com BaiTapHinhHoc12 Tap3 PhuongPhapToaDoTrongKhongGian TranSiTung

8/8/2019 com BaiTapHinhHoc12 Tap3 PhuongPhapToaDoTrongKhongGian TranSiTung

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TRAN S TUNG

---- & ----

BAI TAP HNH HOC 12

TAP 3

ON THI TOT NGHIEP THPT & AI H

Nam 2009


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PP Toa o trong khong gian Tran S Tun

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1. nh ngha va cac phep toan nh ngha, tnh chat, cac phep toan ve vect trong khong gian c xay tng t nh trong mat phang. Lu y:+Qui tac ba iem:Cho ba iem A, B, C bat ky, ta co: AB BC AC + =

uuur uuur uuur

+Qui tac hnh bnh hanh:Cho hnh bnh hanh ABCD, ta co: AB AD AC + = uuur uuur uuur

+Qui tac hnh hop:Cho hnh hop ABCD.ABCD, ta co: AB AD AA AC ' '+ + =

uuur uuur uuur uuuur

+He thc trung iem oan thang:Cho I la trung iem cua oan thang AB, O tuy yTa co: 0 IA IB+ =

uur uur r; 2OA OB OI + =

uuur uuur uur

+He thc trong tam tam giac:Cho G la trong tam cua tam giac ABC, O tuy y.Ta co: 0 3GA GB GC OA OB OC OG;+ + = + + =

uuur uuur uuur uuur uuur uuur uuur r

+He thc trong tam t dien:Cho G la trong tam cua t dien ABCD, O tuy y.Ta co: 0 4GA GB GC GD OA OB OC OD OG;+ + + = + + + =

uuur uuur uuur uuur uuur uuur uuur uuur uuur r

+ieu kien hai vect cung phng: 0a va b cung phng a k R b ka( ) ! : $ = r r r r r r

+iem M chia oan thang AB theo t so k(k 1), O tuy y.

Ta co: 1OA kOB MA kMB OM k ; -= = -

uuur uuur uuur uuur uuur

2. S ong phang cua ba vect

Ba vect c goi la ong phang neu cac gia cua chung cung song sphang. ieu kien e ba vect ong phang:Cho ba vecta b c, , r r r , trong oa va b r r khong cungphng. Khi o:a b c, ,

r r r ong phang $! m, n R:c ma nb= +r r r

Cho ba vecta b c, , r r r khong ong phang, x r tuy y.

Khi o: $! m, n, p R: x ma nb pc= + + r r r r

3. Tch vo hng cua hai vect Goc gia hai vect trong khong gian:

0 00 180 AB u AC v u v BAC BAC , ( , ) ( )= = = uuur uuur r r r r

Tch vo hng cua hai vect trong khong gian: + Cho 0u v,

r r r . Khi o: u v u v u v. . .cos( , )= r r r r r r + Vi 0 0u hoac v= =

r r r r . Qui c: 0u v. = r r + 0u v u v.^ = r r r r + 2u u= r r

CHNG IIIPHNG PHAP TOA O TRONG KHONG GIAN

I. VECT TRONG KHONG GIAN


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Tran S Tung PP Toa o trong khong gian

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1. He toa o ecac vuong goc trong khong gian:

Cho ba truc Ox, Oy, Oz vuong goc vi nhau tng oi mot va chung mot i j k , , r r r

la cac vect n v, tng ng tren cac truc Ox, Oy, Oz. He ba truc nhtoa o ecac vuong goc Oxyz hoac n gian la he toa o Oxyz. Chu y: 2 2 2 1i j k = = =

r r rva 0i j i k k j. . .= = =

r r r r r r.

2. Toa o cua vect:a) nh ngha: ( )u x y z u xi y j zk ; ;= = + +

r r r r r

b) Tnh chat:Cho 1 2 3 1 2 3a a a a b b b b k R( ; ; ), ( ; ; ),= = r r

1 1 2 2 3 3a b a b a b a b( ; ; ) =

r r 1 2 3ka ka ka ka( ; ; )= r

1 1

2 2

3 3

a ba b a b

a b

== = =

r r

0 0 0 0 1 0 0 0 1 0 0 0 1i j k ( ; ; ), ( ; ; ), ( ; ; ), ( ; ; )= = = = r r r r

a r

cung phng 0b b( ) r r r

a kb k R( )= r r

1 1

31 22 2 1 2 3

1 2 33 3

0a kb aa aa kb b b b

b b ba kb, ( , , )

= = = = =

1 1 2 2 3 3a b a b a b a b. . . .= + + r r 1 1 2 2 3 3 0a b a b a b a b^ + + =

r r

2 2 2 21 2 3a a a a= + + r 2 2 21 2 2a a a a= + +

r

1 1 2 2 3 32 2 2 2 2 21 2 3 1 2 3

a b a b a ba ba ba b a a a b b b

.cos( , )

. .+ +

= =+ + + +

r r r r r r (vi 0a b,

r r r )

3. Toa o cua iem:a) nh ngha: M x y z OM x y z( ; ; ) ( ; ; ) =

uuur (x : hoanh o, y : tung o, z : cao o)

Chu y: M (Oxy) z = 0; M (Oyz) x = 0; M (Oxz) y = 0 M Ox y = z = 0; M Oy x = z = 0; M Oz x = y = 0

b) Tnh chat:Cho A A A B B B A x y z B x y z( ; ; ), ( ; ; ) B A B A B A AB x x y y z z( ; ; )= - - -

uuur 2 2 2 B A B A B A AB x x y y z z( ) ( ) ( )= - + - + -

Toa o iem M chia oan AB theo t so k (k 1):1 1 1

A B A B A B x kx y ky z kz M k k k

; ; - - - - - -

Toa o trung iem M cua oan thang AB:2 2 2

A B A B A B x x y y z z M ; ; + + +

Toa o trong tam G cua tam giac ABC:

3 3 3 A B C A B C A B C x x x y y y z z zG ; ; + + + + + +

II. HE TOA O TRONG KHONG GIAN


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Toa o trong tam G cua t dien ABCD:

4 4 4 A B C D A B C D A B C C x x x x y y y y z z z zG ; ; + + + + + + + + +

4. Tch co hng cua hai vect: (Chng trnh nang cao) a) nh ngha: Cho 1 2 3a a a a( , , )=

r, 1 2 3b b b b( , , )=

r.

[ ] ( )2 3 3 1 1 2 2 3 3 2 3 1 1 3 1 2 2 12 3 3 1 1 2

a a a a a aa b a b a b a b a b a b a b a b

b b b b b b, ; ; ; ;

= = = - - -

r r r r

Chu y:Tch co hng cua hai vect la mot vect, tch vo hng cua hai vect la mot so.b) Tnh chat:

[ ]i j k j k i k i j, ; , ; , = = = r r r r r r r r r

a b a a b b[ , ] ; [ , ]^ ^ r r r r r r

( )a b a b a b[ , ] . .sin ,=

r r r r r r a b, r r

cung phng 0a b[ , ] = r r r

c) ng dung cua tch co hng:

ieu kien ong phang cua ba vect:a b, r r

vac r

ong phang 0a b c[ , ]. = r r r

Dien tch hnh bnh hanh ABCD: ABCDS AB AD, = Y

uuur uuur

Dien tch tam giac ABC: 12 ABC

S AB AC ,D = uuur uuur

The tch khoi hop ABCD.A BC D: ABCD A B C DV AB AD AA. ' ' ' ' [ , ]. '= uuur uuur uuur

The tch t dien ABCD: 16 ABCD

V AB AC AD[ , ].= uuur uuur uuur

Chu y: Tch vo hngcua hai vect thng s dung e chng minh hai ng thang vuong goc

tnh goc gia hai ng thang. Tch co hngcua hai vect thng s dung e tnh dien tch tam giac; tnh the tch

khoi t dien, the tch hnh hop; chng minh cac vect ong phang khong ong phang, chminh cac vect cung phng.

[ ][ ]

00

0

a b a ba va b cung phng a ba b c ong phang a b c

.,

, , , .

^ = = =

r r r r r r r r r

r r r r r r

5. Phng trnh mat cau: Phng trnh mat cau (S) tam I(a; b; c), ban knh R:

2 2 2 2 x a y b z c R( ) ( ) ( )- + - + - = Phng trnh2 2 2 2 2 2 0 x y z ax by cz d + + + + + + = vi 2 2 2 0a b c d + + - > la phng trnhmat cau tam I(a; b; c) va ban knh R = 2 2 2a b c d + + - .


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VAN E 1: Cac phep toan ve toa o cua vect va cua ie S dung cac cong thc ve toa o cua vect va cua iem trong khong gian. S dung cac phep toan ve vect trong khong gian.

Bai 1. Viet toa o cua cac vect sau ay:2a i j= - + r r r ; 7 8b i k = -

r r r ; 9c k = -r r ; 3 4 5d i j k = - +

r r r r Bai 2. Viet di dang xi yj zk + + r r r moi vect sau ay:

10 2

2a ; ; =

r ; 4 5 0b ( ; ; )= -

r; 4 10

3 3c ; ; =

r ; 1 1

3 5d ; ;p =

r

Bai 3. Cho: ( ) ( ) ( )2 5 3 0 2 1 1 7 2a b c; ; ; ; ; ;, ,- -= = = r r r . Tm toa o cua cac vectu r vi:a) 14 3

2u a b c= - +

r r r r b) 4 2u a b c= - - r r r r c) 24

3u b c= - +

r r r

d) 3 5u a b c= - + r

r r r e) 1 4 22 3u a b c= - - r

r r r f) 3 24 3u a b c= - - r

r r r Bai 4. Tm toa o cua vect x r , biet rang:

a) 0a x+ =r r r vi ( )1 2 1a ; ;= - r b) 4a x a+ = r r r vi ( )0 2 1a ; ;= - r

c) 2a x b+ =r r r vi ( )5 4 1a ; ;= - r , ( )2 5 3b ; ;= -

r

Bai 5. Cho 1 3 4a ( ; ; )= - r .a) Tm y va z e 2b y z( ; ; )=

rcung phng via r .

b) Tm toa o cua vectc r , biet ranga va c r r ngc hng va2c a= r r .

Bai 6.

Cho ba vect ( ) ( ) ( )1 1 1 4 0 1 3 2 1a b c; ; , ; ; , ; ;= - = - = - r r r

. Tm:a) ( )a b c. r r r b) ( )2a b c.

r r r c) 2 2 2a b b c c a+ + r r r r r r

d) ( ) 23 2a a b b c b.- + r r r r r r e) 2 24 5a c b c. + -

r r r r Bai 7. Tnh goc gia hai vecta r vab

r:

a) ( ) ( )4 3 1 1 2 3a b; ; , ; ;= = - r r b) ( ) ( )2 5 4 6 0 3a b; ; , ; ;= = -

r r c) 2 1 2 0 2 2a b( ; ; ), ( ; ; )= - = -

r r d) 3 2 2 3 3 2 3 1a b( ; ; ), ( ; ; )= = - r r

e) 4 2 4 2 2 2 2 0a b( ; ; ), ( ; ; )= - = - r r f) 3 2 1 2 1 1a b( ; ; ), ( ; ; )= - = -

r r Bai 8. Tm vectu r , biet rang:

a) 2 1 3 1 3 2 3 2 45 11 20

a b ca u u b u c

( ; ; ), ( ; ; ), ( ; ; ). , . , . = - = - = - = - = - =

r r r r r r r r r b) 2 3 1 1 2 3 2 1 1

6a b cu a u b u c

( ; ; ), ( ; ; ), ( ; ; ), , . = - = - = - ^ ^ = -

r r r r r r r r r

c) 2 3 1 1 2 1 2 4 33 4 2

a b ca u b u c u

( ; ; ), ( ; ; ), ( ; ; ). , . , .

= = - - = - = = =

r r r r r r r r r d) 5 3 2 1 4 3 3 2 4

16 9 4a b ca u b u c u

( ; ; ), ( ; ; ), ( ; ; ). , . , .

= - = - = - = = = -

r r r r r r r r r

e) 7 2 3 4 3 5 1 1 15 7

a b ca u b u c u

( ; ; ), ( ; ; ), ( ; ; ). , . ,

= = - = - = - = - ^

r r r r r r r r r

Bai 9. Cho hai vecta b, r r . Tmme:

a) 2 1 2 0 2 22 3

a b

u a mb va v ma b vuong goc

( ; ; ), ( ; ; ) = - = -= + = -

r r r r r r r r b) 3 2 1 2 1 1

3 3 2

a b

u ma b va v a mb vuong goc

( ; ; ), ( ; ; ) = - = -= - = +

r r r r r r r r

c) 3 2 1 2 1 13 3 2

a bu ma b va v a mb cung phng

( ; ; ), ( ; ; ) = - = - = - = +

r r r r r r r r


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Bai 10. Cho hai vecta b, r r . Tnh X, Y khi biet:

a) 4 6a b X a b

, = = = -

r r r r b) 2 1 2 6 4a b a bY a b

( ; ; ), , = - - = - = = +

r r r r r r

c) ( )04 6 120a b a b

X a b Y a b, , ,

,

= = = = - = +

r r r r r r r r d) ( )

