1
Thực hiện:Nguyễn Ngọc Quyến.
Tháng 11 năm 2010
Sở giáo dục - đào tạo lạng sơn
Trờng THPT tràng định
kính chào quý thầy cô
về dự giờ thăm lớp 10A6

2
ôn tập chơng I
I. kiến thức cơ bản chơng I



!!"
II. Mối quan hệ giữa hh tổng hợp - véc tơ - tọa độ
III. Một số bài tập vận dụng.
!2 !1
3
1. VÐc t¬ vµ c¸c kh¸i niÖm:
#$%&
'•
•B
AB
uuur
#!()*+,!"(&
'•
•B
# #/&
=

→ →
'' 00
1
→
=
& 234
'5
• B
#67$%&
AB AB=
uuur
#89:&
a
r
b
r
a b
a b
a, b cung`huong

=

= ⇔



r r
r r
r r
0 0

=
r
4
Tæng vµ hiÖu cña hai vÐc t¬
AB BC AC ,( A, B,C)
+ = ∀
uuur uuur uuur
'•
0
•
•;
#<=>!"?<=>@A
#<=>BCA
AB AC CB ,( A, B,C)
− = ∀
uuur uuur uuur
'•
0•
•;
#
<=>%9%A
;(9'0;D$/EA
'
0
;
D
AB AD AC
+ =
uuur uuur uuur
#!A

FLu ýAG!$
0
r

AB
uuur
;E!$
BA
uuur
1 2 2 3 n 1 n 1 n
A A A A A A A A
−
+ + + + =
uuuuur uuuuur uuuuuuur uuuuur
<=
!"H
B I
5
#JB!"!()*A
K$B!"!()*'0
IA IB 0
⇔ + =
uur uur r
#JB I'0;A
K
'
0
G
M
P

N
A
B
C
GA GB GC 0
+ + =
uuur uuur uuur r
6
3.TÝch cña mét vÐc t¬ víi mét sè
;(
b ka=
ur
r
@L(!7
a
r

b k . a=
ur
r
b
ur
a
r
b ka=
ur
r
9@M(
;24&
a

r

b
ur
N1
-4O4&
P1

(a 0 , k 0)
≠ ≠
r r
F@Quy íc:
0.a 0, k.0 0, ( a, k )
= = ∀ ∀ ∈
r r r r r
R
!3
Q
(G
7

9J%H HHEA
a
r
b
r
k.(a b) k.a kb
(h k).a ha ka
h.(k.a) (h.k)a
1.a a, ( 1).a a

+ = +
+ = +
=
= − = −
r r r r
r r r
r r
r ur r r
 
'•
•0
K
•
R
•
#K$B!"!()*'0% !"R
$/EA
MI (MA MB)= +
→ → →
1
2
8
#;(S$B I'0;R$!"9J%A
1
MG (MA MB MC)
3
= + +
→ → → →
0• •;
•

T
'
•
•<
•S
R•
9
!"#$%&'()&
(b 0) : a, b
≠
r r r r
234
⇔
a k.b ,(k )= ∈
r r
R
*+,-./0.0+($
⇔
AB k.AC ,(k 0)
= ≠
uuur uuur
'
;
0
!12%&3%&#4'()&
;(/234
9J%M!E$/EU3IA
x k.a h.b (k, h )
= + ∈
r r ur

R
a
r
b
r
x
r
10
Tæng hîp VÐc t¬ Täa ®é trªn mp


R$B
!"!()
*'0

'?V

W=

@W0?V
9
W=
9
@R?V

W=

@%A

 6"S$

B I
'0;
 234A 
X34
Y
!"'H0H;
30*

Z
AB
uuur
1) MA MB 0
2) AM MB
3) OA OB 2OM ,( O)
+ =
=
+ = ∀
uuuur uuur r
uuuur uuur
uuur uuur uuuuur
a b
m
a b
m
x x
x
2
y y
y
2

