TUYỂN TẬP HÌNH HỌC GIẢI TÍCH
TRONG KHÔNG GIAN
(ĐÁP ÁN CHI TIẾT)

BIÊN SOẠN: LƯU HUY THƯỞNG
Toàn bộ tài liệu của thầy ở trang:
http://www.Luuhuythuong.blogspot.com HÀ NỘI, 8/2013
HỌ VÀ TÊN: …………………………………………………………………
LỚP :………………………………………………………………….
TRƯỜNG :…………………………………………………………………
 

7
&;$?(2@"%"%$(
Giải
:.
( )/ /( )
Q P
A7&;$?(BC.
( ) : 2 3 0, ( 1)
Q x y z D D
+ − + = ≠

:.$?(2@A"&.
3
D
=

D47&;
( ) : 2 3 3 0
Q x y x
+ − + =


HT 2. :&% =   3 > !
,
Oxyz
 % 78 
1 1 2
:
1 2 1
x y z

− + =


HT 3. :&% =   3 > !
,
Oxyz
 % E * =  
(1;2; 1), ( 1; 0;2), (2; 1;1)
A B C
− − −
 
7&;$@FG(
Giải
:.
( 2; 2;3), (1; 3;2)
AB AC
= − − = −
 

$@FG(!.
[ ]
; (5;7; 8)
n AB AC
= =
  

D47&;
( ) : 5( 1) 7( 2) 8( 1) 0
ABC x y z
− + − + + =



:.
, ( )
( ) ( )
Q
Q P
AB n
A B Q
Q P
n n




⊥
∈


 
⇒
 
 
⊥
⊥
 





3 2
x t
d y t
z t


= − +



=



= − −




Giải
:
(1; 3;2)
BA
=

4B:G
(1;2; 2)
u
= −



⇒7&;$(.
10 4 19 0
x y z
− + − =


HT 6. :&% =   3 > !
,
Oxyz
 %  78  V 
1
2 1
: ;
1 1 2
x y z
d
− +
= =
−
2
1 1 1
:
1 2 1
x y z
d
− − −
= =
−
7&;$(W<78


:.
1 1
2 2
( )
( )
P d n u
P d n u
 
 
⊃ ⊥
 
 
⇒
 
 
⊃ ⊥
 
 
 
 
 

M&4$(
[ ]
1 2
, ( 5; 3;1)
n u u= = − −
  


x y z
d
− − −
= =
YD7&;$(W
1
d

2
d

Giải
:.
1 2
(1; 1;2) ; (4;1;3)
A d B d
− ∈ ∈
4
(3;2;1)
AB
=


J>
1
u

X7
1
d

1
1
( ) :
1 2 3
x y z
d
+
= =
− −

2
1 4
( ) :
1 2 5
x y z
d
− −
= =
 G7[   &_ ]
1 2
, ,
M d d
 ^ ` &Z %Z\ a\ b c 7 &d^ a\
b%[
Giải
:.
1
d
2
1

 
4
1 2
(0;2;4)
M M =

⇒
1 2 1 2
; . 0
u u M M
 
=
 
 

 
⇒
1 2
,
d d
e
J> $(    W
1 2
,
d d
 ⇒ $(  :PQRSJ :TUSP 
(1;2; 1)
n
= −


Giải
:.
( ) / /( )
P Q
M&47&;
( ) : 0 ( 1)
Q x y z D D
+ + + = ≠ −

K$M(h
(1; 2;1)
I
−
4i=j.
5
R
=

$?(+-K$M(=X=.
( ;( ))
5 3
5
5 3
3
I Q
D
D
d R
D


( ) : 4 11 0
x y z
α
+ + − =
+-
$M(
Giải
:.$M(h
(1; 3;2)
I
−
i=j
4
R
=
:PQRSJ:TUSP
( )
α

(1;4;1)
n
=


⇒:PQRSJ:TUSP$(.
, (2; 1;2)
P
n n v
 
= = −

( ) : 2 2 21 0
P x y z
− + − =


HT 11. :&% =   3 %C !
,
Oxyz
4 % 78 
3 3
:
2 2 1
x y z
d
− −
= =
   K
x z
2 2 2
( ) : 2 2 4 2 0
S x y z y
+ + − − − + =
YD7&;$("%"%
d
&l
Ox
4e8
+-K$M(
Giải
 