02 1 2 6 60a b a b X a b Y a b

( ; ; ), , ,

,

= - - = = = - = +

r r r r r r r r

Bai 11. Cho ba vecta b c, , r r r . Tmm, ne [ ]c a b,= r r r :a) ( ) ( ) ( )3 1 2 1 2 5 1 7a b m c; ; , ; ; , ; ;= - - = =

r r r b) ( ) ( ) ( )6 2 5 3 6 33 10a m b n c; ; , ; ; , ; ;= - = - =

r r r c) ( ) ( ) ( )2 3 1 5 6 4 1a b c m n; ; , ; ; , ; ;= = =

r r r Bai 12. Xet s ong phang cua ba vecta b c, , r r r trong moi trng hp sau ay:

a) ( ) ( ) ( )1 1 1 0 1 2 4 2 3a b c; ; , ; ; , ; ;= - = = r r r b) ( ) ( ) ( )4 3 4 2 1 2 1 2 1a b c; ; , ; ; , ; ;= = - =

r r r c) ( ) ( ) ( )3 1 2 1 1 1 2 2 1a b c; ; , ; ; , ; ;= - - = = -

r r r d) ( ) ( ) ( )4 2 5 3 1 3 2 0 1a b c; ; , ; ; , ; ;= = = r r r

e) 2 3 1 1 2 0 3 2 4a b c( ; ; ), ( ; ; ), ( ; ; )= = - = - r r r

f) 5 4 8 2 3 0 1 7 7a b c( ; ; ), ( ; ; ), ( ; ; )= - = - = - r r r

g) 2 4 3 1 2 2 3 2 1a b c( ; ; ), ( ; ; ), ( ; ; )= - = - = - r r r h) 2 4 3 1 3 2 3 2 1a b c( ; ; ), ( ; ; ), ( ; ; )= - = - - = -

r r r Bai 13. Tmm e 3 vecta b c, , r r r ong phang:

a) ( ) ( ) ( )1 2 1 2 1 0 2 2a m b m c m; ; , ; ; , ; ;= = + = - r r r

b) 2 1 1 2 1 1 2 2 2 1 2a m m b m m c m m( ; ; ); ( ; ; ), ( ; ; )= + - = + + = + r r r

c) ( ) ( ) ( )1 2 1 2 1 2 2a m m m b m m m c; ; , ; ; , ; ;= + - = - + = r r r

d) ( ) ( ) ( )1 3 2 1 2 1 0 2 2a b m m m c m; ; , ; ; , ; ;= - = + - - = - r r r

Bai 14. Cho cac vecta b c u, , , r r r r . Chng minh ba vecta b c, , r r r khong ong phang. Bieu die

vectu r

theo cac vecta b c, , r r r

:a) ( ) ( ) ( )2 1 0 1 1 2 2 2 1

3 7 7a b cu

; ; , ; ; , ; ;( ; ; )

= = - = - = -

r r r r b) ( ) ( ) ( )1 7 9 3 6 1 1 7

4 13 6a b c 2u

; ; , ; ; , ; ;( ; ; )

= - = - = - = - -

r r r r

c) ( ) ( ) ( )1 0 1 0 1 1 1 1 08 9 1

a b cu

; ; , ; ; , ; ;( ; ; )

= = - = = -

r r r r d) ( ) ( ) ( )1 0 2 2 3 0 0 3 4

1 6 22a b cu

; ; , ; ; , ; ;( ; ; )

= = - = - = - -

r r r r

e) ( ) ( ) ( )2 3 1 1 2 5 2 2 63 1 2

a b cu

; ; , ; ; , ; ;( ; ; )

= - = - = - =

r r r r f) ( ) ( ) ( )2 1 1 1 3 2 3 2 2

4 3 5a b cu

; ; , ; ; , ; ;( ; ; )

= - = - = - - = -

r r r r

Bai 15. Chng to bon vecta b c d , , , r r r r ong phang:

a) ( ) ( ) ( )2 6 1 4 3 2 4 2 2 2 11 1a b c d ; ; , ; ; , ; ; , ( ; ; )= - - = - - = - - = - - r r r r

b) ( ) ( ) ( )2 6 1 2 1 1 4 3 2 2 11 1a b c d ; ; , ; ; , ; ; , ( ; ; )= - = - = - = - r r r r

Bai 16. Cho ba vecta b c, , r r r khong ong phang va vectd r

. Chng minh bo ba vect sau khongong phang:a) b c d ma nb, , = +

r r r r r (vim, n 0) b) a c d ma nb, , = + r r r r r (vim, n 0)

c) a b d ma nb pc, , = + + r r r r r r, (vim, n, p 0) d) b c d ma nb pc, , = + +

r r r r r r, (vim, n, p 0)e) a c d ma nb pc, , = + +

r r r r r r , (vim, n, p 0)


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VAN E 2: Xac nh iem trong khong gian. Chng minh tnh chat hDien tch The tch.

S dung cac cong thc ve toa o cua vect va cua iem trong khong gian. S dung cac phep toan ve vect trong khong gian. Cong thc xac nh toa o cua cac iem ac biet.

Tnh chat hnh hoc cua cac iem ac biet: A, B, C thang hang AB AC ,

uuur uuurcung phng AB k AC =

uuur uuur 0 AB AC , =

uuur uuur r

ABCD la hnh bnh hanh AB DC = uuur uuur

Cho D ABC co cac chan E, F cua cac ng phan giac trong va ngoai cua goc A cuaD ABC

tren BC. Ta co: AB EB EC AC

.= - uuur uuur

, ABFB FC AC

.= uuur uuur

A, B, C, D khong ong phang AB AC AD, , uuur uuur uuur

khong ong phang 0 AB AC AD, . uuur uuur uuur

Bai 1. Cho iem M. Tm toa o hnh chieu vuong goc cua iem M: Tren cac mat phang toa o: Oxy, Oxz, Oyz Tren cac truc toa o: Ox, Oy, Oz

a) 1 2 3 M ( ; ; ) b) 3 1 2 M ( ; ; )- c) 1 1 3 M ( ; ; )- - d) 1 2 1 M ( ; ; )- e) 2 5 7 M ( ; ; )- f) 22 15 7 M ( ; ; )- g) 11 9 10 M ( ; ; )- h) 3 6 7 M ( ; ; )

Bai 2. Cho iem M. Tm toa o cua iem Moi xng vi iem M: Qua goc toa o Qua mp(Oxy) Qua truc Oy

a) 1 2 3 M ( ; ; ) b) 3 1 2 M ( ; ; )- c) 1 1 3 M ( ; ; )- - d) 1 2 1 M ( ; ; )- e) 2 5 7 M ( ; ; )- f) 22 15 7 M ( ; ; )- g) 11 9 10 M ( ; ; )- h) 3 6 7 M ( ; ; )

Bai 3. Xet tnh thang hang cua cac bo ba iem sau:a) 1 3 1 0 1 2 0 0 1 A B C ( ; ; ), ( ; ; ), ( ; ; ) b)1 1 1 4 3 1 9 5 1 A B C ( ; ; ), ( ; ; ), ( ; ; )- - c) 10 9 12 20 3 4 50 3 4 A B C ( ; ; ), ( ; ; ), ( ; ; )- - - - d) 1 5 10 5 7 8 2 2 7 A B C ( ; ; ), ( ; ; ), ( ; ; )- - - -

Bai 4. Cho ba iem A, B, C. Chng to ba iem A, B, C tao thanh mot tam giac. Tm toa o trong tam G cuaDABC. Xac nh iem D sao cho ABCD la hnh bnh hanh. Xac nh toa o cac chan E, F cua cac ng phan giac trong va ngDABC tren BC. Tnh o dai cac oan phan giac o. Tnh so o cac goc trongDABC. Tnh dien tchDABC. T o suy ra o dai ng cao AH cuaDABC.a) 1 2 3 0 3 7 12 5 0 A B C ( ; ; ), ( ; ; ), ( ; ; )- b) 0 13 21 11 23 17 1 0 19 A B C ( ; ; ), ( ; ; ), ( ; ; )-

c) 3 4 7 5 3 2 1 2 3 A B C ( ; ; ), ( ; ; ), ( ; ; )- - - - d) 4 2 3 2 1 1 3 8 7 A B C ( ; ; ), ( ; ; ), ( ; ; )- - e) 3 1 2 1 2 1 1 1 3 A B C ( ; ; ), ( ; ; ), ( ; ; )- - - - f) 4 1 4 0 7 4 3 1 2 A B C ( ; ; ), ( ; ; ), ( ; ; )- - g) ( ) ( ) ( )1 0 0 0 0 1 2 1 1 A B C ; ; , ; ; , ; ; h) 1 2 6 2 5 1 1 8 4 A B C ( ; ; ), ( ; ; ), ( ; ; )- -

Bai 5. Tren truc Oy(Ox), tm iem cach eu hai iem:a) 3 1 0 A( ; ; ) , 2 4 1 B( ; ; )- b) 1 2 1 11 0 7 A B( ; ; ), ( ; ; )- c) 4 1 4 0 7 4 A B( ; ; ), ( ; ; )- d) 3 1 2 1 2 1 A B( ; ; ), ( ; ; )- - e) 3 4 7 5 3 2 A B( ; ; ), ( ; ; )- - - f) 4 2 3 2 1 1 A B( ; ; ), ( ; ; )- -

Bai 6. Tren mat phang Oxy(Oxz, Oyz), tm iem cach eu ba iem:a) 1 1 1 1 1 0 3 1 1 A B C ( ; ; ), ( ; ; ), ( ; ; )- - b) 3 2 4 0 0 7 5 3 3 A B C ( ; ; ), ( ; ; ), ( ; ; )- -

c) 3 1 2 1 2 1 1 1 3 A B C ( ; ; ), ( ; ; ), ( ; ; )- - - - d) 0 13 21 11 23 17 1 0 19 A B C ( ; ; ), ( ; ; ), ( ; ; )- e) 1 0 2 2 1 1 1 3 2 A B C ( ; ; ), ( ; ; ), ( ; ; )- - - f) 1 2 6 2 5 1 1 8 4 A B C ( ; ; ), ( ; ; ), ( ; ; )- -


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Bai 7. Cho hai iem A, B. ng thang AB cat mat phang Oyz(Oxz, Oxy)tai iem M. iem M chia oan thang AB theo t so nao ? Tm toa o iem M.

a) ( ) ( )2 1 7 4 5 2 A B; ; , ; ;- - b) 4 3 2 2 1 1 A B( ; ; ), ( ; ; )- - c) 10 9 12 20 3 4 A B( ; ; ), ( ; ; )- d) 3 1 2 1 2 1 A B( ; ; ), ( ; ; )- - e) 3 4 7 5 3 2 A B( ; ; ), ( ; ; )- - - f) 4 2 3 2 1 1 A B( ; ; ), ( ; ; )- -

Bai 8. Cho bon iem A, B, C, D. Chng minh A, B, C, D la bon nh cua mot t dien. Tm toa o trong tam G cua t dien ABCD. Tnh goc tao bi cac canh oi dien cua t dien ABCD. Tnh the tch cua khoi t dien ABCD. Tnh dien tch tam giac BCD, t o suy ra o dai ng cao cua t die

a) 2 5 3 1 0 0 3 0 2 3 1 2 A B C D( ; ; ), ( ; ; ), ( ; ; ), ( ; ; )- - - - b) ( ) ( ) ( ) ( )1 0 0 0 1 0 0 0 1 2 1 1 A B C D; ; , ; ; , ; ; , ; ;- - c) ( ) ( ) ( ) ( )1 1 0 0 2 1 1 0 2 1 1 1 A B C D; ; , ; ; , ; ; , ; ; d)( ) ( ) ( ) ( )2 0 0 0 4 0 0 0 6 2 4 6 A B C D; ; , ; ; , ; ; , ; ;e) 2 3 1 4 1 2 6 3 7 5 4 8 A B C D( ; ; ), ( ; ; ), ( ; ; ), ( ; ; )- - - f) 5 7 2 3 1 1 9 4 4 1 5 0 A B C D( ; ; ), ( ; ; ), ( ; ; ), ( ; ; )- - - g) 2 4 1 1 0 1 1 4 2 1 2 1 A B C D( ; ; ), ( ; ; ), ( ; ; ), ( ; ; )- - - h) 3 2 4 2 5 2 1 2 2 4 2 3 A B C D( ; ; ), ( ; ; ), ( ; ; ), ( ; ; )- - - i) 3 4 8 1 2 1 5 2 6 7 4 3 A B C D( ; ; ), ( ; ; ), ( ; ; ), ( ; ; )- - k) 3 2 6 2 4 4 9 9 1 0 0 1 A B C D( ; ; ), ( ; ; ), ( ; ; ), ( ; ; )- - - -

Bai 9. Cho hnh hop ABCD.A'B'C'D'. Tm toa o cac nh con lai. Tnh the tch khoi hop.

a) ( ) ( ) ( ) ( )1 0 1 2 1 2 1 1 1 4 5 5 A B D C ; ; , ; ; , ; ; , ' ; ;- - b) 2 5 3 1 0 0 3 0 2 3 1 2 A B C A( ; ; ), ( ; ; ), ( ; ; ), '( ; ; )- - - - c) 0 2 1 1 1 1 0 0 0 1 1 0 A B D A( ; ; ), ( ; ; ), ( ; ; ;), '( ; ; )- - d) 0 2 2 0 1 2 1 1 1 1 2 1 A B C C ( ; ; ), ( ; ; ), ( ; ; ), '( ; ; )- - -

Bai 10. Cho bon iem S(3; 1; 2), A(5; 3; 1), B(2; 3; 4), C(1; 2; 0).a) Chng minh SA^ (SBC), SB(SAC), SC(SAB).b) Chng minh S.ABC la mot hnh chop eu.c) Xac nh toa o chan ng cao H cua hnh chop. Suy ra o dai ng c

Bai 11. Cho bon iem S(1; 2; 3), A(2; 2; 3), B(1; 3; 3), C(1; 2; 4).a) Chng minh SA^ (SBC), SB(SAC), SC(SAB).b) Goi M, N, P lan lt la trung iem cua BC, CA, AB. Chng minh SMNP lac) Ve SH(ABC). Goi Sla iem oi xng cua H qua S. Chng minh SABC la t dien eu.

Bai 12. Cho hnh hop ch nhat OABC.DEFG. Goi I la tam cua hnh hop.a) Phan tch cac vectOI AG,

uur uuurtheo cac vectOA OC OD, ,

uuur uuur uuur.

b) Phan tch vect BI uur

theo cac vectFE FG FI , , uuur uuur uur

.Bai 13. Cho hnh lap phng ABCD.EFGH.

a) Phan tch vect AE uuur theo cac vect AC AF AH , , uuur uuur uuur .b) Phan tch vect AG

uuurtheo cac vect AC AF AH , ,

uuur uuur uuur.

Bai 14. Cho hnh hop ABCD.A'B'C'D'. Goi M, N lan lt la trung iem cua AD v. Chngminh rang MN^ AC.