+

=



+

=


B A B A
AB (x x ; y y )
= − −
uuur
1) GA GB GC AB
1
2) MA MB MC MG, M
3
+ + =
+ + = ∀
uuur uuur uuur uuur
uuuur uuur uuur uuuur
A B C
G
A B C
G
x x x
x
3

y y y
y
3
+ +

=



+ +

=


a,b
uur
ur
a k.b, (b 0)
⇔ = ≠
r r r r
1 1 2 2
(x ;y ) (x ;y )
a ,b
uur
ur
1 2 2 1
x .y x .y⇔ =
AB k.AC
=
uuur uuur

a b=
r r
<$
9Y
1 1 2 2
(x ;y ) (x ;y )
a ,b
uur
ur
1 2
1 2
x x
a b
y y
=

= ⇔

=

r r
11
1. VÐc t¬ vµ c¸c kh¸i niÖm:
#$%&
'•
•B
AB
uuur
#!()*+,!"(&
'•

•B
##/
=
→ →
'' 00
1
→
=
 234
'5
• B
#67$%&
AB AB=
uuur
#89:&
a
r
b
r
a b
a b
a, b cung`huong

=

= ⇔



r r

r r
r r
0 0
=
r
12
Tæng vµ hiÖu cña hai vÐc t¬
AB BC AC , ( A, B,C)
+ = ∀
uuur uuur uuur
'•
0
•
•;
#<=>!"?<=>@A
#<=>BCA
AB AC CB ,( A, B,C)
− = ∀
uuur uuur uuur
'•
0•
•;
#
<=>%9%A
;(9'0;D$/EA
'
0
;
D
AB AD AC

+ =
uuur uuur uuur
#!A
FLu ýAG!$
0
r

AB
uuur
;E!$
BA
uuur
1 2 2 3 n 1 n 1 n
A A A A A A A A
−
+ + + + =
uuuuur uuuuur uuuuuuur uuuuur
13
#JB!"!()*A
K$B!"!()*'0
IA IB 0
⇔ + =
uur uur r
#JB I'0;A
K
'
0
G
M
P

N
A
B
C
GA GB GC 0
+ + =
uuur uuur uuur r
14
3.TÝch cña mét vÐc t¬ víi mét sè
;(
b ka=
ur
r
@L(!7
a
r

b k . a=
ur
r
b
ur
a
r
b ka=
ur
r
9@M(
;24&
a

r

b
ur
N1
-4O4&
P1

(a 0 , k 0)
≠ ≠
r r
F@Quy íc:
0.a 0, k.0 0, ( a, k )
= = ∀ ∀ ∈
r r r r r
R
!3
Q
(G
15
!"#$%&'()&
(b 0) : a, b
≠
r r r r
234
⇔
a k.b ,(k )= ∈
r r
R
*+,-./0.0+($

⇔
AB k.AC ,(k 0)
= ≠
uuur uuur
'
;
0
!12%&3%&#4'()&
;(/234
9J%M!E$/EU3IA
x k.a h.b (k, h )
= + ∈
r r ur
R
a
r
b
r
x
r
- Gîi ý bai 5: b2
16

9J%H HHEA
a
r
b
r
k.(a b) k.a kb
(h k).a ha ka

h.(k.a) (h.k)a
1.a a, ( 1).a a
+ = +
+ = +
=
= − = −
r r r r
r r r
r r
r ur r r
 
'•
•0
K
•
R
•
#K$B!"!()*'0% !"R
$/EA
MI (MA MB)= +
→ → →
1
2
17
Bµi tËp tæng hîp Bµi tËp täa ®é
18
F
D
E
A

C
B
Bµi 1:8[=\B]3
@;234
9@;24H
@-4O4
7@0:
^@6
B('0;?%_@
Bµi 2:;\BU3IA
0• •;
•
T
'
•
•<
•S
a) AQ QC
b) PB PC
1
c) AG AP
3
d) GB 2GQ
=
=−
=
=−
uuur uuur
uuur uuur
uuur uuur

uuur uuur
19
Bµi 3:;`!*`A
RH-HTH<HaHL9J%
a) MN QP MP QN
+ = +
uuuuuur uuuuuuur
uuuuuuuur uuuuuuur
@ ;EA
;b4`?@?@
!4OA
;EA
;b4?@?@!4
O!3
9@EA
;b4`?@H?@
?@!4OA
?%@cN?!3@
Gi¶iA
(1)
(2)
(3)
MP MS SP
NQ NP PQ
RS RQ QS
= +
= +
= +
uuur uuur uur
uuur uuur uuur

