d = 0.

ia) Diing (b) Sai.

Cdu 42. Mat ciu tam I(a, b, e) ban kfnh r la :

(x - af + (y - b)' + (z - c)' = r

(a) Diing (b) Sai.

140


(x - af + (y - b)' + (z - c)' = r'

(a) Diing (b) Sai.

Cdu 44. Phuang trinh mat cau : Phuong trinh cd dang

x^ + y + z + 2ax + 2by + 2cz + d = 0, • ^ • 2 2 2

vdi dieu kien a + b + e < d, la phuong trinh mat cau cd tam (-a ; - b ; -c) va cd ban

kfnh r = yja^ + b^ + c^ - d

ia) Diing (b) Sai.

Cdu 45. Phuang trinh mat ciu : Phuong trinh cd dang

x^ + y^ + z + 2<3x + 2by + 2cz + d = 0, 2 2 2

vdi dieu kien a + b + c = d, la phuong trinh mat cau cd tam (-a ; - b ; -c) va cd ban

kfnh R = Va^ +b^ +c^ -d

(a) Dung (b) Sai.

Cdu 46. M ± v <=>M.v = 0

(a) Diing (b) Sai.

Cdu 47. Cho i7 = (x ; y ; z) va v = (x ; y' ; z '). Tfch vd hudng cua M va v la sd

u.v = xx'+ yy'+ zz'

ia) Diing (b) Sai.

Cdu 48. Ne'u diem A = (x^ ; y^ ; z^) va diem B = (xg ; y^ ; Zg) thi

AB = (xg-x^;yg-y^;zg- z^ ) .

(a) Diing (b) Sai.

Cau 49. Phucmg trinh chfnh tic :

^ = -—^ = ^ vai abc ^ 0. a b c

141

Dudng thing vdi phuang trinh dd di qua diem (xo ; yo ; ZQ) va cd vecto chi phuong

u = ia; b; c).

(a) Dung (b) Sai.

Cdu 50. Cho (a) cd phuang trinh : Ar + By + Cz + D = 0 va (a') cd phuong trinh :

A'x + B'y + Cz + D'-^O

ia) va ia') cit nhau khi va chi khi A : B : C ?i A': B ' : C

(a) Diing (b) Sai.

Cdu 51. Cho (a) cd phuang trinh : Ax + By + Cz + D = 0 va (a') cd phuang trinh :

A'x + B'y + C'z + B»' = 0

ia) va ia) song song khi va chi khi — = — = — ^ — ^ ^ A' B' C D'

(a) Diing (b) Sai.

Cdu 52. Cho (a) cd phuang trinh : Ax + By + Cz + D = 0 va (a') cd phuang trinh :

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

ia) va ia') triing nhau khi va chi khi — = — = — = — . A' B' C D'

ia) Diing (b) Sai.

Cdu 53. Cho (a) ed phuang trinh : Ax + By + Cz + D = 0 va (a') cd phuang trinh :

A'x + B'y + Cz + D' = 0

ia) va (a') vudng gdc vdi nhau khi va chi khi AA'+ BB'+ CC = 0

(a) Diing (b) Sai.

Cdu 54. Khoang each tiir diem Mo(xo ; yo ; Zo) den mat phang (a) cd phuang trinh

Ax + By + Cz + D = 0 la :

,(Mo,(a)) = K ± £ ^ ^ 3 ^ V A^ + B'^ +C^

ia) Diing (b) Sai.

142

//. DIEN DUNG, SAI VAO 6 THICH HOP

Hay di6n dung, sai vao cac d trdng sau day ma em cho la hpp li nhat.

Cdu 55. Cho hinh chdp S.ABCD, day ABCD la hinh vudng canh a, SA = a

SA l(ABCD).

(a) The tfch hinh chdp la a

(b) The tfch hinh chdp la - a

1 , (c) The tfch hinh chdp la — a


Trd Idi.

D

D

D

•

a

S

b

S

c

D

d

S

Cdu 56. Cho hinh hop chu nhat ABCDA'B'CD' cd AA' = c, AB = a, AD = b B' C

/ / 1 ^[y^

y^

143

(a) The tfch hinh hop la abc

(b) The tich hinh chdp A'.ABCD la abc

(c) The tfch hinh chdp A'.ABCD la - abc

(d)V(ABCD.A'B'C'D') ~ ^ ^ ( A ' . A B C D )

Trd Idi.

U D

• •

a

D

b

S

e

D

d

D

Cdu 57. Cho hinh chdp S.ABCD, day ABCD la hinh vudng canh a, SA = a vudng

gdc vdi day.

S

(a) SB = aV2

(b) SD = aV2

(c) Dien tich tam giac SBD bing

(d) Ca ba cau tren diu sai.

Trd Idi.

a'S

a

D

b

D

c

D

d

S

n D

D

D

144

Cdu 58. Cho hinh lap phuong ABCDA'B'CD' canh a.