Bai 15. Cho hnh lap phng ABCD.A'B'C'D' vi canh bang 1. Tren cac canh BB, CD, ADlanlt lay cac iem M, N, P sao cho BM = CN = DP = x (0 < x < 1). Chng minh ACvuonggoc vi mat phang (MNP).


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VAN E 3: Phng trnh mat cau e viet phng trnh mat cau (S), ta can xac nh tam I va ban knh Rcua mat cau.

Dang 1: (S) co tamI(a; b; c) va ban knhR:(S): 2 2 2 2 x a y b z c R( ) ( ) ( )- + - + - =

Dang 2: (S) co tamI(a; b; c) va i qua iem A:Khi o ban knh R = IA.

Dang 3: (S) nhan oan thang AB cho trc lam ng knh: Tam I la trung iem cua oan thang AB:

2 2 2 A B A B A B

I I I x x y y z z

x y z; ;+ + += = = .

Ban knh R = IA =2

AB.

Dang 4: (S) i qua bon iem A, B, C, D(mat cau ngoai tiep t dien ABCD): Gia s phng trnh mat cau (S) co dang:2 2 2 2 2 2 0 x y z ax by cz d + + + + + + = (*). Thay lan lt toa o cua cac iem A, B, C, D vao (*), ta c 4 phng trnh. Giai he phng trnh o, ta tm c a, b, c, d Phng trnh mat cau (S).

Dang 5: (S) i qua ba iem A, B, C va co tam I nam tren mat phang (P) cho trcGiai tng t nh dang 4.

Dang 6: (S) co tam I va tiep xuc vi mat cau(T) cho trc: Xac nh tam J va ban knh R cua mat cau (T). S dung ieu kien tiep xuc cua hai mat cau e tnh ban knh R cua mat cau (S).

(Xet hai trng hp tiep xuc trong va tiep xuc ngoai)

Chu y:Vi phng trnh mat cau (S):

2 2 2 2 2 2 0 x y z ax by cz d + + + + + + = vi2 2 2 0a b c d + + - > th (S) cotam I(a; b; c) va ban knh R = 2 2 2a b c d + + - .

Bai 1. Tm tam va ban knh cua cac mat cau sau:a) 2 2 2 8 2 1 0 x y z x y+ + - + + = b) 2 2 2 4 8 2 4 0 x y z x y z+ + + + - - = c) 2 2 2 2 4 4 0 x y z x y z+ + - - + = d) 2 2 2 6 4 2 86 0 x y z x y z+ + - + - - = e) 2 2 2 12 4 6 24 0 x y z x y z+ + - + - + = f) 2 2 2 6 12 12 72 0 x y z x y z+ + - - + + = g) 2 2 2 8 4 2 4 0 x y z x y z+ + - + + - = h) 2 2 2 3 4 0 x y z x y+ + - + = i) 2 2 23 3 3 6 3 15 2 0 x y z x y z+ + + - + - = k) 2 2 2 6 2 2 10 0 x y z x y z+ + - + - + =

Bai 2. Xac nhm, t, a , e phng trnh sau xac nh mot mat cau, tm tam va bacac mat cau o:a) 2 2 2 22 2 4 2 5 9 0 x y z m x my mz m( )+ + - + + - + + = b) 2 2 2 22 3 2 1 2 2 7 0 x y z m x m y mz m( ) ( )+ + - - - + - + + = c) 2 2 2 2 1 4 2 2 7 0 x y z x y z(cos ) cos . cosa a a+ + + + - - + + =

d)2 2 2 2 2

2 3 2 4 1 2 4 8 0 x y z x y z( cos ) (sin ) cosa a a+ + + - + - + + + = e) 2 2 2 2 2 6 3 8 0 x y z t x y z t ln . ln+ + - + - + + = f) 2 2 2 22 2 4 2 1 5 8 0 x y z t x t y t z t ( ln ) ln . (ln ) ln+ + + - + + + + + =


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Bai 3. Viet phng trnh mat cau co tam I va ban knh R:a) 1 3 5 3 I R( ; ; ),- = b) 5 3 7 2 I R( ; ; ),- = c) 1 3 2 5 I R( ; ; ),- = d) 2 4 3 3 I R( ; ; ),- =

Bai 4. Viet phng trnh mat cau co tam I va i qua iem A:a) 2 4 1 5 2 3 I A( ; ; ), ( ; ; )- b) 0 3 2 0 0 0 I A( ; ; ), ( ; ; )- c) 3 2 1 2 1 3 I A( ; ; ), ( ; ; )- - d) 4 4 2 0 0 0 I A( ; ; ), ( ; ; )- - e) 4 1 2 1 2 4 I A( ; ; ), ( ; ; )- - -

Bai 5. Viet phng trnh mat cau co ng knh AB, vi:a) 2 4 1 5 2 3 A B( ; ; ), ( ; ; )- b) 0 3 2 2 4 1 A B( ; ; ), ( ; ; )- - c) 3 2 1 2 1 3 A B( ; ; ), ( ; ; )- - d) 4 3 3 2 1 5 A B( ; ; ), ( ; ; )- - e) 2 3 5 4 1 3 A B( ; ; ), ( ; ; )- - f) 6 2 5 4 0 7 A B( ; ; ), ( ; ; )- -

Bai 6. Viet phng trnh mat cau ngoai tiep t dien ABCD, vi:a) ( ) ( ) ( ) ( )1 1 0 0 2 1 1 0 2 1 1 1 A B C D; ; , ; ; , ; ; , ; ; b)( ) ( ) ( ) ( )2 0 0 0 4 0 0 0 6 2 4 6 A B C D; ; , ; ; , ; ; , ; ;c) 2 3 1 4 1 2 6 3 7 5 4 8 A B C D( ; ; ), ( ; ; ), ( ; ; ), ( ; ; )- - - d) 5 7 2 3 1 1 9 4 4 1 5 0 A B C D( ; ; ), ( ; ; ), ( ; ; ), ( ; ; )- - - e) 6 2 3 0 1 6 2 0 1 4 1 0 A B C D( ; ; ), ( ; ; ), ( ; ; ), ( ; ; )- - f) 0 1 0 2 3 1 2 2 2 1 1 2 A B C D( ; ; ), ( ; ; ), ( ; ; ), ( ; ; )- -

Bai 7. Viet phng trnh mat cau i qua ba iem A, B, C va co tam nam trong

cho trc, vi:a) 1 2 0 1 1 3 2 0 1 A B C

P Oxz( ; ; ), ( ; ; ), ( ; ; )

( ) ( ) - -

b) 2 0 1 1 3 2 3 2 0 A B C P Oxy( ; ; ), ( ; ; ), ( ; ; )

( ) ( )

Bai 8. Viet phng trnh mat cau (S) co tam I va tiep xuc vi mat cau (T), va) 2 2 2

5 112 4 6 5 0

I T x y z x y z( ; ; )

( ) : - + + - + - + =

b) 2 2 23 2 2

2 4 8 5 0 I

T x y z x y z( ; ; )

( ) : - + + - + - + =

VAN E 4: V tr tng oi gia hai mat cau mat cauCho hai mat cau S 1(I 1 , R1) va S 2(I 2 , R2).

1 2 1 2 I I R R< - (S 1), (S 2) trong nhau 1 2 1 2 I I R R> + (S 1), (S 2) ngoai nhau

1 2 1 2 I I R R= - (S 1), (S 2) tiep xuc trong 1 2 1 2 I I R R= + (S 1), (S 2) tiep xuc ngoai

1 2 1 2 1 2 R R I I R R- < < + (S 1), (S 2) cat nhau theo mot ng tron.

Bai 1. Xet v tr tng oi cua hai mat cau:

a)2 2 2

2 2 28 4 2 4 04 2 4 5 0

x y z x y z x y z x y z

+ + - + - - = + + + - - + =b)

2 2 2

2 2 21 2 3 9

6 10 6 21 0 x y z

x y z x y z( ) ( ) ( ) + + - + - = + + - - - - =

c)2 2 2

2 2 22 4 10 5 04 6 2 2 0


+ + - + - + = + + - - + - =d)

2 2 2

2 2 28 4 2 15 04 12 2 25 0


+ + - + - - = + + + - - + =

e)2 2 2

2 2 22 6 4 5 06 2 4 2 0


+ + - - + + = + + - + - - =f)

2 2 2

2 2 24 2 2 3 06 4 2 2 0


+ + + - + - = + + - + - - =

Bai 2. Bien luan theom v tr tng oi cua hai mat cau:

a)2 2 2

2 2 2 22 1 3 644 2 3 2

x y z x y z m( ) ( ) ( )( ) ( ) ( ) ( )

- + - + + = - + + + - = +b)

2 2 2

2 2 2 23 2 1 811 2 3 3

x y z x y z m( ) ( ) ( )( ) ( ) ( ) ( )

- + + + + = - + - + - = -

c)2 2 2

2 2 2 22 2 1 251 2 3 1 x y z x y z m( ) ( ) ( )( ) ( ) ( ) ( )

+ + - + - = + + + + + = - d)

2 2 2

2 2 2 23 2 1 161 2 3 3 x y z x y z m( ) ( ) ( )( ) ( ) ( ) ( )

+ + + + + = - + - + - = +


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1. Vect phap tuyen Cap vect ch phng cua mat phang

Vect 0n r r

la VTPT cua (a) neu gia cuan r

vuong goc vi (a). Hai vecta b,r r khong cung phng la cap VTCP cua (a) neu cac gia cua chung song so

hoac nam tren (a).Chu y: Neun r la mot VTPT cua (a ) th kn r (k 0) cung la VTPT cua (a ).

Neu a b,r r la mot cap VTCP cua (a ) th [ ]n a b,=

r r r la mot VTPT cua (a ). 2. Phng trnh tong quat cua mat phang

2 2 20 0 Ax By Cz D vi A B C + + + = + + > Neu (a) co phng trnh 0 Ax By Cz D+ + + = thn A B C ( ; ; )= r la mot VTPT cua (a). Phng trnh mat phang i qua

0 0 0 0 M x y z( ; ; ) va co mot VTPTn A B C ( ; ; )= r la:

0 0 0 0 A x x B y y C z z( ) ( ) ( )- + - + - = 3. Cac trng hp rieng

Chu y: Neu trong phng trnh cua (a ) khong cha an nao th (a ) song song hoac chatruc tng ng.

Phng trnh mat phang theo oan chan: 1 x y za b c

+ + =

(a ) cat cac truc toa o tai cac iem (a; 0; 0), (0; b; 0), (0; 0; c)

4. V tr tng oi cua hai mat phangCho hai mat phang (a), (b) co phng trnh: (a): 1 1 1 1 0 A x B y C z D+ + + =

(b): 2 2 2 2 0 A x B y C z D+ + + = (a ), (b ) cat nhau 1 1 1 2 2 2 A B C A B C : : : :

(a ) // (b ) 1 1 1 12 2 2 2

A B C D A B C D

= = (a ) (b ) 1 1 1 12 2 2 2

A B C D A B C D

= = =

(a ) ^ (b ) 1 2 1 2 1 2 0 A A B B C C + + = 5. Khoang cach t iem M 0(x0 ; y0 ; z0 ) en mat phang( a ): Ax + By + Cz + D = 0

( ) 0 0 00 2 2 2 Ax By Cz D

d M A B C

,( )a+ + +

=+ +

III. PHNG TRNH MAT PHANG

Cac he so Phng trnh mat phang (a ) Tnh chat mat phang (a )D = 0 0 Ax By Cz+ + = (a) i qua goc toa o OA = 0 0 By Cz D+ + = (a) // Ox hoac (a) OxB = 0 0 Ax Cz D+ + = (a) // Oy hoac (a) OyC = 0 0 Ax By D+ + = (a) // Oz hoac (a) OzA = B = 0 0Cz D+ = (a) // (Oxy) hoac (a) (Oxy)A = C = 0 0 By D+ = (a) // (Oxz) hoac (a) (Oxz)B = C = 0 0 Ax D+ = (a) // (Oyz) hoac (a) (Oyz)


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VAN E 1: Viet phng trnh mat phang e lap phng trnh mat phang (a ) ta can xac nh mot iemthuoc (a ) va mot VTPT cua no.

Dang 1: (a ) i qua iem( )0 0 0M x ; y ; z co VTPT ( )n A; B;C =r

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

Dang 2: (a ) i qua iem( )0 0 0M x ; y ; z co cap VTCPa b,r r :

Khi o mot VTPT cua (a ) la [ ]n a b,= r r r .Dang 3:(a ) i qua iem( )0 0 0M x ; y ; z va song song vi mat phang(b ): Ax + By + Cz + D = 0:

(a ): ( ) ( ) ( )0 0 0 0 A x x B y y C z z - + - + - = Dang 4:(a ) i qua 3 iem khong thang hang A, B, C:

Khi o ta co the xac nh mot VTPT cua (a ) la: n AB AC , = uuur uuur r

Dang 5: (a ) i qua mot iem M va mot ng thang (d) khong cha M: Tren (d) lay iem A va VTCPu r . Mot VTPT cua (a ) la: n AM u

, =

uuur r r

Dang 6: (a ) i qua mot iem M va vuong goc vi mot ng thang (d):VTCP u r cua ng thang (d) la mot VTPT cua (a ).

Dang 7: (a ) i qua 2 ng thang cat nhau d1, d2: Xac nh cac VTCP a b,

r r cua cac ng thang d 1 , d 2. Mot VTPT cua (a ) la: [ ]n a b,=

r r r . Lay mot iem M thuoc d 1 hoac d 2 M (a ).

Dang 8: (a ) cha ng thang d1 va song song vi ng thang d2 (d 1 , d 2 cheo nhau): Xac nh cac VTCP a b,

r r cua cac ng thang d 1 , d 2.

Mot VTPT cua (a ) la: [ ]n a b,=r r r

. Lay mot iem M thuoc d 1 M (a ).

Dang 9: (a ) i qua iem M va song song vi hai ng thang cheo nhau d1, d2: Xac nh cac VTCP a b,

r r cua cac ng thang d 1 , d 2. Mot VTPT cua (a ) la: [ ]n a b,=

r r r . Dang 10: (a ) i qua mot ng thang (d) va vuong goc vi mot mat phang (b):

Xac nh VTCPu r cua (d) va VTPT nb r cua (b ).