$(+-$M(⇔
( ,( ))
d I P R
=
⇔
2 2
1 4
2
1 2
D− +
=
+
⇔
3 2 5
D
− =
⇔
3 2 5
3 2 5
D
D

= +



= −



Chú ý: Đối với dạng này, chúng ta không tìm được vec-tơ pháp tuyến của mặt phẳng dưới dạng trực tiếp. Chính vì vậy, ta
phải dùng phương trình tổng quát của mặt phẳng để viết.
Giải
:.$M(h
( 1;2;0)
I
−
i=j
3
R
=
4$(:PQRSJ:TUSP
(1;0;1)
P
n =


PQRSJ:TUSP$?(2 BC.
2 2 2
( 3) ( 1) ( 1) 0, 0
A x B y C z A B C
− + − + + = + + ≠

$?(+-$M(⇔
2 2 2
( ,( )) 4 3
d I Q R A B C A B C
= ⇔ − + + = + +
$m(
( ) ( ) . 0 0

G>Fo6p4@oI4Go6I⇒
( ) : 4 7 4 9 0
Q x y z
− − − =


HT 13. :&%=3&l
,
Oxyz
%K
2 2 2
( ) : – 2 4 2 – 3 0
S x y z x y z
+ + + + =
7&;
$(W&l
Ox
VK$M(%!78&qi=j
3
r
=

Giải
:.$M(h
(1; 2; 1)
I
− −
4i=j
3
R

S x y z x y z
+ + + − + =
78
 
GV.Lưu Huy Thưởng 0968.393.899

BỂ HỌC VÔ BỜ - CHUYÊN CẦN SẼ ĐẾN BẾNs
2 2
:
1 1 2
x y z
d
− +
= =
7&;$(WBVK$M(%!78&qi=j
1
r
=

Giải
:.$M(h
( 1;1; 1)
I
− −
4i=j
2
R
=

PQRSJ:TUSP




⇔
,2 ( ), 3 (1)
17 7 ,2 ( ), 3 (2)
a b c a b d a b
a b c a b d a b

= = − + = − −


= − = − + = − −



 /$(⇒
( ) : 4 0
P x y z
+ − − =
 /$<(⇒
( ) : 7 17 5 4 0
P x y z
− + − =


HT 15. :&%=3%C!
,
Oxyz
%K

=
)78&quπAi=j
3
r
=

f%vnr$β(
2 2 2 2
5 3 4
h R r
= − = − =

0%
(loaïi)
2 2 2
2.1 2( 2) 3
7
4 5 12
17
2 2 ( 1)
D
D
D
D

+ − − +
= −

= ⇔ − + = ⇔



 :.
( )
( )
( ,( )) 3
A P
B P
d I P


∈




∈




=



⇔
,2 , (1)
5 7 ,2 , (2)
a b c a b d a b
a b c a b d a b



=



= − +



 =


*
( 1;2;3)
A
−
7
&;$(W78$B("%%=%vn*@$(i_E
Giải
:.$B(2*
(0; 1;1)
M
−
:G:
(1;2; 0)
u
=

J>
( ; ; )

− + + +
= ⇔ = ⇔ = ⇔ + = +
+ + +


(
)
2
2 2
4 4 0 2 0 2
b bc c b c c b
⇔ − + = ⇔ − = ⇔ =
 $E(
 :n$<($E(4>
1
b
= −
⇒
2, 2
a c
= = −
⇒7&;$(.
2 2 1 0
x y z
− − + =


HT 18. :&%=3&l>!
Oxyz
4%*

P ax by cz
+ + =
4
2 2 2
0
a b c
+ + ≠

 0%@∈$(⇒
2 3 0
a b c
+ + =
 $(
( ,( )) ( ,( )) 2
d B P d C P b c a b c
= ⇔ − + = + +
$<(
 :n$($<(⇒
0
b
=
%
0
c
=