B' C'

/ \

A' 1

/ B /

/ /

D'

(a) The tfch khd'i lap phuong la a

(b) The tfch khdi chdp A'.DD'C la - a ^

(c) The tfch khdi lang tru AA'B'.DD'C la - a^

6

(d) Ca ba eau tren deu sai

Trd Idi. a

D

b

D

c

D

d

S


(a) AB = ( 1 ; - 1 ;1)

(b) AB .= 1

(c) \^= S

(d) Ca ba eau tren deu sai.

Trd Idi.

a

D

b

S

c

D

d

S

•

•

D D D D

H hoc 12/2 145


(a) Phuang trinh dudng thing AB la x = t

y = l - t

z = t

(b) Phucmg trinh dudng thing AB la

(c) Phuang trinh dudng thing AB la


Trd Idi.

x = l + t

y = - t

z = l + t

x - 1 _ y _ z - 1

a

D

b

D

c

D

d

S

Cdu 61. Cho phuang trinh x +y^ +z^ 2x +4y - 2z = 6

(a) Day la mot phuong trinh mat cau

(b) Day la mot phuang trinh mat cau tam I (1 ; -2 ; 1)

(c) Day la mdt phuang trinh mat ciu cd ban kinh r = 6

(d) Day la mot phuang trinh mat cau ban kfnh r = V6

Trd Idi.

a

D

b

D

c

S

d

D

•

•

D

D

• D D D

146


(a) Dudng thing A'D' ed phuong trinh

(b) Dudng thing CC'cd phuang trinh

(c) Dudng thing A'C'cd phuang trinh


Trd Idi.

x = 0

y = t

z = l

x = l

y = l

z = t

x = t

y = t

z = l

a

D

b

D

e

D

d

S

•

n

/ / / . CAU HOI DA Ll/A CHON

Chon cau tra Idi dung trong cac bai tap sau:

Cdu 64. Cho hinh chdp SABCD, day ABCD la hinh thang vudng tai A,

SA l(ABCD), SA = a, AB = 2a, AD = DC = a. Khoang each tir A den (SBC) la

H.hoc 12/2 147

(a) a;

(c) aV3 ;

Trdldi (d).

Cdu 65. Cho hinh chdp SABCD, day ABCD la hinh vudng canh a, SA ±(ABCD),

SA = a. Khoang each giiia AB va SD la S

(a) a;

(c) aV2 ;

Trd Idi (d).

Cdu 66. Cho hinh chdp SABCD, day ABCD la hinh thang vu6ng tai A,

SA ±(ABCD), SA = a, AB = 2a, AD = DC = a. Khoang each tur A den mat phang (SBC) la

148

(a) a^;

(c) a^N/i" ; (d) 1^76

Trdldi (d).

Cdu 67. Cho hinh lang tru luc giac deu canh day la 2v3 ndi tiep mot hinh tru ed dudng cao la 3

C

, ; » -

F' D' E'

Dudng sinh ciia hinh tru la

(a)2V3;

(c)3;

Trd Idi. (c).

(b)3V3

(d)6.

149

Cdu 68. Mdt hinh eau cd dudng trdn Idn ngoai tiep mdt tam giic deu canh 1 co

dien tfch toan phin la :

(a)2V3;

( c ) - | - ;

V3

Trd Idi. (c).

Cdu 69: Gpi d la khoang each tii O ciia mat ciu S(0 ; r) de'n mat phing (P)

Dien vao chd trdng sau : d

r

Vi tri tuang

ddi cua (P) vd

(S)

6

5

5

4

4

4

8

8

Cdu 70. Cho ba diem M(l ;0 ;0) , A^(0;-2;0), B(0;0;l). Neu MNPQ la mdt hinh

binh hanh thi PQ ed phuang trinh la

(a)

x = t

y = 2t;

z = l

(b)

x = 2t

y = t

z = l

150

(c)

x = l

y = 2t ;

z = t

(d)

x = t

y = 2t

z = - l

Trd Idi. (a).

CdM 7i. Cho ba diem ^( l ;2 ;0 ) , B( l ;0 ; - l ) , C(0 ; - l ; 2 ) .

Dd dai AB la :

(a) 2; (b) 2

( c ) l ; id)yf5

Trd Idi. (d).

CdM 72. Cho ba diem ^ ( l ;2 ;0 ) , B( l ;0 ; - l ) , C (0 ; - l ; 2 ) .

Dp dai BC la : i

ia) 2; (b) VTT

( c ) l ; (d)V5

Trdldi (b).

CdM 73. Cho tam giac ABC cd ^ = ( l ; l ; l ) , B = ( 0 ; - 2 ; 3 ) , C = ( 2 ; l ; 0 ) .

Phucmg trinh mat phing (ABC) la

(a) 3x + y +3z +7 = 0 (b) 3x + y +3z -7 = 0

(c) 3x + y +3z +5 = 0; (d) 3x + y +3z - 5 = 0

Trdldi (b).