Mot VTPT cua (a ) la: n u n, b = r r r .

Lay mot iem M thuoc d M (a ).Dang 11: (a ) i qua iem M va vuong goc vi hai mat phang cat nhau (b), (g): Xac nh cac VTPT n n,b g

r r cua (b ) va (g ).

Mot VTPT cua (a ) la: n u n,b g = r r r .

Dang 12: (a ) i qua ng thang (d) cho trc va cach iem M cho trc mot trc:

Gia s (a ) co phng trnh: 0 Ax By Cz+D+ + = ( )2 2 2 0 A B C + + . Lay 2 iem A, B (d) A, B (a ) (ta c hai phng trnh (1), (2)).

T ieu kien khoang cach d M k ( ,( ))a = , ta c phng trnh (3). Giai he phng trnh (1), (2), (3) (bang cach cho gia tr mot an, tm cac an con lai).Dang 13: (a ) la tiep xuc vi mat cau (S) tai iem H:

Gia s mat cau (S) co tam I va ban knh R.


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Mot VTPT cua (a ) la:n IH = uur r

Chu y:e viet phng trnh mat phang can nam vng cac cach xac nh mat phang a hlp 11.

Bai 1. Viet phng trnh mat phang (P) i qua iem M va co VTPTnr cho trc:a) ( ) ( )= -M 3;1;1 , n 1;1;2

r

b) ( ) ( )- =M 2; 7; 0 , n 3; 0;1r

c) ( ) ( )- - =M 4; 1; 2 , n 0;1; 3r

d) ( ) ( )- =M 2;1; 2 , n 1;0;0r e) ( ) ( )= - -M 3; 4; 5 , n 1; 3; 7r f) ( ) ( )= -M 10;1;9 , n 7;10;1r

Bai 2. Viet phng trnh mat phang trung trc cua oan thang AB cho trc, va) 2 1 1 2 1 1 A B( ; ; ), ( ; ; )- - b) 1 1 4 2 0 5 A B( ; ; ), ( ; ; )- - c) 2 3 4 4 1 0 A B( ; ; ), ( ; ; )- -

d) 1 1A ; 1; 0 , B 1; ;52 2

- - e) 2 1 1A 1; ; , B 3; ;1

3 2 3

- f) 2 5 6 1 3 2 A B( ; ; ), ( ; ; )- - -

Bai 3. Viet phng trnh mat phang i qua iem M va co cap VTCPa b, r r cho trc, vi:a) 1 2 3 2 1 2 3 2 1 M a b( ; ; ), ( ; ; ), ( ; ; )- = = -

r r b) 1 2 3 3 1 2 0 3 4 M a b( ; ; ), ; ; ), ( ; ; )- = - - = r r

c) 1 3 4 2 7 2 3 2 4 M a b( ; ; ), ( ; ; ), ( ; ; )- = = r r

d) 4 0 5 6 1 3 3 2 1 M a b( ; ; ), ( ; ; ); ( ; ; )- = - = r r

Bai 4. Viet phng trnh mat phang (a) i qua iem M va song song vi mat phang( )b chotrc, vi:a) ( ) ( ) ( )2 1 5 M Oxy; ; , b = b) ( ) ( )1 2 1 2 3 0 M x y; ; , :b - - + = c) ( ) ( )1 1 0 2 10 0 M x y z; ; , :b - - + - = d) ( ) ( )3 6 5 1 0 M x z; ; , :b - - + - = e) 2 3 5 2 5 0 M x y z( ; ; ), ( ) :b - + - + = f) 1 1 1 10 10 20 40 0 M x y z( ; ; ), ( ) :b - + - =

Bai 5. Viet phng trnh mat phang (a) i qua iem M va lan lt song song vi cac matoa o, vi:a) ( )2 1 5 M ; ; b) ( )1 2 1 M ; ;- c) ( )1 1 0 M ; ;- d) ( )3 6 5 M ; ;- e) 2 3 5 M ( ; ; )- f) 1 1 1 M ( ; ; ) g) 1 1 0 M ( ; ; )- h) 3 6 5 M ( ; ; )-

Bai 6. Viet phng trnh mat phang (a) i qua ba iem A, B, C khong thang hang cho tvi:a) 1 2 4 3 2 1 2 1 3 A B C ( ; ; ), ( ; ; ), ( ; ; )- - - - b) 0 0 0 2 1 3 4 2 1 A B C ( ; ; ), ( ; ; ), ( ; ; )- - - c) 1 2 3 2 4 3 4 5 6 A B C ( ; ; ), ( ; ; ), ( ; ; )- - d) 3 5 2 1 2 0 0 3 7 A B C ( ; ; ), ( ; ; ), ( ; ; )- - - e) 2 4 0 5 1 7 1 1 1 A B C ( ; ; ), ( ; ; ), ( ; ; )- - - - f) 3 0 0 0 5 0 0 0 7 A B C ( ; ; ), ( ; ; ), ( ; ; )- -

Bai 7. Viet phng trnh mat phang (a) i qua iem A va vuong goc vi ng thang iem B, C cho trc, vi:

a) 1 2 4 3 2 1 2 1 3 A B C ( ; ; ), ( ; ; ), ( ; ; )- - - - b) 0 0 0 2 1 3 4 2 1 A B C ( ; ; ), ( ; ; ), ( ; ; )- - - c) 1 2 3 2 4 3 4 5 6 A B C ( ; ; ), ( ; ; ), ( ; ; )- - d) 3 5 2 1 2 0 0 3 7 A B C ( ; ; ), ( ; ; ), ( ; ; )- - - e) 2 4 0 5 1 7 1 1 1 A B C ( ; ; ), ( ; ; ), ( ; ; )- - - - f) 3 0 0 0 5 0 0 0 7 A B C ( ; ; ), ( ; ; ), ( ; ; )- -

Bai 8. Viet phng trnh mat phang (a) i qua hai iem A, B va vuong goc vi mat phb)cho trc, vi:a) ( )

3 1 1 2 1 42 3 1 0

A B x y z

( ; ; ), ( ; ; ):b

- - - + - =

b) ( )2 1 3 4 2 1

2 3 2 5 0 A B

x y z( ; ; ), ( ; ; )

:b - - - + - + =

c) ( )2 1 3 4 7 9

3 4 8 5 0 A B

x y z( ; ; ), ( ; ; )

:b - - - + - - =

d) ( )3 1 2 3 1 2

2 2 2 5 0 A B

x y z( ; ; ), ( ; ; )

:b - - - - - + =

Bai 9. Viet phng trnh mat phang (a) i qua iem M va vuong goc vi hai mat phanb), (g)cho trc, vi:a) ( ) ( )1 2 5 2 3 1 0 2 3 1 0 M x y z x y z( ; ; ), : , :b g - - + - + = - + + =


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b) ( ) ( )1 0 2 2 2 0 3 0 M x y z x y z( ; ; ), : , :b g - + - - = - - - = c) ( ) ( )2 4 0 2 3 2 5 0 3 4 8 5 0 M x y z x y z( ; ; ), : , :b g - + - + = + - - = d) ( ) ( )5 1 7 3 4 3 6 0 3 2 5 3 0 M x y z x y z( ; ; ), : , :b g - + + = - + - =

Bai 10. Viet phng trnh mat phang (a) i qua iem M va giao tuyen cua hai mat pha(Q) cho trc, vi:a) ( ) ( ) ( )1 2 3 2 3 5 0 3 2 5 1 0 M P x y z Q : x y z; ; , : ,- - + - = - + - = b) ( ) ( ) ( )2 1 1 4 0 3 1 0 M P x y z Q : x y z; ; , : ,- - + - = - + - = c) ( ) ( ) ( )3 4 1 19 6 4 27 0 42 8 3 11 0 M P x y z Q : x y z; ; , : ,- - + = - + + = d) ( ) ( ) ( )0 0 1 5 3 2 5 0 2 1 0 M P x y z Q x y z; ; , : , :- + - = - - - =

Bai 11. Viet phng trnh mat phang (a) qua giao tuyen cua hai mat phang (P), (Q), onsong song vi mat phang (R) cho trc, vi:a) 2 4 0 3 0 2 0P y z Q x y z R x y z( ) : , ( ) : , ( ) :+ - = + - - = + + - = b) 4 2 5 0 4 5 0 2 19 0P x y z Q y z R x y( ) : , ( ) : , ( ) :- + - = + - = - + = c) 3 2 0 4 5 0 2 7 0P x y z Q x y R x z( ) : , ( ) : , ( ) :- + - = + - = - + =

Bai 12. Viet phng trnh mat phang (a) qua giao tuyen cua hai mat phang (P), (Q), onvuong goc vi mat phang (R) cho trc, vi:a) 2 3 4 0 2 3 5 0 2 3 2 0P x y Q y z R x y z( ) : , ( ) : , ( ) :+ - = - - = + - - = b) 2 4 0 3 0 2 0P y z Q x y z R x y z( ) : , ( ) : , ( ) :+ - = + - + = + + - = c) 2 4 0 2 5 0 2 3 6 0P x y z Q x y z R x y z( ) : , ( ) : , ( ) :+ - - = + + + = - - + = d) 3 2 0 4 5 0 2 7 0P x y z Q x y R x z( ) : , ( ) : , ( ) :- + - = + - = - + =

Bai 13. Viet phng trnh mat phang (a) qua giao tuyen cua hai mat phang (P), (Q), on

cach iem M cho trc mot khoang bangk , vi:a) 2 0 5 13 2 0 1 2 3 2P x y Q x y z M k ( ): , ( ) : , ( ; ; ),- - = - + = =

VAN E 2: V tr tng oi cua hai mat phang

Bai 1. Xet v tr tng oi cua cac cap mat phang sau:a) 2 3 2 5 0

3 4 8 5 0 x y z x y z

+ - + = + - - =b) 3 4 3 6 0

3 2 5 3 0 x y z x y z

- + + = - + - =c) 5 5 5 1 0

3 3 3 7 0 x y z x y z

+ - - = + - + =

d) 6 4 6 5 012 8 12 5 0

x y z x y z - - + = - - - =

e) 2 2 4 5 0255 5 10 0

2

x y z x y z

- - + = - - + =

f) 3 2 6 23 03 2 6 33 0 x y z x y z - - - = - - + =

Bai 2. Xac nhm, ne cac cap mat phang sau: song song cat nhau trung nhaua) 3 2 7 0

7 6 4 0 x my znx y z

+ - - = + - + =b) 5 2 11 0

3 5 0 x y mz

x ny z - + - = + + - =

c) 2 3 5 06 6 2 0

x my znx y z

+ + - = - - + =

d) 3 9 02 2 3 0 x y mz x ny z

- + - = + + - =e) 2 3 5 0

6 6 2 0 x y z

mx y z + + - = - - - =

f) 3 5 3 02 3 1 0

x y mz x y z

- + - = + - + =

g) 2 02 4 3 0 x my z

x y nz + - + = + + - =

h) 2 2 1 03 2 0 x ny z x y mz

- + - = - + - =

i) 3 3 2 5 02 2 10 0

x m y zm x y mz

( )( )

- - + - = + - + - =

Bai 3. Xac nhm e cac cap mat phang sau vuong goc vi nhaua) 2 7 2 0

3 2 15 0 x y mz x y z

- + + = + - + =b) 2 1 3 2 3 0

1 4 5 0m x my zmx m y z

( )( )

- - + + = + - + - =


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c) 2 12 07 0

mx y mz x my z

+ + - = + + + =d) 3 3 2 5 0

2 2 10 0 x m y zm x y mz

( )( )

- - + - = + - + - =

e) 4 3 3 02 7 1 0

x y zmx y z

- - = + - - =f) 3 5 3 0

3 2 5 0 x y mz x y z

- + - = + + + =

VAN E 3: Khoang cach t mot iem en mot mat phangKhoang cach gia hai mat phang song song.

Hnh chieu cua mot iem tren mat phang . iem oi xng cua mot ie Khoang cach t iem M 0(x0; y0; z0) en mat phang (a ): Ax + By + Cz + D = 0

( ) 0 0 00 2 2 2 Ax By Cz D

d M A B C

,( )a+ + +

=+ +

Khoang cach gia hai mat phang song song bang khoang cach t mot iem bat k tren phang nay en mat phang kia.