 •
0
b
=

2

Giải
:.7&;$(2NABC.
0
Ax By Cz
+ + =
$
2 2 2
0
A B C
+ + ≠
(
;$(⊥$?(A.
1. 1. 1. 0
A B C
+ + =
⇔
C A B
= − −
 $(
( ,( )) 2
d M P
=
⇔
2 2 2
2
2
A B C
A B C

( ) : 0
P x z
− =

 
GV.Lưu Huy Thưởng 0968.393.899

BỂ HỌC VÔ BỜ - CHUYÊN CẦN SẼ ĐẾN BẾNp
:n$I(.x@/sFowG>@os4Fo6x⇒GoE⇒
( ) : 5 8 3 0
P x y z
− + =

HT 20. :&%=3&l>!
,
Oxyz
%78∆.
1 3
1 1 4
x y z
− −
= =
* $wH6<Hw(
7&;$(2* 4"%"%78∆4e8=%v578∆
$(i_I
Giải
:.7&;$(2 $wH6<Hw(BC.
2 0
ax by cz b
+ + + =


∆


 
+
⇔
 
=
 
=
 



+ +




⇔
4
2
a c
a c


=



+ − + =
x y z


HT 21. :&%=3&l>!
Oxyz
4%i*
(1;1; 1)
A
−
4
(1;1;2)
B
4
( 1;2; 2)
C
− −
$(.
2 2 1 0
x y z
− + + =
7&;
( )
α
2@4$(4V78FGCr
"%%
2
IB IC
=


⊥
A
2 2 0
a b c
− + =
$<(

2
IB IC
=
⇒
( , ( ;
( )) 2 ( ))
d B d C
α α
=
⇒
2 2 2 2 2 2
2 2 2
2
a b c d a b c d
a b c a b c
+ + + − + − +
=
+ + + +

 
3 3 6 0
(3)
5 2 3 0



 − + − =



  G>
2 1; 2; 3
a b c d
= ⇒ = − = − = −
⇒
( )
α
.
2 2 3 0
x y z
− − − =

 :P<.
0
3 3
2 2 0 ; ;
2 2
5 2 3 0
a b c d
a b c b a c a d a
a b c d


+ − + =

− − − =
%
( )
α
.
2 3 2 3 0
x y z
+ + − =


HT 22. :&% =%Z  [  Z\ %\ %Z\
,
Oxyz
%  7^  b
1 2
,
d d
 y 7\ %[ 7 &d^
1
2 2 3
:
2 1 3
x y z
d
− − −
= =
4
2
1 2 1
:


2
(2; 1;4)
d
u = −


 
GV.Lưu Huy Thưởng 0968.393.899  
BỂ HỌC VÔ BỜ - CHUYÊN CẦN SẼ ĐẾN BẾNx
0%$(z
1 2
,
d d
A$("%"%
1 2
,
d d
⇒
1 2
, (7; 2; 4)
P d d
n u u
 
= = − −
 
 
  

⇒7&;$(BC.


HT 23. :&%=%Z[ Z\%\%Z\
,
Oxyz
%7^ b
1 2
,
d d
y7\%[7&d^
1
1
: 2
1
x t
d y t
z


= +



= −



 =


4

:G
1
(1; 1;0)
u = −


2
d
2
(2;1; 1)
B
−
:G
2
(1; 2;2)
u = −


 J>
n

$(4;$("%"%
1
d

2
d
A
1 2
, ( 2; 2; 1)



1 2
( ,( )) 2 ( ,( ))
d d P d d P
=
7 2. 5
m m
⇔ + = +

7 2(5 )
7 2(5 )
m m
m m

+ = +

⇔

+ = − +


17
3;
3
m m⇔ = − = −

 /
3
m

2 2 2
( 1) ( 2) ( 1) 2
x y z
− + − + + =

 Giải
:.$M(h
(1;2; 1)
I
−
4i=j
2
R
=

7&;$(BC.
2 2 2
0 ( 0)
ax by cz d a b c+ + + = + + ≠

:.
( )
( )
( ,( ))
A P
B P
d I P R


∈

x y
− − =

/$<(⇒7&;$(.
8 3 5 7 0
x y z
− − + =

http://www.Luuhuythuong.blogspot.com

Dạng 4: Viết phương trình mặt phẳng liên quan đến góc

HT 25. :&%=3%C!
,
Oxyz
%$α(W78$∆(.
1
1 1 2
x y z
−
= =
− −
C%
$(.
2 2 1 0
x y z
− − + =
!uw
w
:;>!%* $α(&lNO