uuur uuur uuur
(1)
(2)
MN MP PN
QP QN NP
=
=
+
+
uuuuuuuur uuuuuuur uuuuuur
uuuuuur uuuuuuur
uuuuuur
(dpcm)
MN QP MP QN PN NP
MN QP MP QN
+ +
⇔
+ = +
+ = +
uuuuuur uuuuuuur
uuuuuuuur uuuuuuur uuuuuur uuuuuur
uuuuuur uuuuuuur
uuuuuuuur uuuuuuur
MP NQ RS MS NP RQ SP PQ QS
+ + = + + + + +
uuuuur uuuur uuuur uuuur uuur uuuur uuuur
uuuuur uuuur
MP NQ RS MS NP RQ
b)
+ + = + +

uuuuur uuuur uuuuur uuuuur
uuuuur uuuur
MP NQ RS MS NP RQ
= + +
⇔ + +
uuuuuuur uuuuuur uuuuuuur uuuuuuur
uuuuuuur uuuuuur
SP PQ QS 0
+ + =
uuur uuuur uuuur ur
(1)
MN QN QM
(2)
QP MP MQ
= −
= −
uuuuur uuuuuur
uuuuuur
uuuur uuuuuur
uuuuur
<$
9
S
d
20

Bµi 4: ;`B:bSSe$f$4O$B I
'0;'e0e;e%A
- Gîi ý
3GG ' AA ' BB' CC '

= + +
uuuur uuuur uuuur uuuur
Gi¶i
Ph©n tÝchA
?@H?@H?@!4OA

%SSe$f$4O$B I'0;'e0e;e.
L=B6T;R
(1)
(2)
(3)
GG' GA AA' A'G'
GG' GB BB' B'G'
GG' GC CC' C'G'









= + +
= + +
= + +
uuuuuuur uuuuuur uuuuuuur uuuuuuuur
uuuuuuur uuuuur uuuuuur uuuuuuur
uuuuuuur uuuuur uuuuuur uuuuuuur
3GG ' AA ' BB' CC '

GA GB GC
A 'G ' B'G ' C'G '
= + +
+ + +
+ + +
uuuur uuuur uuuur uuuur
uuur uuur uuur
uuuuur uuuuur uuuuur
GA GB GC 0
G 'A ' G 'B' G 'C ' 0 A 'G ' B'G ' C 'G ' 0
+ + =
+ + = ⇒ + + =
uuur uuur uuur r
uuuuur uuuuur uuuur r uuuuur uuuuur uuuuur r
céng t¬ng øng hai vÕ
9

!
21
)3GG ' GG ' GG ' GG '
)GA GB GC 0
)G 'A ' G 'B' G 'C' 0 A 'G ' B'G ' C'G ' 0
+ = + +
+ + + =
+ + + = ⇒ + + =
uuuur uuuur uuuur uuuur
uuur uuur uuur r
uuuuur uuuuur uuuur r uuuuur uuuuur uuuuur r
<$9


22
- Để chứng minh một đẳng thức véc tơ ta
thờng sử dụng các quy tắc tổng, hiệu
của hai véc tơ hoặc tìm véc tơ đối để
biến đổi vế này thành vế kia của đẳng
thức hoặc biến đổi cả hai vế của đẳng
thức để đợc hai vế bằng nhau. Ta cũng
có thể biến đổi đẳng thức cần chứng
minh đó tơng đơng với một đẳng
thức véc tơ đợc công nhận là đúng.
(Trang 18 SBT cơ bản).
23
Bµi 5: ;(!"'?W@W0?W#@W;?#W@
@ !
9@;`B:!"'H0H;/*
@% !B IS'0;
7@% !BK!"!()'0
^@% !!"D((`D'0;$%9%

AB, AC
uuuuuur uuuuuuur
- Gîi ý Gi¶i
!
24
- Gîi ý a
- Gîi ý d
- Gîi ý b1
- Gîi ý c
- Gîi ý e
<$9

Y
25
;
0
'
D
Ta cã: 
BC AD
=
uuur uuur
BA CD
=
uuur uuur
- Gîi ý e2