Chu y:Neu hai mat phang khong song song th khoang cach gia chung bang 0.

iem H la hnh chieu cua iem M tren (P) MH n cung phng H P

,( )

uuuur r

iem M oi xng vi iem M qua (P) 2 MM MH = uuuuur uuuur

Bai 1. Cho mat phang (P) va iem M. Tnh khoang cach t M en (P). Tm toa o hnh chieu H cua M tren (P). Tm toa o iem Moi xng vi M qua (P).

a) 2 2 6 0 2 3 5P x y z M ( ) : , ( ; ; )- + - = - b) 5 14 0 1 4 2P x y z M ( ) : , ( ; ; )+ + - = - - c) 6 2 3 12 0 3 1 2P x y z M ( ) : , ( ; ; )- + + = - d) 2 4 4 3 0 2 3 4P x y z M ( ) : , ( ; ; )- + + = - e) 4 0 2 1 1P x y z M ( ) : , ( ; ; )- + - = - f) 3 2 0 1 2 4P x y z M ( ) : , ( ; ; )- + - =

Bai 2. Tm khoang cach gia hai mat phang:a) 2 3 1 0

2 3 5 0 x y z

x y z - + + = - + + =

b) 6 2 1 06 2 3 0 x y z x y z

- + + = - + - =c) 2 4 5 0

3 5 1 0 x y z x y z

- + + = + - - =

d) 4 8 1 04 8 5 0 x y z x y z

- + + = - + + =e) 2 4 5 0

3 5 1 0 x y z x y z

- + + = + - - =f) 3 6 3 7 0

2 1 0 x y z

x y z + - + = + - + =

Bai 3. Tm tap hp cac iem cach mat phang mot khoang bangk cho trc:

a) 6 3 2 7 0 3 x y z k ,- + - = = b) 3 2 6 5 0 4 x y z k ,- - + = = c) 6 2 3 12 0 2 x y z k ,- + + = = d) 2 4 4 14 0 3 x y z k ,- + - = = Bai 4. Tm tap hp cac iem cach eu hai mat phang:

a) 2 3 1 02 3 5 0 x y z

x y z - + + = - + + =

b) 6 2 1 06 2 3 0 x y z x y z

- + + = - + - =c) 2 4 5 0

3 5 1 0 x y z x y z

- + + = + - - =

d) 4 8 1 04 8 5 0 x y z x y z

- + + = - + + =e) 2 4 5 0

3 5 1 0 x y z x y z

- + + = + - - =f) 3 6 3 7 0

2 1 0 x y z

x y z + - + = + - + =

Bai 5. Tm tap hp cac iem co ty so cac khoang cach en hai mat phang bak cho trc:

a)2 2 10 0

2 4 4 3 023

x y z

x y zk

+ - - =

+ - + = =

b)6 2 1 0

6 2 3 012

x y z

x y zk

- + + =

- + - = =

c)6 3 2 1 0

2 2 6 047

x y z

x y zk

+ - - =

+ - + = =

Bai 6. Tm iem M tren trucOx (Oy, Oz)cach eu iem N va mat phang (P):


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a) 2 2 5 0 1 2 2P x y z N ( ) : , ( ; ; )+ + - = - b) 5 14 0 1 4 2P x y z N ( ) : , ( ; ; )+ + - = - - c) 6 2 3 12 0 3 1 2P x y z N ( ) : , ( ; ; )- + + = - d) 2 4 4 3 0 2 3 4P x y z N ( ) : , ( ; ; )- + + = - e) 4 0 2 1 1P x y z N ( ) : , ( ; ; )- + - = - f) 3 2 0 1 2 4P x y z N ( ) : , ( ; ; )- + - =

Bai 7. Tm iem M tren trucOx (Oy, Oz)cach eu hai mat phang:

a)1 05 0

x y z x y z

+ - + = - + - = b)

2 2 1 02 2 5 0 x y z x y z

+ - + = + + - = c)

2 4 5 04 2 1 0 x y z x y z

- + + = + - - =

d) 4 8 1 04 8 5 0 x y z x y z

- + + = - + + =e) 2 4 5 0

3 5 1 0 x y z x y z

- + + = + - - =f) 3 6 3 7 0

2 1 0 x y z

x y z + - + = + - + =

Bai 8. Tm phng trnh tong quat cua mat phang (P) i qua iem A va song song(Q) cho trc. Tnh khoang cach gia (P) va (Q):a) ( )1 2 3 2 4 4 0 A Q x y z; ; , ( ) : - - + = . b) ( )3 1 2 6 2 3 12 0 A Q x y z; ; , ( ) : - + + = .

Bai 9. Tm phng trnh tong quat cua mat phang (P) song song vi mat phang (A mot khoangk cho trc:a) 2 2 5 0 2 1 4 4Q x y z A k ( ) : , ( ; ; ),+ - + = - = b) 2 4 4 3 0 2 3 4 3Q x y z A k ( ) : , ( ; ; ),- + + = - =

Bai 10. Tm phng trnh tong quat cua mat phang (P) cach mat phang (Q) mot k :a) 3 2 3 0 14Q x y z k ( ) : ,- + - = = b) 4 3 2 5 0 29Q x y z k ( ) : ,+ - + = =

VAN E 4: Goc gia hai mat phangCho hai mat phang (a ), (b ) co phng trnh: (a ): 1 1 1 1 0 A x B y C z D+ + + =

(b ): 2 2 2 2 0 A x B y C z D+ + + = Goc gia (a ), (b ) banghoac buvi goc gia hai VTPT 1 2n n,

r r .

( ) 1 2 1 2 1 2 1 22 2 2 2 2 21 2 1 1 1 2 2 2

n n A A B B C C n n A B C A B C

.cos ( ),( )

. .a b + += =

+ + + +

r r r r

Chu y:

( )0 00 90( ),( )a b . 1 2 1 2 1 2 0 A A B B C C ( ) ( )a b ^ + + =

Bai 1. Tnh goc gia hai mat phang:a) 1 0

5 0 x y z x y z

+ - + = - + - =b) 2 2 1 0

2 2 5 0 x y z x y z

+ - + = + + - =c) 2 4 5 0

4 2 1 0 x y z x y z

- + + = + - - =

d)4 4 2 7 02 4 5 0 x y z x z

+ - + = + - = e)

2 2 3 02 2 12 0

x y z y z

- - + = + + = f) 3 3 3 2 04 2 4 9 0 x y z x y z

- + + = + + - =

Bai 2. Tmme goc gia hai mat phang sau banga cho trc:

a)0

2 1 3 2 3 01 4 5 0

90

m x my zmx m y z( )

( )a

- - + + = + - + - = =

b)0

2 12 07 0

45

mx y mz x my za

+ + - = + + + = =

c)0

2 2 5 03 2 3 0

90

m x my mzmx m y z( )

( )a

+ + - + = + - + - = =

d)0

3 02 1 1 1 6 0

30

mx y mzm x m y m z( ) ( ) ( )

a

- + + = + + - + - - = =

Bai 3. Cho t dien OABC co cac canh OA, OB, OC vuong goc vi nhau tng b a ,, lan lt la cac goc hp bi cac mat phang (OAB), (OBC), (OCA)(ABC). Bang phng phap toa o, chng minh rang:a) Tam giac ABC co ba goc nhon b) 1coscoscos 222 =++ g b a


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VAN E 5: V tr tng oi gia mat phang va mat cau.Phng trnh mat phang tiep xuc vi mat cau

Cho mat phang (a ): 0 Ax By Cz D+ + + = va mat cau (S): 2 2 2 2 x a y b z c R( ) ( ) ( )- + - + - = (a ) va (S) khong co iem chung d I R( ,( ))a > (a ) tiep xuc vi (S) d I R( ,( ))a = (a ) la tiep dien

e tm toa o tiep iem ta co the thc hien nh sau: Viet phng trnh ng thang d i qua tam I cua (S) va vuong goc vi (a ). Tm toa o giao iem H cua d va (a ). H la tiep iem cua (S) vi (a ).

(a ) cat (S) theo mot ng tron d I R( ,( ))a < e xac nh tam H va ban knh r cua ng tron giao tuyen ta co the thc hien nh sa Viet phng trnh ng thang d i qua tam I cua (S) va vuong goc vi (a ). Tm toa o giao iem H cua d va (a ). H la tam cua ng tron giao tuyen cua (S) vi (a ).

Ban knh r cua ng tron giao tuyen: 2 2r R IH = -

Bai 1. Xet v tr tng oi gia mat phang (P) va mat cau (S):a) 2 2 2

2 2 1 06 2 4 5 0

P x y zS x y z x y z

( ) :( ) :

+ + - = + + - - + + =

b) 2 2 22 3 6 9 0

1 3 2 16P x y zS x y z

( ) :( ) : ( ) ( ) ( )

- + - = - + - + + =

c) 2 2 22 11 0

2 4 2 2 0P x y zS x y z x y z

( ) :( ) :

+ - - = + + + - - + =

d) 2 2 22 2 5 0

6 4 8 13 0P x y zS x y z x y z

( ) :( ) :

- + + = + + - - - + =

e) P x y zS x y z x y z2 2 2( ) : 2 2 0( ) : 6 2 2 10 0 + + = + + - + - + = f)P zS x y z x y z2 2 2( ) : 3 0( ) : 6 2 16 22 0 - = + + - + - + =

Bai 2. Bien luan theom, v tr tng oi gia mat phang (P) va mat cau (S):a) 2 2 22 2 4 0 2 1 4 4 8 0P x y z S x y z m x my z m( ) : ; ( ) : ( )- - - = + + - - + + + = b) 2 2 2 24 2 4 5 0 1 2 3 1P x y z S x y z m( ) : ; ( ) : ( ) ( ) ( ) ( )- + - = - + + + - = - c) 2 2 2 23 2 6 7 0 2 1 1 2P x y z S x y z m( ) : ; ( ) : ( ) ( ) ( ) ( )+ - + = - + - + + = + d) 2 2 2 22 3 6 10 0 4 2 1 2 3 5 4 0P x y z S x y z mx m y z m m( ) : ; ( ) : ( )- + - = + + + - + - + + + - =

Bai 3. Viet phng trnh mat cau (S) co tam I va tiep xuc vi mat phang (P) ca) 3 5 2 2 3 1 0 I P x y z( ; ; ), ( ) :- - - - + = b) 1 4 7 6 6 7 42 0 I P x y z( ; ; ), ( ) : + - + = c) 1 1 2 2 2 3 0 I P x y z( ; ; ), ( ) : + + + = d) 2 1 1 2 2 5 0 I P x y z( ; ; ), ( ) :- + - + =

Bai 4. Viet phng trnh mat phang (P) tiep xuc vi mat cau (S) cho trc:a) S x y z2 2 2( ) : ( 3) ( 1) ( 2) 24- + - + + = tai 1 3 0 M ( ; ; )- b) S x y z x y z2 2 2( ) : 6 2 4 5 0+ + - - + + = tai 4 3 0 M ( ; ; )c) 2 2 21 3 2 49S x y z( ) : ( ) ( ) ( )- + + + - = tai 7 1 5 M ( ; ; )- d) 2 2 2 2 2 2 22 0S x y z x y z( ) : + + - - - - = va song song vi mat phang3 2 6 14 0 x y z- + + = .e) 2 2 2 6 4 2 11 0S x y z x y z( ) : + + - + + - = va song song vi mat phang4 3 17 0 x z+ - = .f) 2 2 2 2 4 4 0S x y z x y z( ) : + + - - + = va song song vi mat phang2 2 5 0 x y z+ + + = .g) 2 2 2 2 6 2 8 0S x y z x y z( ) : + + - + + + = va cha ng thang4 4 3 1 1d x t y t z t : , ,= + = + = +


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Trang 43

h) Tiep xuc vi mat cau ngoai tiep t dien ABCD tai A vi A(6; 2; 3), B1), D(4; 1; 0).i) Tiep xuc vi mat cau: 011326210222 =-++-++ z y x z y x va song song vi 2 ng

thang:15 1 13

2 3 2 x y zd : + - += =

-, 1

7 1 83 2 0

x y zd : + + -= =-

.

Bai tap on: Phng trnh mat phang

Bai 1. Cho t dien ABCD. Viet phng trnh cac mat cua t dien. Viet phng trnh mat phang cha mot canh va song song vi canh oi Viet phng trnh mat phang i qua mot nh va song song vi mat oi d Viet phng trnh mat phang i qua canh AB va vuong goc vi (BCD). Viet phng trnh mat phang trung trc cua cac canh t dien.

Tm toa o cac iem A, B, C, Dlan lt la cac iem oi xng vi cac iemD qua cac mat oi dien. Tnh khoang cach t mot nh cua t dien en mat oi dien. Viet phng trnh mat cau (S) ngoai tiep t dien ABCD. Xac nh tam cua (S). Viet phng trnh cac tiep dien cua (S) tai cac nh A, B, C, D cua t d Viet phng trnh cac tiep dien cua (S) song song vi cac mat cua t di

a) ( ) ( ) ( ) ( )5 1 3 1 6 2 5 0 4 4 0 6 A B C D; ; , ; ; , ; ; , ; ; b)( ) ( ) ( ) ( )1 1 0 0 2 1 1 0 2 1 1 1 A B C D; ; , ; ; , ; ; , ; ;c) ( ) ( ) ( ) ( )2 0 0 0 4 0 0 0 6 2 4 6 A B C D; ; , ; ; , ; ; , ; ; d) 2 3 1 4 1 2 6 3 7 5 4 8 A B C D( ; ; ), ( ; ; ), ( ; ; ), ( ; ; )- - -

e) 5 7 2 3 1 1 9 4 4 1 5 0 A B C D( ; ; ), ( ; ; ), ( ; ; ), ( ; ; )- - - f) 0 1 0 2 3 1 2 2 2 1 1 2 A B C D( ; ; ), ( ; ; ), ( ; ; ), ( ; ; )- - Bai 2. Cho hai mat phang (P), (Q) lan lt cat ba truc toa o tai cac iem: A(1;C(0; 0; 3) va E(2; 0; 0), F(0; 1; 0), G(0; 0; 1).a) Tm phng trnh tong quat cua (P) va (Q).b) Tnh o dai ng cao cua hnh chop O.ABC.c) Tnh goc gia hai mat phang (P), (Q).

Bai 3. Cho bon iem: A(1; 1; 1), B(3; 3; 1), C(3; 1; 3) va D(1; 3; 3).a) Chng minh ABCD la mot t dien eu.b) Chng minh t dien ABCD co cac cap canh oi oi mot vuong goc.c) Tm phng trnh tong quat cua cac mat phang (ABC), (ABD), (ACD), (BCd) Tnh goc gia cac cap mat phang: (ABC) va (ABD), (BCD) va (ACD).


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1. Phng trnh tham so cua ng thang Phng trnh tham so cua ng thangd i qua iem0 0 0 0 M x y z( ; ; ) va co VTCP

1 2 3a a a a( ; ; )= r :

1

2

3

o

o

o

x x a t d y y a t t R

z z a t ( ) : ( )

= + = + = +

Neu1 2 3 0a a a th 0 0 01 2 3

x x y y z zd

a a a( ) : - - -= = gl phng trnh chnh taccua d.