 
 


$α($(.
2 2 1 0
x y z
− − + =
C%uw
w
A.
(
)
2
2
1 1 1
cos , 2 4 1 0
2 2
2 4 5
n n m m
m m
′
= ⇔ = ⇔ − + =
− +
 
⇔
2 2
m = −

2 2

2 2
cos
9
ϕ =

Giải
Yg

(0;1;0), (1;3;2)
A B d
∈
$(2@⇒7&;$(BC.
– 0
Ax By Cz B
+ + =

$(2FA.
3 2 – 0
A B C B
+ + =
⇒
(2 2 )
A B C
= − +

⇒
( ) : (2 2 ) – 0
P B C x By Cz B
− + + + =


⇒
( ) : 4 – 1 0
P x y z
− + + =

 /
,
5
1
13
B C
= =
⇒
( ) : 23 5 13 – 5 0
P x y z
− + + =


HT 27. :&% =   3 > !
,
Oxyz
 %  *
( 1;2; 3), (2; 1; 6)
A B
− − − −
   
 
GV.Lưu Huy Thưởng 0968.393.899  
BỂ HỌC VÔ BỜ - CHUYÊN CẦN SẼ ĐẾN BẾNw
( ) : 2 3 0







=




⇔
a
2 2 2
2 3 0
2 6 0
2 3
6
1 4 1
a b c d
b c d
a b c
a b c


− + − + =



− − + =

%$?(.
3 0
x y
− − =


HT 28. :&% =   3 > !
,
Oxyz
 %   
( ) : 5 2 5 1 0
P x y z
− + − =
 
( ) : 4 8 12 0
Q x y z
− − + =
YD7&;
( )
R
2*#>!N4$(
C%$?(!
0
45
=
α

Giải
Jv"}7&;
2 2 2


= −

+ − = ⇔

=



 •
a c
= −
.>
1, 0, 1
a b c
= = = −
⇒7&;
( ) : 0
R x z
− =

 •
7
c a
=
.>
1, 20, 7
a b c
= = =
⇒7&;

⇒
4 5 6
1
5 6 0
4 6 0
a b c
b c
a c



+ + =




− + =



− + =


⇒
⇒⇒
⇒
77 77 77
; ;
4 5 6
a b c= = =

M P
∈
A
1 1 1
1
2
b c
+ + =
⇔
2
bc
b c+ =

:
( 2; ; 0)
AB b
−

4
( 2;0; ).
AC c
−

f
2 2 2
( )
S b c b c
= + + +

;



HT 31. :&%=%C!
,
Oxyz
%*
(2;2;4)
A

( ) :
P
4 0
x y z
+ + + =
7&;
$?("%"%$($?(V
,
Ox

Oy
C<*F4G"%%@FGB3ji_u
 Giải
;$?(tt$(A$?(.
0 ( 4)
x y z d d
+ + + = ≠
Jv"}
Ox
( ) , ( )
B Q C Q Oy

HT 32. :&%=3>!
,
Oxyz
%*
(2; 1;1)
A
−
7&;$(2*@
#>!N!=%vg
 Giải
:
( ,( ))
d O P OA
≤

0%
max
( ,( ))
d O P OA
=
+v&
( )
OA P
⇔ ⊥
A$(K;2@N@
:
(2; 1;1)
OA
= −


D$(K;2@D
AH

⇒$(.
7 5 77 0
x y z
+ − − =

HT 34. :&%=3>!
,
Oxyz
%78
2 2
: .
1 2 2
x y z
d
+ −
= =
−
J>∆782
*@$IHwH6("%"%
d
r$6<HwH<(;@&A
d
7&;
W∆=%v
d
g
Giải




