chương viii : tích vô hướng và ứng dụng · pdf filechương 2....

Trn Thnh Minh Phan Lu Bin - Trn Quang Ngha
H NH H C 10 Ch ng 2.
Tch V Hng V ng Dng
http://www.saosangsong.com.vn/Save Your Time and Money
Sharpen Your Self-Study Skill Suit Your Pace
http://www.saosangsong.com.vn/

Chng 2. Tch v hng v ng dng
www.saosangsong.com.vn/
2
2
1.Tch v hng ca hai vect A .Tm tt gio khoa : 1 . Gc gia hai vect : a) Gc hnh hc : Gc hnh hc l hnh to bi hai tia c chung gc .S o a ( tnh bng ) ca mt gc hnh hc tha : 0 180o oa
Nu v a khng phi l gc c bit cc gi tr lng gic ca a c tnh bng my tnh b ti
0 9o a 0o
o
(0 ;30 ;45 ;60 ;90 )o o o o o
Nu , ta dng gc b tnh gi
tr lng gic ca a : 90 180o a<
sin sin(180 )cos cos(180 )tan tan(180 )cot cot(180 )
o
o
o
o
a aa aa aa a
=
=
=
=
b) Gc gia hai vect : Cho 2 vect ; ( 0a b ) ; V cc vect OA Gc AOB c gi l gc gia 2 vect ;a OB b= = ;a b K hiu : ( , )a b2 . Tch v hng ca hai vect : a ) nh ngha : Tch v hng ca hai vect ,a b .a b k hiu l l mt s xc nh bi :
. cos(a b a b a b= , )
b) Tnh cht :
. .
.( ) .
( ) ( . ) .( )
a b b a
a b c a b ac
ka b k a b a kb
=
+ = +
= =
Ta cng c cc kt qa sau :
22
; . 0a a a b a b= =
2 22
2 2
( ) 2 .
( )( )
a b a a b b
a b a b a b
+ = + +
+ =
Ch : S dng cc tnh cht ta s c cc h thc :
c) Cng thc hnh chiu : Cho hai vect bt k , ;AB CD . Gi E , F ln lt l hnh chiu vung gc ca C , D xung ng thng AB . Ta c cng thc :
d) Cng thc v ta : Cho cc vect : . Ta c cc cng thc : 1 2 1 2( , ) ; ( , )a a a b b b= =
. .AB CD AB EF=
O x
y
a
b
A B
C
D
E F

3
3
2 21 2
1 1 2 2
1 1 2 2
1 1 2 22 2 2
1 2 1 2
.
0
cos( , ).
a a a
a b a b a b
a b a b a ba b a ba b
a a b b
= +
= +
+ =+
=+ +
2
3 . p dng : Bi ton 1 : Tm tp hp im M tha : . (1)MA MB k= ( A , B c nh ; k l hng s ) Gi I l trung im ca AB , ta c :
2 2
2 2
(1) ( )( )MI IA MI IB k MI IA kIM k IA
+ + = =
= +
Tp hp cc im M l ng trn ( I , 2 0:k IA + > 2k IA+ ) Tp hp cc im M l : 2 0:k IA + = { }I : Tp hp cc im M l tp rng 2 0k IA +
Suy ra : I( 1 , 1 ) ; Do : Vy im M ngoi ng trn (ABC) Ta cng c : 2 /( ) 12 12 2 3MMT ABC MT= = = =
. ; . ; ( 2 )
.
C . Bi tp rn luyn : 2 .1 Cho tam gic u ABC cnh bng a . Tinh cc tch v hung sau : AB GB AB CM AB AB AC
n c c ( , ) ; ( , )
( G l trng tm tam gic ABC v M l trung im ca BC )
2 . 2 .Cho tam gic ABC vung ti A : AB = 3 ; AC = 4. T h c gAB BC AC BC v cc tch v hng sau : . ; .AB BC AC BC
2. 3 .Cho tam gic ABC vung ti A ; AB = 3 , AC = 4 . Trn tia AB ly im D sao cho BD = 4 Tnh cc tch v hng sau : . ; .BC BD AC BI
, c nh ng a , G l trn)
( I l trung im ca CD )
2 .4 . Cho tam gic ABC u b g tm tam gic ; M l mt im bt k . Chng minh rng T = ( . . .MA GB MB GC MC GA+ + c gi tr khng i . Tnh gi tr
au :
ny .
2 .5 . Cho hnh vung ABCD , cnh bng a . Dng nh l hnh chiu tnh cc tch v hng s
. ; ( ).( ) ; ( ).AB BD AB AD BD BC OA OB OC AB+ + + ( O l tm hnh vung )
* 2 .6 . Cho tam gic ABC u , cnh bng a . Tm tp hp cc im M tha : 23( 2 ).
4aCA BC CM+ =
2 . 7 .Cho tam gic ABC c trng tm l G .Chng minh rng :

