CHƯƠNG I: VÉC TƠ
Bài 1: Các khái niệm cơ bản
và các phép toán cộng, trừ véc tơ, phép nhân véc tơ với một số



I. Mục tiêu:
  

 !"#$$
%%&$'$()*$$+
,  /& 0+
!"
1*23(45"6"0"7)289($5
",#$+
#$%&
:73;-,3,+
<-=!(4,
II. Chuẩn bị của giáo viên và học sinh:
>+ '()*+ ,-$./
,,-?0/(@A
BC!%!D4
'()*+ ,-0 
E"F
BC!%7
III. Phương pháp:
G(45-?HA+
IV. Tiến trình bài dạy:
 1%+2  345
 6789-* :
/(@

Z1$Q2" A5B<5(5(
),,,[-%$(+
ZI5($Q%?,[
%&\&+
e) Hai vectơ bằng nhau
1$Q2"bằng nhau5(
%&"% !"+
I5($Q
a
r
"
b
r
'$(?$
5
a b=
r r
+
f) Góc giữa hai vectơ:
, $    
a

 " 
b

T(      
+G. =]",)$^
_L 5`
_L 5`
  3!a>,?(

b

+U)0,#$
)Y]b2"0,#$)*$$
a

"
b

$42")*$$
a

"
b

"3(2"V
a


b

W+
I5(V
a


b

Wfhi
i

b
thì
AC
được gọi là véctơ tổng của hai véc tơ
a
và
b
.
U3(
AC a b= +
uuur r r
<2H4j#$$Q2"
 Q+
Tính chất:
ZG3H$,, :
a b b a+ = +
r r r r
* G3H5:
V W V Wa b c a b c+ + = + +
r r r r r r
ZG3H#$Q:
ia a+ =
r r r
Các quy tắc:
* C(D *-%78
&$=HkA, B, C+G$)

AB BC AC+ =
uuur uuur uuur
?C(D E*E

+

Y
b

AB AD AC+ =
uuur uuur uuur?C(D E&5
,? ABCD.A’B’C’D’ . G$)

n nAC AB AD AA= + +
uuuur uuur uuur uuur
b) Phép trừ véc tơ
?F <%: I5(
a

≠

i
?
∃

x
$,,
x
o
a
f

<2H4(#$$Q2" 
 Q+
* C(D 9G : &$=Hk]Y
b$)
OB

OA
f
AB
c, Phép nhân véc tơ với 1 số
* Định nghĩaG3 
a
 & 0
62" 3(2"k+
a
;
p$(
I5( k qi?Q k
a
 %&&
Q 
a
+  I5( k r  i  ?  Q k
a
 
Ví dụ 4: ,  ?   
Ybc+Ysbscss&/]+
1M4  N  -$  K  `  $
-,K`$(/4
Y+

Y
c

b
Y
c
Ys
cs

s
bs
b
&&Q
a
+
B 2&
+ +k a k a=
r r
?H I
o+
i
f
i

oV+
a
WfV+W+
a
oV
a

ADACABAG ++=
+
c+
WV
l
>
ADACABAG ++=
+
Câu 2. ,`!Ybc+2"-
/#$`!"Ys2"-/#$$
bc+G-,Kp$(K
p",2"$d
Y+
i=+++ GDGCGBGA
+
b+
n
m
e
AAGA −=
+
+
MGMDMCMBMA =+++
&R2"
 =H?+
c+
m nAB AC AD AA+ + =
uuur uuur uuur uuur
Câu 3.,u=Ybcvw+BK
`",!&/4d

?  ^        '
<y QR RP
uuur uuuruuur
+
Bài 2. ,  $  
Yb+2"-/#$
$Yb"R2"=
(k+`-'
$W
iGA GB GC+ + =
uuur uuur uuur r
+
          W
mMA MB MC MG+ + =
uuur uuur uuuur uuuur
+
 c+
cw ADEEBCFACDAB =+++++
+
Câu 4.BT(",!&/42"P"#
==]2"-(=#$,DYb+
Y+]Yf]b+
b+
OBOA =
+
+
BOAO =
+
c+
i=+ OBOA