:&%$(4
IH IA
≤
HB%
axIH = IA H A
m
⇔ ≡
Y-$(ƒ,&j$
w
(⊥r@C@
$
w
(
(
)
6; 0; 3
n IA
= = −
 
4'7
(
)
2; 0; 1
v
= −



( ; ; )
n a b c
=

4
d
2*
(1;0;2)
M
:G
(2;1;2)
u
=


;$(⊃BA
( )
. 0
M P
n u


∈




=







„k<&78L.
 :P.Siow;$(.
1 0
x z
− + =
f.
( ,( )) 0
d A P
=

 :P<.Si≠wG>
1
b
=
7L$(.
2 2 (2 1) 2 2 0
ax y a z a
+ − + + + =

 f.
2 2
9 9
( ,( )) 3 2
8 4 5
1 3
2 2



 
GV.Lưu Huy Thưởng 0968.393.899

BỂ HỌC VÔ BỜ - CHUYÊN CẦN SẼ ĐẾN BẾNE
HT 36. :&%=%C!
,
Oxyz
%*
(0; 1;2)
M
−

( 1;1;3)
N
−
7&;$(
2 4S"%%=%vn*
(0;0;2)
K
$(g
 Giải
7&;$(BC.
( 1) ( 2) 0 2 0
Ax B y C z Ax By Cz B C
+ + + − = ⇔ + + + − =

         
2 2 2

2
4 2 4
2 1 2
B
d K P
B C BC
C
B
= = ≤
+ +  



+ +




 

0g…o†+v&=Fo6GG>Gof7&;
( ) : – 3 0
P x y z
+ + =

HT 37. :&% =   3 > !
,
Oxyz
 %   $?(.
2 5 0

( ) 7 4
M P c a b
N P d a b
 
 
∈ = − −
 
 
⇒
 
 
∈ = +
 
 
 

⇒$(.
( 2 ) 7 4 0
ax by a b z a b
+ + − − + + =
⇒
2 2
3
cos .
6
5 4 2
a b
a ab b
α
+

α
+
=
 



+ +




 
)
b
x
a
=

2
( ) cos
f x
α
=

 „k"#
2
2
9 2 1
( ) .

HT 38. :&%=3%C!
,
Oxyz
7&;$(2*
(9;1;1)
M
4V
Ox
4
,
Oy
Oz
C@4F4G"%%*jWB3N@FG&,•g
 Giải
J"}
( ;0;0) , (0; ; 0) , (0; 0; )
A a Ox B b Oy C c Oz
∈ ∈ ∈

( , , 0)
a b c
>

f7&;$(BC.
1
x y z
a b c
+ + =

:.

0g‚o‚+v&⇔
27
9
3
9 1 1
1
3
a
bc ac ab
b
c
a b c



=

= =






⇔ =




+ + =

+ +
&,•g
 Giải
J"}
( ;0;0) , (0; ; 0) , (0; 0; )
A a Ox B b Oy C c Oz
∈ ∈ ∈

( , , 0)
a b c
>

f7&;$(BC.
1
x y z
a b c
+ + =

:.
(1;2; 3) ( )
M P
∈
⇒
1 2 3
1
a b c
+ + =

:.
2 2 2 2 2 2



 
 
2 2 2
1 1 1 1
14
a b c
⇒ + + ≥

0g…o†+v&=
2 2 2
1 2 3
1
1 1 1
2 3
1 1 1 1
14
a b c
a b c
a b c



+ + =






⇔ =






=




D47&;.
( ) : 2 3 14 0
P x y z
+ + − =

HT 40. :&%=3%C!
,
Oxyz
7&;$(2*
(2;5;3)
M
4V
Ox
4
,
Oy
Oz
C@4F4G"%%i*W

a b c
+ + =

( )
( ) ( ) ( )
2 2 2
2 2 2
2 5 3 2 5 3
a b c a b c
a b c a b c
 

     

 
 


  
  




  

  