11
11
2 21. . . (2
GA GB GB GC GC GA GA GB GC+ + = + + 2 )
2 .8. Cho hnh vung ABCD cnh bng a ; I l trung im ca CD . Tnh cc tch v hng sau : . ; .BD BI BI BG ( G l trng tm tam gic ABD )
2 .9 .Cho hnh ch nht ABCD c AB = 4 ; AD = 3 v im M tha AM k AB= .nh k 2 ng thng AC v DM vung gc
2 . 10 . Cho tam gic ABC vung ti A . Trn tia i ca tia AB ,ly im D sao cho AD = AC ; trn tia i ca tia AC , ly im E saocho AE = AB . Chng minh rng ng trung tuyn ca tam gic ADE th vung gc vi BC
2 . 11 . Cho : 6 ; 3a b= = .nh x hai vect sau vung gc vi nhau ( ) ; ( )a xb a xb+
2 . 12. Cho tam gic ABC vung ti A ; D thuc tia AC v AD = 3AC Chng minh rng 2 21 ( 16
92 )AG AB AC= + (G l trng tm tam gic BCD )
* 2 .13. Cho t gic ABCD a) Chng minh rng ; 2 2 2 2 2AB BC CD DA ACDB + = b)Suy ra rng iu kin cn v t gic ABCD c 2 ng cho vung gc l 2 2 2 2AB CD BC AD+ = +
2 . 14 . Cho tam gic ABC . Tm tp hp cc im M tha :
*
) ( ). 0
) .( 2 )
a AB AC AM
b MA MA MB MC
+ =
+ + = 0
* 2 .15 . Cho tam gic ABC u , cnh bng a . Tm tp hp cc im M tha : 2( ).MA MB MC a+ =
2 . 16 . Cho hai im A( 1 , 2 ) ; B( 6 , 3 ) . Tm ta im C nm trn trc Ox bit rng tam gic ABC vung ti C .
2 . 17 . Cho 4 im A( - 1 , 0 ) ; B( 0 , 3 ) ; C( 3 , 2 ) ; D( 5 , - 2) . Chng minh rng t gic ABCD l mt hnh thang vung . Tnh din tch ca hnh thang ny
.
D. Hng dn gii hay p s 23 3. . .cos30 . .
3 2 2o a aAB GB AB GB a= = =2 .1
21. . .cos60 . .
2 2 4o a aAB CM AB CM a= = = .
2 2 2 2 1.( 2 ) 2 . 2 . .cos 60 2 . 02
oAB AB AC AB AB AC a a a a a = = = =
2 .2 . ( , )AB BC v gc ABC b nhau ;cosABC = (3 : 5) = 0,6 Suy ra ABC = 53 7 '48" ( , ) 180 53 7 '48" 126 52 '12"o o oAB BC = = o