2. V tr tng oi gia hai ng thangCho hai ng thangd , d co phng trnh tham so lan lt la:

0 1

0 2

0 3

x x tad y y ta

z z ta: = + = + = +

va0 1

0 2

0 3

x x t ad y y t a

z z t a:

= + = + = +

d // d 0 1 0 10 2 0 2

0 3 0 3

a a cung phng x ta x t a

he y ta y t a an t t vo nghiem z ta z t a

,

( , )

+ = + + = + + = +

r r

0 0 0 0

a a cung phng M x y z d , ( ; ; )

r r

0 0

a a cung phnga M M khong cung phng,,

r r

uuuuuur r [ ]

0 0

0

0

a aa M M ,,

=

r r r

uuuuuur r r

d d 0 1 0 1

0 2 0 2

0 3 0 3

x ta x t ahe y ta y t a an t t co vo so nghiem

z ta z t a( , )

+ = + + = + + = +

0 0 0 0

a a cung phng M x y z d

,( ; ; )

r r 0 0a a M M oi mot cung phng, ,

uuuuuur r r

[ ] 0 0 0a a a M M , , = = uuuuuur r r r r

d, d cat nhau he0 1 0 1

0 2 0 2

0 3 0 3

x ta x t a y ta y t a z ta z t a

+ = + + = + + = +

(an t, t ) co ung mot nghiem

0 0

a a khong cung phnga a M M ong phang,, ,

r r uuuuuur r r

[ ][ ] 0 0

00

a aa a M M ,, .

=

r r r uuuuuur r r

d, d cheo nhau 0 1 0 10 2 0 2

0 3 0 3

a a khong cung phng x ta x t a

he y ta y t a an t t vo nghiem z ta z t a

,

( , )

+ = + + = + + = +

r r

0 0a a M M khong ong phang, , uuuuuur r r [ ] 0 0 0a a M M , .

uuuuuur r r d ^ d a a^ r r 0a a. = r r

IV. PHNG TRNH NG THANG


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3. V tr tng oi gia mot ng thang va mot mat phang

Cho mat phang (a): 0 Ax By Cz D+ + + = va ng thangd :0 1

0 2

0 3

x x ta y y ta z z ta

= + = + = +

Xet phng trnh: 0 1 0 2 0 3 0 A x ta B y ta C z ta D( ) ( ) ( )+ + + + + + = (ant ) (*) d // (a ) (*) vo nghiem d cat (a ) (*) co ung mot nghiem d (a ) (*) co vo so nghiem

4. V tr tng oi gia mot ng thang va mot mat cau

Cho ng thangd :0 1

0 2

0 3

x x ta y y ta z z ta

= + = + = +

(1) va mat cau(S): 2 2 2 2 x a y b z c R( ) ( ) ( )- + - + - = (2)

e xet VTT cuad va(S) ta thay (1) vao (2), c mot phng trnh (*). d va (S) khong co iem chung (*) vo nghiem d(I, d) > R d tiep xuc vi (S) (*) co ung mot nghiem d(I, d) = R d cat (S) tai hai iem phan biet (*) co hai nghiem phan biet d(I, d) < R

5. Khoang cach t mot iem en mot ng thang(chng trnh nang cao)Cho ng thangd i qua M 0 va co VTCPa r va iem M.

0 M M ad M d a

,( , )

=

uuuuur r

r

6. Khoang cach gia hai ng thang cheo nhau (chng trnh nang cao)Cho hai ng thang cheo nhaud 1 vad 2.

d 1 i qua iem M 1 va co VTCP1a r

, d 2 i qua iem M 2 va co VTCP2a r

1 2 1 2

1 21 2

a a M M d d d

a a

, .( , )

, =

uuuuuur r r

r r

Chu y:Khoang cach gia hai ng thang cheo nhau d 1 , d 2 bang khoang cach gia d 1 vimat phang (a ) cha d 2 va song song vi d 1.

7. Khoang cach gia mot ng thang va mot mat phang song song Khoang cach gia ng thang d vi mat phang (a ) song song vi no bang khoang cach tmot iem M bat k tren d en mat phang (a ).

8. Goc gia hai ng thangCho hai ng thang d 1 , d 2 lan lt co cac VTCP1 2a a, r r .Goc gia d 1 , d 2 banghoac buvi goc gia1 2a a,

r r .

( ) 1 21 21 2

a aa a

a a.cos ,.

= r r

r r r r

9. Goc gia mot ng thang va mot mat phangCho ng thang d co VTCP 1 2 3a a a a( ; ; )=

r va mat phang (a ) co VTPT n A B C ( ; ; )= r .Goc gia ng thang d va mat phang (a ) bang goc gia ng thang d vi hnh chieu d cuano tren (a ).

( ) 1 2 32 2 2 2 2 2

1 2 3

Aa Ba Cad

A B C a a asin ,( )

.a

+ +=

+ + + +


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VAN E 1: Lap phng trnh ng thang e lap phng trnh ng thang d ta can xac nh mot iemthuoc d va mot VTCPcua no.

Dang 1: d i qua iem0 0 0 0 M x y z( ; ; ) va co VTCP1 2 3a a a a( ; ; )= r :

1

2

3

o

o

o

x x a t d y y a t t R

z z a t ( ) : ( )

= + = + = +

Dang 2: d i qua hai iem A, B: Mot VTCP cua d la AB

uuur.

Dang 3: d i qua iem0 0 0 0 M x y z( ; ; ) va song song vi ng thangDcho trc:V d // D nen VTCP cuaD cung la VTCP cua d.

Dang 4: d i qua iem0 0 0 0 M x y z( ; ; ) va vuong goc vi mat phang (P) cho trc:V d (P) nen VTPT cua (P) cung la VTCP cua d.

Dang 5: d la giao tuyen cua hai mat phang (P), (Q): Cach 1: Tm mot iem va mot VTCP.

Tm toa o mot iem A d: bang cach giai he phng trnhPQ( )( )

(vi viec chon gia tr

cho mot an) Tm mot VTCP cua d: P Qa n n, =

r r r

Cach 2: Tm hai iem A, B thuoc d, roi viet phng trnh ng thang i qua hai iem o.Dang 6: d i qua iem0 0 0 0 M x y z( ; ; ) va vuong goc vi hai ng thangd 1 , d 2:

V d d 1 , d d 2 nen mot VTCP cua d la:1 2d d

a a a, = r r r

Dang 7: d i qua iem0 0 0 0 M x y z( ; ; ) , vuong goc va cat ng thangD. Cach 1: Goi H la hnh chieu vuong goc cua M 0 tren ng thangD.

0

H M H u D ^ V uuuuur r

Khi o ng thang d la ng thang i qua M 0 , H. Cach 2: Goi (P) la mat phang i qua A va vuong goc vi d; (Q) la mat phang i qua Acha d. Khi o d = (P) (Q)

Dang 8: d i qua iem0 0 0 0 M x y z( ; ; ) va cat hai ng thangd 1 , d 2: Cach 1: Goi M 1 d 1 , M 2 d 2. T ieu kien M, M 1 , M 2 thang hang ta tm c M 1 , M 2. To suy ra phng trnh ng thang d. Cach 2: Goi (P) = 0 1 M d ( , ) , (Q) = 0 2 M d ( , ). Khi o d = (P) (Q). Do o, mot VTCP cua d co the chon la P Qa n n, =

r r r .

Dang 9: d nam trong mat phang (P) va cat ca hai ng thangd 1 , d 2:Tm cac giao iem A = d 1 (P), B = d 2 (P). Khi o d chnh la ng thang AB.

Dang 10: d song song viDva cat ca hai ng thangd 1 , d 2:Viet phng trnh mat phang (P) chaD va d 1 , mat phang (Q) chaD va d 2.

Khi o d = (P) (Q). Dang 11:

d

la ng vuong goc chung cua hai ng thangd

1 , d

2 cheo nhau:

Cach 1: Goi M d 1 , N d 2. T ieu kien 12

MN d MN d ^ ^

, ta tm c M, N.

Khi o, d la ng thang MN.


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Cach 2: V d d 1 va d d 2 nen mot VTCP cua d co the la:

1 2d d a a a, = r r r .

Lap phng trnh mat phang (P) cha d va d 1 , bang cach:+ Lay mot iem A tren d 1.+ Mot VTPT cua (P) co the la:

1P d n a a, =

r r r .

Tng t lap phng trnh mat phang (Q) cha d va d 2.Khi o d = (P) (Q).

Dang 12: d la hnh chieu cua ng thangDlen mat phang (P): Lap phng trnh mat phang (Q) chaD va vuong goc vi mat phang (P) bang cach:

Lay M D. V (Q) chaD va vuong goc vi (P) nenQ Pn a n,D =

r r r .

Khi o d = (P) (Q).Dang 13: d i qua iem M, vuong goc vid 1 va catd 2:

Cach 1: Goi N la giao iem cua d va d 2. T ieu kien MN ^ d 1 , ta tm c N.Khi o, d la ng thang MN. Cach 2:

Viet phng trnh mat phang (P) qua M va vuong goc vi d 1. Viet phng trnh mat phang (Q) cha M va d 2.

Khi o d = (P) (Q).

Bai 1. Viet phng trnh tham so cua ng thang i qua iem M va co VTCPa r cho trc:a) M a(1;2; 3), ( 1;3;5)- = - r b) M a(0; 2;5), (0;1;4)- = r c) M a(1;3; 1), (1;2; 1)- = - r d) M a(3; 1; 3), (1; 2;0)- - = - r e) M a(3; 2;5), ( 2;0;4)- = - r f) M a(4;3; 2), ( 3;0;0)- = - r

Bai 2. Viet phng trnh tham so cua ng thang i qua hai iem A, B cho trca) ( ) ( )2 3 1 1 2 4 A , B; ; ; ;- b) ( ) ( )1 1 0 0 1 2 A , B; ; ; ;- c) ( ) ( )3 1 5 2 1 1 A , B; ; ; ;- - d) ( ) ( )2 1 0 0 1 2 A , B; ; ; ; e) ( ) ( )1 2 7 1 2 4 A , B; ; ; ;- f) ( ) ( )2 1 3 4 2 2 A , B; ; ; ;- -

Bai 3. Viet phng trnh tham so cua ng thang i qua iem A va song song vD cho trc:a) ( )3 2 4 A , Ox; ; D- b) ( )2 5 3 5 3 2 2 1 2 A i qua M N ; ; , ( ; ; ), ( ; ; )D- -

c)

2 3

2 5 3 3 45 2

x t

A y t z t ( ; ; ), :D = -

- = + = - d)2 5 2

4 2 2 4 2 3 x y z

A( ; ; ), :D+ - -

- = =

e)3 4

1 3 2 2 23 1

x t A y t

z t ( ; ; ), :D

= +- = - = -

f) 3 1 25 2 32 3 4

x y z A( ; ; ), :D + - +- = =

Bai 4. Viet phng trnh tham so cua ng thang i qua iem A va vuong go(P) cho trc:a) ( )2 4 3 2 3 6 19 0 A , (P) x y z; ; :- - + + = b) ( )1 1 0 A , P cac mp toa o; ; ( ) :- c) ( )3 2 1 2 5 4 0 A P x y; ; , ( ) : - + = d) 2 3 6 2 3 6 19 0 A P x y z( ; ; ), ( ) :- - + + =

Bai 5. Viet phng trnh tham so cua ng thang la giao tuyen cua hai matcho trc:a) 6 2 2 3 0

3 5 2 1 0P x y zQ x y z

( ) :( ) :

+ + + = - - - =b) 2 3 3 4 0

2 3 0P x y zQ x y z

( ) :( ) :

- + - = + - + =c) 3 3 4 7 0

6 2 6 0P x y zQ x y z

( ) :( ) :

+ - + = + + - =


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d) 2 3 01 0

P x y zQ x y z

( ) :( ) :

+ - + = + + - =e) 1 0

2 0P x zQ y

( ) :( ) :

+ - = - =f) 2 1 0

1 0P x y zQ x z

( ) :( ) :

+ + - = + - =

Bai 6. Viet phng trnh tham so cua ng thang i qua iem A va vuong gothangd 1 , d 2 cho trc:

a) 1 21 2 1

1 0 5 3 2 21 1 3

x t x t

A d y t d y t z t z t ( ; ; ), : , :

= + = - = - = + = + = - b)

1 2

1 1 3

2 1 1 2 23 3

x t x t

A d y t d y t z z t ( ; ; ), : , :

= + = + - = - + = - + = = +

c) 1 21 1

1 2 3 2 2 23 3 3

x t x A d y t d y t

z t z t ( ; ; ), : , :

= - = - = - - = - + = - = +

d) 1 27 3 1

4 1 4 4 2 9 24 3 12

x t x t A d y t d y t

z t z t ( ; ; ), : , :

= - + = + = - = - + = + = - -

e) 1 21 3 2

2 1 3 1 3 42 2 2


z t z t ( ; ; ), : , :

= + = - - = + = - + = - + = -

f) 1 23 1 4 1 1 22 0


z t z( ; ; ), : , :

= = - = - = - = - =

Bai 7. Viet phng trnh tham so cua ng thang i qua iem A, vuong goc

thangDcho trc:a) 1 2 2 1

2

x t A y t

z t ( ; ; ), :D

=- = - =

b)3 2

4 2 4 11 4

x t A d y t

z t ( ; ; ), :

= - +- - = - = - +

c)1 3

2 1 3 12 2

x t A y t

z t ( ; ; ), :D

= +- - = + = - +

d) 3 1 4 12

x t A y t

z t ( ; ; ), :D

=- = - = -

e)1

1 2 3 2 23 3

x t A y t

z t ( ; ; ), :D

= -- = - - = -

f)1

2 1 1 23

x t A y t

z( ; ; ), :D

= +- = - + =

Bai 8. Viet phng trnh tham so cua ng thang i qua iem A va cat ca haid 1 ,d 2 cho trc:

a) 1 21 2 1

1 0 5 3 2 21 1 3


z t z t ( ; ; ), : , :

= + = - = - = + = + = -

b) 1 21 1 3

2 1 1 2 23 3


z z t ( ; ; ), : , :

= + = + - = - + = - + = = +

c) 1 21 3 2 2

4 5 3 3 2 1 32 1 5


z t z t ( ; ; ), : , :

= - + = + - - = - - = - + = - = -

d) 1 21 3

2 1 1 2 43 5 2


z t z t ( ; ; ), : , :

= + = - - = - + = = - + =

e) 1 22 4 3

2 3 1 1 2 11 3 2 3

x t x t

A d y t d y t z t z t ( ; ; ), : , : = + = - +

- = - = + = + = - + f)1 2

3 3 3 2

3 2 5 1 4 12 2 2 3

x t x t

A d y t d y t z t z t ( ; ; ), : , : = - + = +

- = + = - = + = - Bai 9. Viet phng trnh tham so cua ng thang nam trong mat phang(P) va cat ca hai

ng thangd 1 , d 2 cho trc:

a)1 2

2 0214 2

1 1 4 1

P y z x t x y zd d y t z

( ) :