CdM 74. Cho tam giac ABC cd ^ = ( l ; l ; l ) , B = ( 0 ; - 2 ; 3 ) , C = ( 2 ; l ; 0 ) .

Phuang trinh mat phing di qua M (1 ; 2 ; -7) va song song vdi mat phing (ABC) la

(a)3x + y+3z+12 = 0 (b) 3x + y+3z-32 = 0

(c)3x + y+3z+16 = 0; (d) 3x + y+3z -22 = 0

Trdldi (c).

151

Cdu 75. Cho tam giac ABC cd ^ = ( l ; l ; l ) , B = ( 0 ; - 2 ; 3 ) , C = ( 2 ; l ; 0 ) .

Phucmg trinh dudng thing di qua M (1 ; 2 ; -7) va vudng gdc vdi mat phing

(ABC) la

(a)

(c)

x = l - 3 t

y = 2 + t

z = -7 + 2t

x = l + 3t

y = 2 + t

z = 7 + 2t

(b)

X = 1 + 3t

y = 2 - t

z = -7 + 2t

(d)

X - 1 + 3t

y = 2+ t

z = - 7 + 2t

Trdldi (d).

Cdu 76. Cho tam giac ABC cd ^ = ( l ; l ; l ) , B = ( 0 ; - 2 ; 3 ) , C = ( 2 ; l ; 0 ) . Mat

ciu tam / ( l ; l ; - l ) tie'p xiic vdi mat phing toa dp (ABC) cd phucmg trinh la :

\2 / .\2 / ,N2 36 ( a ) ( x - l f + ( y - l f + ( z + l ) ^ -

( b ) ( x - l f + ( y - l f + ( z - l )

19

2 36

19

\2 / ,\2 / .\2 36

19 (c)(x + l) + ( y - l ) +(z + l) =

( d ) ( x - l ) ^ + ( y - l ) ^ + ( z + l f = - |

Trd Idi ia).

HOATDQNC 2

2. Hi/dng dan tra Idi cau hoi va bai tap on tap cuoi nam

Bai 1. Hudng ddn. Gpi I la trung diem cua OO' Chiing minh O la tam ddi xiing

cua phep ddi xiing tam O bien lang tru nay thanh lang tru kia.

152

Bai 2. Hudng ddn.

R

J

A

' 1 ' s / ' ^ / 1

/ 1 / 1

/ 1

^ \ IA'

N

D

/ /

F

I

Trudc het tim the tich tii dien A.A'MN rdi trft di 2 lin the tfch khdi tii dien J. MEB'.

Ddp sd. Nn^^ = —a^ 72

Bai 3. Hudng ddn.

Caua.TaedV= -7tAH^IH = -n(2r-h)h2 3 3

caub. h= - r 3

Bai 4. Hudng ddn. Dua true tiep vao phucmg trinh dudng thing:

cau a. Lap phuang trinh dudng thing AB va chiing minh AB cit d hoac AB // d.

cau b. Tim A' ddi xiing vdi A qua d. A' B cit d tai I. Ta cd I la diem cin tim.

153

MLJC LUC

C/jifo-«g 2 - MAT NON, MAT TRU, MAT CAU 3

B/idn 7 - GIOI THIEU CHUONG 3

P/ia« 2 - CAC BAI SOAN 5

§2. Mat cdu 5

Or) tap chuang II 28

Mgt so de kiem tra chuang II 38

Chuang 3 - PHUONG P H A P TOA DO TRONG KHONG GIAN 39

B/iaw/-GIOI THIEU CHUONG 39

Bftdn 2 - CAC BAl SOAN 41

§ 1. He toa do trong khong gian 41

§ 2. Phuang trinh mat phang 58

§ 3. Phuang trinh duang thang trong khong gian 82

On tap chuang II 103

Gidi thieu mgt so de kiem tra chuang 3 123

On tap cuoi nam 129

154

Thiet ke bai giang

HlNH HOC 12-TAP HAI

TRAN VINH

N H A XUAT BAN H A NOI

Chiu trdch nhiem xudt bdn:

NGUYEN KHAC OANH

Bien tap:

PHAM QUOC TUAN

Vebia:

NGUYfiN TUAN

Trinh bdy:

QUYNH TRANG

5M'a bdn in:

PHAM QUOC TUAN

In 2000 cud'n, khd 17x24 cm, tai Cdng ty Co phan in Khoa hpc Cdng nghe

mdi. Giay phep xuat ban sd: 127 - 2008/CXB/lOO h TK - 05/HN

In xong va nop luu chieu nam 2008

156

Sach lien ket v6i Cong ty CO phan In va Phat hanh sach Viet Nam

• INPHAVI Phat hanh tai Cong ty cd phan In va Phat hanh sach Viet Nam

Dia chi : 1 78 - Dong Cac - Ddng Da - Ha Noi

DT : (04) 5.115921 -Fax : (04)5.115921 Thiet k^ 'nr H- i, i, , • met KC UO Hmh hoc 12T2 ^"^ mm -c

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