(
)
2
2 5 3 10 2 10 2 6 2 15
≥ + + = + + +

2 5 3
a b c
⇒ + + ≥ + +

0g…o†+v&=.
2 5 3
1
2 6 10
2 5 3
5 10 15
3 6 15
10 2 10 2 6 2 15
a b c
a
b
a b c
c
a b c



+ + =



P
+ + =
+ + + + + +


 
GV.Lưu Huy Thưởng 0968.393.899  
BỂ HỌC VÔ BỜ - CHUYÊN CẦN SẼ ĐẾN BẾNu
PHẦN II VIẾT PHƯƠNG TRÌNH ĐƯỜNG THẲNG

Dạng 1: Viết phương trình đường thẳng bằng cách xác định vectơ chỉ phương
X778"%"%%&'78
 !78"#X7
)*;X778-"#".
+ Trực tiếp:01%#A35#.):):4  ˆ:P‰SJ
+ Gián tiếp::;='7'X778
BÀI TẬP
HT 41. :&% =3>!
,
Oxyz
%78
1 1 2
:
2 1 2
x y z
d
− + −
= =
−
*

2 1 2
x y z
+ − −
∆ = =
−


HT 42. :&%=3>!
,
Oxyz
%
( ) : 1 0
P x y z
+ + − =
*
(1;2; 3)
A
7
&;78
d
2@$(
Giải
:4
( )
d P
⊥
A
d
!X7.
(1;1;1)

:
2 1 1
x y z
d
+ − +
= =
−
*
(1;2; 3)
A
7&;78
∆
2@v
1 2
, .
d d

Giải
J>
1 2
, ,
u u u
  
K7LX7
1 2
, ,
d d
∆



, (1; 3; 3)
u u u
= = −
  

D47&;78
1 2 3
:
1 3 3
x y z
− − −
∆ = =
−


HT 44. :&%=3>!
,
Oxyz
%
( ) : 1 0
P x y z
+ + + =
H
( ) : 2 0
Q x y z
+ − =
*
(1;2; 3)
A
7&;78

d Q u n
 
 
⊥
 
 
⇒
 
 
⊥
 
 
 
 
 
M&4
d
X7
[ ]=(-3;2;1)
1 2
;u n n=
  

D47&;78
1 2 3
:
3 2 1
x y z
d
− − −

Giải

J>
u

X778
.
∆

(2;1; 3)
d
u =

X7
.
d

(1; 1; 1)
P
n
= − −

$(
:.
/ /( )
d
P
d u u
P u n
 

1 1 2
:
2 5 3
x y z
− − +
∆ = =
−


HT 46. :&% =   3 %C !
,
Oxyz
% 78 
1 2
:
1 2 1
x y z
d
+ −
= =
−
 $
t R
∈
(   
( ) : 2 2 3 0
P x y z
− − − =
7&;"#78∆_&A$(4V$B(
Giải

∆ ⊂ ⊥
 
 
⇒
 
 
∆ ⊥ ⊥
 
 
 
 
 
M&4
∆
X7.
[ ]
, (3;0;3)
P d
u n u
∆
= =
  

D47&;78∆.
1
3
1
x t
y
z t

(2;1; 1)
u
∆
= −

J>PoB∩∆Jv"}
(1 2 ; 1 ; )
H t t t
+ − + −
⇒
(2 1; 2; )
MH t t t
= − − −


:%zi4
d
⊥ ∆ ⇒
MH u
∆
⊥


⇔
2(2 1) ( 2) ( ) 0
t t t
− + − − − =
⇔
2
3



 
GV.Lưu Huy Thưởng 0968.393.899  
BỂ HỌC VÔ BỜ - CHUYÊN CẦN SẼ ĐẾN BẾNx
HT 48. :&% =3&l%C !
,
Oxyz
%
( ) : 2 2 1 0
P x y z
+ − + =
*@$HpH6(4
F$IH<Hw(YD7&;78
d
;78@F&A$(
Giải
J>G;@&A$(
J>
∆
782
A
$(
M&4
( ).
C P
= ∆ ∩

:.
( )