:73;-,3,+
<-=!(4,
II. Chuẩn bị của giáo viên và học sinh:
 '()*+ ,-$./
,,-?0/(@A
BC!%!D4
'()*+ ,-0 
E"F
BC!%7
III. Phương pháp:
G(45-?HA+
IV. Tiến trình bài dạy:
 1%+2  345
 6789-* :
/(@
I8(9(4#$ y(4m=9(4?
?"9(4-(=9(4-/+
#84
Hoạt động của giáo viên Hoạt động của học sinh
1. Vectơ cùng phương
* Định nghĩa1$Q2"%
5(),,,[
'$(+
* Định lýQ
b
%&Q
a
V
a


a
g
b
 (45?
$
a

b

c
CK"N
CD!(4H[02$,,

c
f+
a
o2
b
+
     I5($ 
a

b

c
  C
_L 5`
_L 5`
3!a>,$Yb+
  Ys  bss  2P  2  2"

f+
a
o2
b
o+
c
+

Z Hệ quảb0=]Yb%'
-8 [K"N
OCOBOA 
CK+

Z{8(P(2"3!a!a
9(+
Bài tập:
Câu 1:G-,T$(T
",2"$d
Y+,$%
ba
+U)$
cba 
CK
"N)[0$,,
bnamc +=
+
b+I5()
i=++ cpbnam
" -,
$0i?$

=
+ =
uur uur
uur uuur r
`-'~•9($
-  /   #$ $ 
Yb+
3!al1M4?T
$-,T$(
Y+b$     
cba 
C
K5() -,$
)'
i
+
b+b$
cba 
CK
5()$-,$)
%+
+  G-,  ?   
Ybc+Ysbsscs$    
nnnn DAACAB
CK+
c+     
cbax ++=
 2(
2(CK&$


+c+
NM
"
NP
+
Câu 4:,$T(Yb&J
$,Y1+BK`",!&/4d
Y+
HCHB =
+b+
HCAC e=
+
+
BC
e
m
Y1 =
+c+
ACAB =
+
Câu 5: ,?$Ybc&D
42"Ybfm$cfu$+U)
CDAB +
'$,8(d
Y+h$+b+m$+
+m$+c+i+
Câu 6: ,$T(Yb)D
'$+-p
CAAB −
'$,8(d

+
  =  R  I  2P  2
( DYcbb
>
$,
,YRfbI+`
-' 
DBABMN
>

 C
K+
3!au ,2\-a$
Yb+Ynbnn+~U2"
-(=#$bbn"Ynn+
R  2" =  $ ,D  bnn
Q,•0
e
>
+`
-'0=YU~R
%(  [K+
b"7
Câu 10:,$=0
pY"b/+G7
=R,MK
`
ABkMBMA =+
&>r
r>2"

c+Ufe?R0;`&b
9($Y+
Câu 11: &  $  =
Yb    K  "+
G7=R,M
MCnMBmMA +=
 &o
fi2"
Y+G7-S+
b+BJKb+
+BJK9($Y
",,&b+
c+BJK9($Y
"()&b+
V.Củng cố và hướng dẫn học ở nhà:
 U/(2D5`T$%$C
K+
 {8(P(2""7|G
E72D5`MT3&#$$Q,+

Bài 3: Tích vô hướng của hai véc tơ và áp dụng



I. Mục tiêu:
  

G3&#$$BpL$3H+
b?&+
!"