: , :

+ = = -- = = = + - =

b)1 2

6 2 2 3 01 2 13 2 21 1 3

P x y z x t x t

d y t d y t z t z t

( ) :

: , :

+ + + = = + = - = - = + = + = -

c) 1 2

2 3 3 4 0

7 3 14 2 9 24 3 12

P x y z

x t x t d y t d y t z t z t

( ) :

: , :

- + - = = - + = + = - = - + = + = - -

d) 1 2

3 3 4 7 0

1 12 2 23 3 3

P x y z

x t xd y t d y t z t z t

( ) :

: , :

+ - + = = - = = - - = - + = - = +


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Bai 10. Viet phng trnh tham so cua ng thang song song vi ng thangDva cat ca haing thangd 1 , d 2 cho trc:

a) 1

2

1 12 1 2

1 1

1 2 12 1 33 2 1

x y z

x y zd

x y zd

:

:

:

D - -= = - + -= =

- - + + = =

b) 1

2

1 53 1 1

1 2 2

1 4 34 75 9 1

x y z

x y zd

x y zd

:

:

:

D - -= = - - + -= = + + = =

c) 1

2

1 2 2:

1 4 31 2 2

:1 4 3

4 7:

5 9 1

- + - D = = - + - = =

+ + = =

x y z

x y z d

x y z d

d) 1

2

1 3 23 2 1

2 2 13 4 1

7 3 91 2 1

x y z

x y zd

x y zd

:

:

:

D + + -= = - - - + -= =

- - - = = -

Bai 11. Viet phng trnh tham so cua ng thang vuong goc chung cua hai nhaud 1 , d 2 cho trc:

a) 1 23 2 2 31 4 4

2 4 1 2

x t x t d y t d y t

z t z t : , : = - = + = + = - = - + = -

b) 1 21 2 2 3

3 1 22 3 4 4

x t x t d y t d y t

z t z t : , : = + = - + = - + = + = + = - +

c) 1 22 2 11 33 1 2

x t x t d y t d y t

z t z t : , : = + = + = + = + = - = +

d) 1 22 3 1 2

3 1 21 2 2

x t x t d y t d y t

z t z t : , : = + = - + = - - = - = + = +

Bai 12. Viet phng trnh tham so cua ng thangd la hnh chieu cua ng thangDtren matphang (P) cho trc:

a) 2 3 12 1 32 2 3 0

x y z

P x y z

:( ) :D + - - = = -

- + + =b) 3 2 21 2 3

3 4 2 3 0

x y z

P x y z

:( ) :D - - + = = -

+ - + =

c)1 1 3

1 2 22 2 3 0

x y z

P x y z

:( ) :D

+ - - = = - - + - =

d)1

2 1 11 0

x y z

P x y z

:( ) :D

- = = - + - + =

e)2 2 1

3 4 12 3 4 0

x y z

P x y z

:( ) :D

- + - = = + + + =

f)1 2

1 2 12 3 5 0

x y z

P x y z

:( ) :D

- - = = - - - - + =

g) 5 4 2 5 02 2 02 1 0

x y z x zP x y z:

( ) :D - - - = + - =

- + - =h) 1 02 2 0

2 1 0

x y z x zP x y z:

( ) :D - - - = + - =

+ - - =

Bai 13. Viet phng trnh tham so cua ng thang i qua iem A, vuong goc vd 1 va cat ng thangd 2 cho trc:

a) 1 211 2

0 1 13 1 1 1

x x y z A d d y t z t

( ; ; ), : , : = -- -= = = = +

b)1 2

21 11 1 1 1 2

2 1 1 1

x x y z A d d y t z t

( ; ; ), : , : =- += = = +

- = - -

c) 1 21 4 1 1 3

1 2 36 2 3 3 2 5

x y z x y z A d d ( ; ; ), : , :+ - - + -- - = = = =- - -


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Bai 14. Cho t dien ABCD co A(1; 0; 0), B(0; 1; 0), C(0; 0; 1); D(1; 1; 1). Viet phnso cua cac ng thang sau:a) Cha cac canh cua t dien t dien ABCD.b) ng thang qua C va vuong goc vi mp(ABD).c) ng thang qua A va qua trong tam cua tam giac BCD.

Bai 15. Cho tam giac ABC co A(1; 2; 5) va hai trung tuyen: 1 32 623 :)( 1 -=-=-- z y xd ,

12

42

14

:)( 2-=

--=- z y xd . Viet phng trnh tham so cua cac ng thang sau:

a) Cha cac canh cua tam giac ABC.b) ng phan giac trong cua goc A.

Bai 16. Cho tam giac ABC co3 1 1 1 2 7 5 14 3 A B C ( ; ; ), ( ; ; ), ( ; ; )- - - - - . Viet phng trnh tham so cuacac ng thang sau:a) Trung tuyen AM. b) ng cao BH.c) ng phan giac trong BK. d) ng trung trc cua BC trongDABC.

Bai 17. Cho bon iem1 2 1 3 4 1 1 4 1 3 2 1S A B C ( ; ; ), ( ; ; ), ( ; ; ), ( ; ; )- - .a) Chng minh S.ABC la mot hnh chop.b) Viet phng trnh tham so cua cac ng thang cha cac canh cua hnh c) Viet phng trnh ng vuong goc chung cua SA va BC.

Bai 18. Cho bon iem1 2 3 2 2 3 1 1 3 1 2 5S A B C ( ; ; ), ( ; ; ), ( ; ; ), ( ; ; )- - - - .a) Chng minh S.ABC la mot t dien.b) Viet phng trnh cac hnh chieu cua SA, SB tren mat phang (ABC).


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VAN E 2: V tr tng oi gia hai ng thang e xet VTT gia hai ng thang, ta co the s dung mot trong cac phng phap sau:

Phng phap hnh hoc: Da vao moi quan he gia cac VTCP va cac iem thuoc cac thang. Phng phap ai so: Da vao so nghiem cua he phng trnh cac ng thang.

Bai 1. Xet v tr tng oi gia hai ng thangd 1 , d 2 cho trc:a) {1 2

1 2 41 2 3

2 1 3 x y zd d x t y t z t : ; : ; ;- + -= = = - + = - = - +-

b) { {1 25 2 1 5 3 2 3 1d x t y t z t d x t y t z t : ; ; ; : '; '; '= + = - = - = + = - - = - c) { {1 22 2 1 1 1 1 3d x t y t z d x y t z t : ; ; ; : ; ;= + = - + = = = + = -

d) 1 21 2 3 7 6 5

9 6 3 6 4 2 x y z x y zd d : ; :- - - - - -= = = =

e) 1 21 5 3 6 1 3

2 1 4 3 2 1 x y z x y z

d d : ; :- + - - + +

= = = = f) 1 2

2 1 7 24 6 8 6 9 12

x y z x y zd d : ; :- + - -= = = =- - -

g) 1 22 2 2 0 2 2 0

2 2 4 0 2 1 0 x y z x y zd d x y z x y z

: ; : - + - = + - + = + - + = - + - =

h) {1 22 3 3 9 09 5 3

2 3 0 x y zd x t y t z t d x y z

: ; ; ; : - - - == = = - - + + =

Bai 2. Chng to rang cac cap ng thang sau ay cheo nhau. Viet phng trgoc chung cua chung:a) { {1 21 2 3 2 3 2 1 3 2d x t y t z t d x t y t z t : ; ; ; : '; '; '= - = + = - - = = + = - b) { {1 21 2 2 2 2 5 3 4d x t y t z t d x t y t z: ; ; ; : '; ';= + = - = - = = - = c) { {1 23 2 1 4 4 2 2 3 4 1 2d x t y t z t d x t y t z t : ; ; ; : '; '; '= - = + = - = + = - = -

d) 1 22 1 1 1

3 2 2 1 2 4 x y z x y zd d : ; :- + - += = = =

-

e) 1 27 3 9 3 1 1

1 2 1 7 2 3 x y z x y zd d : ; :- - - - - -= = = =

- -

f) 1 22 1 3 3 1 1

2 1 2 2 2 1

x y z x y zd d : ; :

- - - - + -= = = =- -

g) 1 22 2 2 0 2 2 0

2 2 4 0 2 1 0 x y z x y zd d x y z x y z

: ; : - + - = + - + = + - + = - + - =

Bai 3. Tm giao iem cua hai ng thangd 1 va d 2:a) { {1 23 1 2 3 1 2 4d x t y t z t d x t y t z t : ; ; ; : '; '; '= = - = + = + = = +

b) {1 23 0 1 2 3

2 1 0 x y zd d x t y t z t x y

: ; : ; ; + + + = = + = - + = - - + =

c) 1 22 4 0 2 0

2 6 0 2 7 0 x y z x zd d x y z y z

: ; : - - - = - - = + + + = + + =

d) 1 22 1 0 3 3 0

1 0 2 1 0 x y x y zd d x y z x y

: ; : + + = + - + = - + - = - + =


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Bai 4. Tmm e hai ng thangd 1 va d 2 cat nhau. Khi o tm toa o giao iem cua chua) { {1 21 1 2 1 2 2 3d x mt y t z t d x t y t z t : ; ; ; : '; '; '= + = = - + = - = + = - b) { {1 21 3 2 2 1 2 3d x t y t z m t d x t y t z t : ; ; ; : '; '; '= - = + = + = + = + = -

c) 1 22 4 0 2 3 0

3 0 2 6 0 x y z x y mzd d x y x y z

: ; : + - - = + + - = + - = + + - =

VAN E 3: V tr tng oi gia ng thang va mat phang e xet VTT gia ng thang va mat phang, ta co the s dung mot trong cac phng pha

Phng phap hnh hoc: Da vao moi quan he gia VTCP cua ng thang va VTPT cumat phang. Phng phap ai so: Da vao so nghiem cua he phng trnh ng thang va mat phan

Bai 1. Xet v tr tng oi gia ng thangd vamat phang(P). Tm giao iem (neu co) cuachung:a) { 2 1 3 10 0d x t y t z t P x y z: ; ; ; ( ) := = - = + + + - = b) { 3 2 1 4 4 5 4 3 6 5 0d x t y t z t P x y z: ; ; ; ( ) := - = - = - - - - =

c) 12 9 1 3 5 2 04 3 1

x y zd P x y z: ; ( ) :- - -= = + - - =

d) 11 3 3 3 2 5 02 4 3

x y zd P x y z: ; ( ) :+ -= = - + - =

e) 13 1 4 2 4 1 08 2 3

x y zd P x y z: ; ( ) :- - -= = + - + =

f) 3 5 7 16 0 5 4 02 6 0 x y zd P x z x y z

: ; ( ) : + + + = - - = - + - =

g) 2 3 6 10 0 4 17 05 0

x y zd P y z x y z: ; ( ) : + + - = + + = + + + =

Bai 2. Cho ng thangd vamat phang(P). Tmm, n e:i)d cat(P). ii)d // (P). iii)d (P). iv)d (P).

a) 1 2 3 3 2 5 02 1 2

x y zd P x y zm m

: ; ( ) :- + += = + - - =-

b) 1 3 1 3 2 5 02 2 x y zd P x y zm m: ; ( ) :+ - -= = + + - =-

c) 3 2 3 0 2 3 2 04 3 4 2 0 x y zd P x y m z x y z

: ; ( ) : ( ) - + + = - + + - = - + + =

d) { 3 4 1 4 3 1 2 4 9 0d x t y t z t P m x y z n: ; ; ; ( ) : ( )= + = - = - + - + - + - = e) { 3 2 5 3 2 2 2 3 3 5 0d x t y t z t P m x n y z: ; ; ; ( ) : ( ) ( )= + = - = - + + + + - =

Bai 3. Cho ng thangd vamat phang(P). Tmm, n e:a) { 2 3d x m t y t z t : ; ;= + = - = ct 2 5 0P x y z( ) : - + - = tai iem co tung o bang 3.

b)2 3 02 5 0

x yd y z:

- - = + + = cat 2 2 2 0P x y z m( ) : + + - = tai iem co cao o bang 1.

c) 2 3 03 2 7 0 x yd x z

: + - = - - =ct 0P x y z m( ) : + + + =


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VAN E 4: V tr tng oi gia ng thang va mat cau e xet VTT gia ng thang va mat cau ta co the s dung cac phng phap sau:

Phng phap hnh hoc: Da vao khoang cach t tam mat cau en ng thang va ban Phng phap ai so: Da vao so nghiem cua he phng trnh ng thang va mat cau.