12
12
AC 4cosACB= 0,8BC 5
( , ) 90 36 52 '123 3. . .( ) 3.5.( ) 95 5
4 4. . . 4.5. 165 5
o oAC BC ACB ABC
AB BC AB BC
AC BC AC BC
= =
= = =
= = =
= = =
C
AB D
I
"
2
2.3 . . .cos35.4.( ) 125
. ( ) . ( .1 1. ( ) 82 2
BC BD BC BD CBD
AC BI AC AI AB AC AI AC AB
AC AC AD AC
=
= =
= = =
= + = =
0)
2 . 4 .Ta c :
MA
B C
I
M'
2
; ;
. . . ( ). .cos120 . .cos120 . .cos120 0
3 13( ) .( )3 2
o o o
MA GA GM MB GB GM MC GC GM
T GA GB GB GC GC GA GM GA GB GCGA GB GB GC GC GA
a a
= = =
= + + + +
= + +
= =
2 2 3 3( .
2 3 2 3a ado GA GB GC= = = =
2
2
22 2
2.5. . .
( )( ) . .
2( ). . . ( )2 2
AB BD AB BA a
AB AD BD BC AC CD DC CD a
a aOA OB OC AB OB AB OB OB OB
= =
+ = = =
+ + = = = = =
2 ; 2
*2 .6 . Ta c : V AI BC CA BC CA AI CI= + = + =
( 2 ). .
. '
CA BC CM CI CM
CI CM
+ =
=
( I c nh v tam gic ACI l na tam gic u)
Theo gi thit : 23 3 1. ' '
4 2 2a aCM CM CI= = =
. 7 a :
CI
Vy tphp cc im M l ng trung trc ca on CI 2 .T c
2 2 2
2 2 2
0 2 . 2 . . 01. . . ( )2
GA GB GC GA GB GC GA GB GB GC GC GA
GA GB GB GC GC GA GA GB GC
+ + = + + + + + =
+ + = + +
2 .8 .

13
13
.
2
22 2 2 2
22 2 2
1 1 1. . ( ) .(2 2 2
1 1 1 1 30 ( 2)2 2 2 2 21 1. ( ) ( )2 31 2(0 2 )6 3
)BD BI BD BD BC BD BC BC CD
aBD BC a a
BI BG BC BD BA BD BB
aa a a
= + = + +
= + + = + =
= + + +
= + + + =
2 2
2.9. . 0
( )( ) 0
( )( ) 090 .16 9 0
16
AC DM AC DM
AB AD AM AD
AB AD k AB AD
kAB AD k k
=
+ =
+ =
= = =
2 . 10 . Gi AI l trung tuyn ca tam gic ADE , ta c : 1 ( )2
1 1. ( ).( ) (0 . . 0)2 21 ( . . ) 02
AI AD AE
AI BC AD AE AC AB AD AB AE AC
AC AB AB AC AI BC
= +
= + = +
= =
2. 11 . x = 2 2 .12 .Ta c :
22 2 2
2 2
1 1( ) ( 3 )3 3
1 ( 16 8 . )9
1 ( 16 )9
AG AB AC AD AB AC AC
AG AG AB AC AB AC
AB AC
= + + = + +
= = + +
= +
2 2 2 22 2 2 22.13. )
( )( ) ( )( )
( ) ( )
2 .
a AB BC CD DA AB BC CD DA
AB BC AB BC CD DA CD DA
AC AB BC CD DA AC DB BD
AC DB
+ = +
= + + +
= + =
=2 2 2 2
2 2 2 2
) . 0 0b AC BD AC BD AB BC CD DAAB CD BC AD
= + =
+ = +
2 .14 . a) Tp hp cc im M l ng thng d qua A v vung gc vi trung tuyn AI ca tam gic ABC
b) ( 2 ) 0 .( ) 0
.(2 2 ) 0 4 . 0
MA MA MB MC MA MA MB MB MC
MA MJ MI MA MK MA MK
+ + = + + + =
+ = = ( J , I , K ln lt l trung im ca AB , BC , IJ) .Vy tp hp cc im M l ng trn dng knh AK

14
14
22 2 2
2 2 2 22 2
2.15.( ). 2 .2
3 8 3 11 11( )2 4 16 16 4
aMA MB MC a MI MC a MJ JC
a a a a a aJM JM
+ = = =
+= + = = =
2
.
( I , J ln lt l trung im ca AB , CI ) . Vy tp hp cc im M l ng trn ( J , 11)
4a
2 16 , C hai im C : C( 2 , 0 ) ; C( 7 , 0 ) 2 17 . Hnh thang ABCD vung ti A v D . Din tch ca hnh thang ny bng 15
2. H thc lng trong tam gic
A . Tm tt gio khoa 1 .nh l cosin : Trong mt tam gic ABC , bnh phng mt cnh bng tng cc bnh phng ca hai cnh cn li tr i tch ca hai cnh nhn vi cosin ca gc xen gia chng .
2 2 2
2 2 2
2

chương viii : tích vô hướng và ứng dụng · pdf filechương 2....

Documents