r
"
b
r
2"
 03(2" 
a
r
.
b
r
;pA
( )
+ + ab a b cos a b=
r r r r r r
+
Tính ch\t:
*
a
.
b
=
b

a
* (k.
a
.).
b
= k.(

a
.
c
*
a

⊥

b

⇔

a
.
b
= 0
,848(P(2"3!a"
&!„,,2"3!aQ,p
L$"3H+
B0c
3  !a  >:,  $     
 AAB
)
i
mi
…
eYb === AaAC
+UKp",!&
/4d
Y+

e
h
$
e
+
+h$
e
+
c+
e
h
−
$
e
+
3!am,?(YbcD'
> /  ]+  &    3& 
BCOB+

DCOB+
 
ODOB+
 " 
OAOB+
 ?  0  3  
&)59('
e
>
−
2"

+ G3    3   &
$(
AB
+
AC
g
BCCA


+
g
BAGA


+
g
CGBG


+
g
AGGB

+
g
CBAG

+
+
b">12""Q,

eCF CA CB= +
uuur uuur uuur
G.)!%3H#$3&
(4-$T(`+
Bài 2. ,,DKAB) !"2a"
k
2
+  G?  7      = M $,  ,
e
+MA MB k=
uuur uuur
+
,8&!„2""+
‡?&j
e
e
V WV WAB AM MB MA MB
a
= + +
=
uuur uuuur uuur uuur uuur
G5a/&=?7
=R+
Bài tập
Câu 1.,J-X(O;R)"=R
0p+R JK
∆
$4j2(
9($RJ-XD$=Y"
b+`-'

V WV W
e +
e
AB AM MB MA MB
a
AM AM MB MB a
MA MB a k
= + +
=
⇔ + + =
⇔ + = +
uuur uuuur uuur uuur uuur
uuuur uuur
G.)(4-$/=R
@$MT(-8+
Câu 7: ,$=0
p  Y  "  b+G7    
=R$,, 
aAMAB =+
V$2"'0W2"
Y+ R   J  K
,,&Yb+
b+ BJ  -X  7
/  2"  -(  =
#$Yb+
+BJK(
)&Yb+
c+ BJKYb+
Câu 8: I5(  ?  ,
Ybc)Ybf!"

e
v
uur
+UKp",
!&/4d
Y+
> >
e e
+ iv v v v= ⇔ ⊥
r uur r uur
+
b+
>
e > e
+ + ,v v v v
α
=
r uur ur uur
+
+
e e
> >
e e
v v v v= ⇒ =
r uur r uur
+
c+
e e
V + Wv v v
uur uur r

c+
>
v
r
%&&
e
v
uur
> e > e
+ +v v v v⇒ =
ur uur ur uur
+
Câu 6:,$Yb(/D
b+I5(Ybf?K`",!&/4
d
Y
e
+ kACAB =
+b+
e
+ kACAB −=
+
 
kACAB e+ =
+c+
kACAB e+ −=
+

c+
e

e
a
AGAB =
+
+
u
+
e
a
BGAG −=
+
c+
l
+
e
a
GCGA −=
+
V.Củng cố và hướng dẫn học ở nhà:
 U/(2D5`T3&#$$
?&+
 {8(P(2""7|G


Bài 4: Ôn tập



I. Mục tiêu:
  

tập sau:
Câu 1:,
v

>
v

e
v
2"$H?2"
06+<@,",!&/4d
 Y+
e>e>>
++ vvvvvv =⇒=
+
b+
e>
W+V vvv
f
W+V
e>
vvv
+
 +
WV
e>
vvv −
f
e>
++ vvvv +

+
c+
m>
vv
,,+
Câu 3 : _\-a$Yb+Ysbss+B[
n   AA a AB b AC c BC d= = = =
uuur uuur uuur uuur
+G-,=(
`$(=(`",2"d
Y+
cba +=
+
b+
i=+++ dcba
+
+
i=−+ cdb
+
c+
dcba =++
+
Câu 4 : ,?*7Ybc+G-,
K`!&/4K`",d
Y+
CDAB =
+
b+
DABC =
+