Bai 1. Xet v tr tng oi gia ng thangd vamat cau(S). Tm giao iem (neu co) cuachung:a) 2 2 21 2 2 4 1 0

2 1 1 x y zd S x y z x z: ; ( ) :- -= = + + - + + =

-

b) 2 2 22 1 0 1 2 162 3 0

x y zd S x y z x z: ; ( ) : ( ) ( ) + - - = - + - + = - - =

c) 2 2 22 1 0 2 2 14 02 0

x y zd S x y z x y x y: ; ( ) : - - - = + + - + - = + + =

d) 2 2 22 1 0 4 2 10 8 02 0

x y zd S x y z x y z x y

: ; ( ) : - - - = + + + - - - =+ + =

e) { 2 2 22 3 2 4 2 2 0d x t y t z t S x y z x y z: ; ; ; ( ) := - - = = - + + - - + - = f) { 2 2 21 2 2 3 2 4 6 2 0d x t y t z t S x y z x y z: ; ; ; ( ) := - = + = + + + - - + - = g) { 2 2 21 2 4 2 4 6 2 0d x t y t z S x y z x y z: ; ; ; ( ) := - = - = + + - - + - =

Bai 2. Bien luan theom, v tr tng oi gia ng thangd vamat cau(S):a) 2 2 22 0 1 2 1 8

2 0 x y z md S x y z x y

: ; ( ) : ( ) ( ) ( ) - - + = - + - + + = + + =

b) { 2 2 21 2 2 4 1 0d x t y m t z t S x y z x z: ; ; ; ( ) := - = + = + + + - + + =

c) 2 2 22 3 0 2 2 4 02 1 0 x yd S x y z x y z m x z

: ; ( ) : - - = + + + - + + = + - =

Bai 3. Viet phng trnh mat cau(S)co tam I va tiep xuc vi ng thangd :a) {1 2 1 1 4 3 2 4 2 I d x t y t z t ( ; ; ); : ; ;- = + = - = - b) {1 2 1 1 2 2 I d x t y z t ( ; ; ); : ; ;- = - = =

c) 2 1 14 2 12 1 2

x y z I d ( ; ; ); : - + -- = =

d) 1 21 2 12 1 3

x y z I d ( ; ; ); : - -- = =-

e) 2 1 01 2 11 0

x y I d z( ; ; ); : - - =- - =

Bai 4. Cho mat cau(S) co tam I(2; 1; 3) va ban knh R = 3. Viet phng trnh tiep tud cua(S), biet:a) d i qua A(0; 0; 5) (S) va co VTCP1 2 2a ( ; ; )= r .b)d i qua A(0; 0; 5) (S)va vuong goc vi mat phang:3 2 2 3 0 x y z( ) : .a - + + =

Bai 5. Cho t dien ABCD. Viet phng trnh mat cau tiep xuc vi cac canh cua) A(1; 1; 1), B(3; 3; 1), C(3; 1; 3), D(1; 3; 3).

b) A(1; 0; 2), B(2; 1; 1), C(0; 2; 1), D(1; 3; 0).c) A(3; 2; 1), B(1; 2; 1), C(2; 2; 2), D(1; 1; 1).d) A(1; 0; 11), B(0; 1; 10), C(1; 1; 8), D(3; 1; 2).


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VAN E 5: Khoang cach1. Khoang cach t iem M en ng thang d

Cach 1: Cho ng thang d i qua M 0 va co VTCPa r .

0 M M ad M d a

,( , )

=

uuuuur r

r

Cach 2: Tm hnh chieu vuong goc H cua M tren ng thang d. d(M,d) = MH.

Cach 3: Goi N(x; y; z) d. Tnh MN 2 theo t (t tham so trong phng trnh ng thang d). Tm t e MN 2 nho nhat. Khi o N H. Do o d(M,d) = MH.

2. Khoang cach gia hai ng thang cheo nhau Cho hai ng thang cheo nhau d 1 va d 2.

d 1 i qua iem M 1 va co VTCP1a r , d 2 i qua iem M 2 va co VTCP2a

r

1 2 1 21 21 2

a a M M d d d a a

, .( , ) ,

=

uuuuuur r r

r r

Chu y:Khoang cach gia hai ng thang cheo nhau d 1 , d 2 bang khoang cach gia d 1 vimat phang (a ) cha d 2 va song song vi d 1.

3. Khoang cach gia hai ng thang song songbang khoang cach t mot iem thuoc ngthang nay en ng thang kia.

4. Khoang cach gia mot ng thang va mot mat phang song song Khoang cach gia ng thang d vi mat phang (a ) song song vi no bang khoang cach tmot iem M bat k tren d en mat phang (a ).

Bai 1. Tnh khoang cach t iem A en ng thangd :

a)1 4

2 3 1 2 24 1

x t A d y t

z t ( ; ; ), :

= - = + = -

b)2 2

1 2 6 13

x t A d y t

z t ( ; ; ), :

= +- = - = -

c) 2 11 0 01 2 1

x y z A d ( ; ; ), : - -= = d) 2 1 12 3 11 2 2

x y z A d ( ; ; ), : + - += =-

e) 2 1 11 1 11 2 2

x y z A d ( ; ; ), : + - +- = =-

f) 2 1 02 3 13 2 2 0

x y z A d x y z( ; ; ), : + - - =- + + + =

Bai 2. Chng minh hai ng thangd 1 , d 2 cheo nhau. Tnh khoang cach gia chung:a) { {1 21 2 3 2 3 2 1 3 2d x t y t z t d x t y t z t : ; ; ; : '; '; '= - = + = - - = = + = -

b) { {1 21 2 2 2 2 5 3 4d x t y t z t d x t y t z: ; ; ; : '; ';= + = - = - = = - = c) { {1 23 2 1 4 4 2 2 3 4 1 2d x t y t z t d x t y t z t : ; ; ; : '; '; '= - = + = - = + = - = -

d) 1 22 1 1 1

3 2 2 1 2 4 x y z x y zd d : ; :- + - += = = =

-

e) 1 27 3 9 3 1 1

1 2 1 7 2 3 x y z x y zd d : ; :- - - - - -= = = =

- -

f) 1 22 1 3 3 1 1

2 1 2 2 2 1 x y z x y z

d d : ; :- - - - + -

= = = =- - g) 1 2

2 2 2 0 2 2 02 2 4 0 2 1 0 x y z x y zd d x y z x y z

: ; : - + - = + - + = + - + = - + - =


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Bai 3. Chng minh hai ng thangd 1 , d 2 song song vi nhau. Tnh khoang cach gia chuna) { {1 23 2 4 3 2 4 4 5 6 3 2d x t y t z t d x t y t z t : , , ; : , ,= + = + = + = + = + = +

b) 1 21 2 3 2 3 1

2 6 8 3 9 12 x y z x y zd d : ; :- + - + - += = = =

- - -

c) 1 13 1 2 1 5 1

2 1 3 4 2 6 x y z x y z

d d : ; :- - + + + -

= = = = d) 1 2

7 5 92 2 10 022 0 3 1 4

x y z x y zd d x y z: ; : + - - + - - = = = - - - = -

Bai 4. Chng minh ng thangd song song vi mat phang (P). Tnh khoang cach gia a) { 3 2 1 4 4 5 4 3 6 5 0d x t y t z t P x y z: ; ; ; ( ) := - = - = - - - - = b) { 1 2 2 2 8 0d x t y t z t P x z: ; ; ; ( ) := - = = + + + =

c) 2 1 0 2 2 4 5 02 3 0 x y zd P x y z x y z

: ; ( ) : - + + = - + + = + - - =

d) 3 2 3 0 2 2 2 04 3 4 2 0 x y zd P x y z x y z: ; ( ) : - + + = - - - = - + + =

VAN E 6: Goc1. Goc gia hai ng thang

Cho hai ng thang d 1 , d 2 lan lt co cac VTCP1 2a a, r r .

Goc gia d 1 , d 2 banghoac buvi goc gia1 2a a, r r .

( )1 2

1 21 2

a aa a a a

.cos , .=

r r r r

r r

2. Goc gia mot ng thang va mot mat phangCho ng thang d co VTCP 1 2 3a a a a( ; ; )=

r va mat phang (a ) co VTPT n A B C ( ; ; )= r .Goc gia ng thang d va mat phang (a ) bang goc gia ng thang d vi hnh chieu d cuano tren (a ).

( ) 1 2 32 2 2 2 2 2

1 2 3

Aa Ba Cad

A B C a a asin ,( )

.a

+ +=

+ + + +

Bai 1. Tnh goc gia hai ng thang:a) { {1 21 2 1 3 4 2 1 3 4 2d x t y t z t d x t y t z t : , , ; : , ,= + = + = + = = + = +

b) 1 21 2 4 2 3 4

2 1 2 3 6 2 x y z x y zd d : ; :- + - + - += = = =

- -

c) {1 22 3 3 9 0 9 5 3

2 3 0 x y zd d x t y t z t x y z

: ; : ; ; - - - = = = = + - + + =

d) {1 22 2 0 2 3 1 4

7 3 17 0 x zd d x t y z t x y z

: ; : ; ; - + = = + = = - + - =

e) 1 21 2 2 2 1 02 3 2 03 1 4 x y z x y zd d x z: ; :- + + + - - == = + - =

f) 13 1 2

2 1 1 x y zd : + - -= = vad 2 la cac truc toa o.


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g) 1 24 0 2 3 1 0

2 1 0 0 x y z x y zd d x y z x y z

: ; : - + - = - + - = - + + = + + =

h) 1 22 3 4 0 2 3 03 2 7 0 4 3 7 0 x y z x y zd d x y z x y z

: ; : - + - = + - + = + - + = - + + =

Bai 2. Chng minh hai ng thang sau vuong goc vi nhau:a) 1 27 2 15 0 7 07 5 34 0 3 4 11 0

x z x y zd d y z x y: ; : - - = - - - = + + = - - =

b)Bai 3. Tmme goc gia hai ng thang sau banga :

a) { { 01 21 2 2 2 1 2 2 60d x t y t z t d x t y t z mt : ; ; ; : ; ; ;a= - + = - = + = + = + = + = .b)

Bai 4. Tnh goc gia ng thangd va mat phang(P)::

a)1 1 3

2 2 10 01 2 3 x y z

d P x y z: ; ( ) : - - +

= = =- .b) { 4 41 2 5 3 5 4 0d x y t z t P x z: ; ; ; ( ) := = + = + + + = c) 4 2 7 0 3 1 0

3 7 2 0 x y zd P x y z x y z

: ; ( ) : + - + = + + = + - =

d) 2 3 0 3 4 2 5 02 3 5 0 x y zd P x y z x y z

: ; ( ) : + - + = + = - + + =

Bai 5. Cho t dien ABCD co A(3; 2; 6), B(3; 1; 0), C(0; 7; 3), D(2; 1; 1).a) Chng minh cac cap canh oi cua t dien oi mot vuong goc vi nha

b) Tnh goc gia AD va mat phang (ABC).c) Tnh goc gia AB va trung tuyen AM cua tam giac ACD.d) Chng minh AB vuong goc vi mat phang (BCD). Tnh the tch cua t d

Bai 6. Cho t dien SABC co S(1; 2; 1), A(3; 2; 1), B(1; 3; 1), C(1; 2; 5).a) Viet phng trnh cua cac mat phang (ABC), (SAB), (SAC).b) Tnh goc tao bi SC va (ABC) va goc tao bi SC va AB.c) Tnh cac khoang cach t C en (SAB) va t B en (SAC).d) Tnh khoang cach t C en AB va khoang cach gia SA va BC.

Bai 7. Cho t dien SABC co S(1; 2; 3), A(2; 2; 3), B(1; 1; 3), C(1; 2; 5).a) Tm phng trnh cac hnh chieu cua SA, SB tren mat phang (ABC).b) Goi M, N, P lan lt la trung iem cua BC, CA, AB. Tnh goc tao bi SMtao bi SM va (ABC).c) Tnh cac khoang cach gia SM va NP, SP va MN.


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VAN E 7: Mot so van e khac1. Viet phng trnh mat phang

Dang 1:Mat phang (P) i qua iem A va ng thang d: Tren ng thang d lay hai iem B, C. Mot VTPT cua (P) la: n AB AC , =

uuur uuur r .

Dang 2:Mat phang (P) cha hai ng thang song song d 1 , d 2: Xac nh VTCPa r cua d 1 (hoac d 2). Tren d 1 lay iem A, tren d 2 lay iem B. Suy ra A, B (P). Mot VTPT cua (P) la: n a AB, =

uuur r r . Dang 3: M at phang (P) cha hai ng thang cat nhau d 1 , d 2:

Lay iem A d 1 (hoac A d 2) A (P). Xac nh VTCPa r cua d 1 , b

rcua d 2.

Mot VTPT cua (P) la: [ ]n a b,=r r r .

Dang 4:Mat phang (P) cha ng thang d 1 va song song vi ng thang d 2 (d 1 , d 2 cheonhau):

Xac nh cac VTCP a b,r r cua cac ng thang d 1 , d 2.


Lay mot iem M thuoc d 1 M (P). Dang 5:Mat phang (P) i qua iem M va song song vi hai ng thang cheo nhau d 1 , d 2:

Xac nh cac VTCP a b,r r cua cac ng thang d 1 , d 2.


2. Xac nh hnh chieu H cua mot iem M len ng thang d

Cach 1: Viet phng trnh mat phang (P) qua M va vuong goc vi d. Khi o: H = d (P)

Cach 2: iem H c xac nh bi:d

H d MH a

^ uuuur r

3. iem oi xng M' cua mot iem M qua ng thang d Cach 1: Tm iem H la hnh chieu cua M tren d.

Xac nh iem M sao cho H la trung iem cua oan MM . Cach 2: Goi H la trung iem cua oan MM . Tnh toa o iem H theo toa o cua M, M .

Khi o toa o cua iem M c xac nh bi: d MM a

H d

' ^

uuuuur r.

4. Xc nh hnh chieu H cua mot iem M len mat phang (P) Cach 1: Viet phng trnh ng thang d qua M va vuong goc vi (P).

Khi o: H = d (P)

Cach 2: iem H c xac nh bi:P

H P MH n cung phng

( ),

uuuur r

5. iem oi xng M' cua mot iem M qua mat phang (P) Cach 1: Tm iem H la hnh chieu cua M tren (P).

Xac nh iem M sao cho H la trung iem cua oan MM .

Cach 2: Goi H la trung iem cua oan MM . Tnh toa o iem H theo toa o cua M, M . Khi o toa o cua iem M c xac nh bi:

P

H P MH n cung phng

( ),

uuuur r .


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Bai 1. Viet phng trnh cua mat phang(P) i qua iem A va ng thangd :

a)4 2

2 3 1 2 33

x t A d y t

z t ( ; ; ), :

= +- = - = +

b)2

1 4 3 1 21 3

x t A d y t

z t ( ; ; ), :

= -- = - + = -

c) 1 2 54 2 33 4 2

x y z A d ( ; ; ), : - + -- = = d) 3 2 12 1 52 1 3

x y z A d ( ; ; ), : + + -- = =

e) 2 1 02 1 42 2 5 0

x y z A d x y z( ; ; ), : - + - =- + + + =

f) 3 2 1 03 2 42 3 0 x y z A d x y z

( ; ; ), : + - + =- - + - =

Bai 2. Viet phng trnh cua mat phang(P) i qua hai ng thang song songd 1 , d 2: a) {1 2

2 1 32 3 4 2 1

3 2 1 x y zd x t y t z t d : ;

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