+b+e$++$+c+i+
Câu 7:,$$Yb"Ysbss2P
2)-/2""s+BK`",
!&/42"$d
Y+
nnYnnm CCBBAGG ++=
+
b+
nnbnnm CABCAGG ++=
+
+
nnnnm CBBAAGG ++=
+
c+
nnYnnm CCBBAGG ++=
+
Câu 8:I5(2"-/$Yb?
K`",!&/4d
Y+
e
ACAB
AG
+
=
+b+
m
ACAB
AG
+
=

Câu 7
Bc
Câu 8
Bb
Câu 9
BY
Câu 10
B
Y+I5(
OBOA u=
?
OBOAOBOA ++ =
+
b+I5(
OBOA m−=
?
OBOAOBOA ++ −=
+
+I5(
OBOA e=
?
e
e+ OBOBOA =
+
c+R -,$Kp-8$+
Câu 11: G-,Kp$(K
p",d
Y+I5(
e>
vv

Câu 12:UKp",!&/4d
Y+
e>
W+V vvv
2"+
b+
e>
WV vvv +
2"+
+
W+WV+V
e>
vvvv
2"+
c+
i+v
2"+
Câu 11
Bc
Câu 12
BY
V.Củng cố và hướng dẫn học ở nhà:
 E72DH2}(45#$+
 {8(P(,""5"7,48(P(2""
*"DMˆ$+

CHƯƠNG II: Đường thẳng và mặt phẳng trong không gian
Bài 1: Đại cương về đường thẳng và mặt phẳng
NN#
OJ#N.

 6789-* :
/(@
G??[K$("$JK,
$(-,2&d
# 84
Hoạt động của giáo viên Hoạt động của học sinh
1. Các tính ch\t thừa nhận của hhkg:
Tính ch\t 1:) "N JK
9($$=/+
Tính ch\t 2:) "N [K
9($$=K",-&.
Tính ch\t 3: GCDl=%'
-8 [K+
Tính ch\t 4:I5($[K/)
 =(?)JK
(!(4H`$H=(
#$$[K)+
Tính ch\t 5:G-,S[K5
9(M5#$?KT(
2. Điều kiện xác định của mặt phẳng
_L 5`
_L 5`
_L 5`+
$WR[K,",";p5(
5)9($$=K"+
WR[K,",";p5(
5  )     = " `$     J
K9($=)+
WR[K,",";p5(
5)`$$JK$(+

A
>
. 1?C$
Y
>
Y
e
+++Y

"$|Y
>
Y
e
|Y
e
Y
m
+++
|Y

Y
>
 2"?)"3(2"
|+Y
>
Y
e
+++Y

+

D8#$?)+
_L 5`
&5`

T$ SA
>
A
e
, SA
e
A
m
, , SA

A
>
2"[8#$?)+
I5(4#$?)2" T$
`+++??)`2"?
)$?)`+++
WG`!
 ,0=YbcC
K+1?C0$YbYc
Ybc"bc2"?`!V$4
2"`!W"}(2"Ybc+
=|bc2"N#$`
!+
,DKYbbccYYbc
2"D#$`!+
1$D)=(2"$

VYscW"VYbcW2"c
VYscW"V|YbW2"YsvVv2"$,#$Yb
"cW
VYscW"V|bW2"wVw2"$,#$Ysv
"|bW
VYscW"V|cW2"c
VYscW"V|YcW2"Ysc
"    = O ' ,"
[  K  VABCW+  A’,
B’, C’ 2"=2P2
-8,DKOA, OB,
OC "-%&
P(#$,DK
)+`-'5(
[JKA’B’ "ABg
B’C’"BCgC’A’"CA
$(2P2DD, E, F?
$=D, E, FK"+
_L 5`+
Ví dụ 2. ,?)`
S.ABCDAB"CD
$(+   A’ 2"     =
' *$ S " A+G? 
$,  (45  #$  VA’CDW
&[KVABCDW
VSABWVSBCWVSCDWVSDAW+