:4
C
∈ ∆
A"&
(1 , 7 2 , 1 2 )
C c c c
+ + − −

=4
( )
C P
∈ ⇒
1 14 4 2 4 1 0
c c c
+ + + + + + =
2
c
⇔ = −

D4
( 1;3;3)
C
−

J>0;F&A$(:71&A.
(3; 0;2)
D




− =




− + − =



&A
: 2 5 0
P x y z
− + + =

Giải
7&;"#B.
4
3
7
2
2
x t
y t
z t


=





 
:
3 3
0; ;0 , 0; ;0 ( )
2 2
B d B P
   
 
 
 
− ∈ − ∉
 
 
 
 
   

J>
( ; ; )
H x y z
;F&A$(:;7L
4 7 4
; ;
3 6 3
H
 




 = +



 = +




HT 50. :&%=3>!
,
Oxyz
>@4F4GK7L%*
(
)

: 6 2 3 6 0
P x y z
+ + − =


, ,
Ox Oy Oz
YD7&;78B2h78&q%C@FGe8
$(
Giải
:.
( ) (1; 0;0); ( ) (0; 3; 0); ( ) (0; 0;2)

⇒7&;78B.
1
6
2
3
2
2
1 3
x t
y t
z t



= +






= +





= +




⇒7&;
( ) : 5 2 9 0
ABC x y z
+ + − =

J>&1h@FG
( ; ; )
H a b c
4=3.
 
( )
. 0
2 3 2
. 0 3 0 1 (2;1;1)
5 2 9 1
BH AC
a b c a
CH AB a b c b H
a b c c
H ABC


 
=

 
− + = =

 

∆


⊥


 
⇒ = = −

 
 

⊥



 
  
 

D7&;78
2 1 1
:
12 2 11
x y z
− − −
∆ = =
−

http://www.Luuhuythuong.blogspot.com


= − +



 = −


BX7
(2;1; 1)
u
= −


J>P; &AB⇒
(1 2 ; 1 ; )
H t t t
+ − + −
⇒
(2 1; 2 ; )
MH t t t
= − − + −


: P⊥B⇔
. 0
MH u
=








 


7&;78∆.
2 1
1 4 2
x y z
− −
= =
− −

 
GV.Lưu Huy Thưởng 0968.393.899  
BỂ HỌC VÔ BỜ - CHUYÊN CẦN SẼ ĐẾN BẾN<w
J> ′*#+W 2B⇒P&* ′⇒
8 5 4
; ;
3 3 3
M
 


′

− −

x t
y t
z t


= − +



= −



 = +


 $(::

(1; 3; 2)
n
=


Jv"}S$−/EH<−<H</<(∈B⇒
(3 3; 2 ;2 2)
MN t t t
= − − −


)* Stt$(;

(1; 2;1), (1; 3; 3) , ( 3; 2; 1)
P Q P Q
n n n n
 
= − = − ⇒ = − − −
 
 
   

::M$B(.
1 2 , , 1
x t y t z t
= + = = +
J>@o$B(∩$∆(⇒
(1 2 ; ;1 )
A t t t
+ +

0%@⊂$(A.
1 2 2 1 0 2
t t t t
+ − + + = ⇔ = −
⇒
( 3; 2; 1)
A
− − −

:%v.
, ( 3; 2; 1)
P

3 2 1
x y z
+ + +
∆ = =


HT 55. :&% =   3 %C !
,
Oxyz
% E *
(1;2; 1), (2;1;1), (0;1;2)
A B C
−
  78 
1 1 2
( ) :
2 1 2
x y z
d
− + +
= =
−
YD7&;78∆2&1h@FG4_&%
$@FG(78$B(
Giải
:
(1; 1;2), ( 1; 1;3) , ( 1; 5; 2)
AB AC AB AC
 
= − = − − ⇒ = − − −

 

 

= ⇔ + − = ⇔ = ⇒
  
  
  
  ∈ + + = =
  
 



 
 

 
GV.Lưu Huy Thưởng 0968.393.899

BỂ HỌC VÔ BỜ - CHUYÊN CẦN SẼ ĐẾN BẾN<
0%$∆(⊂$@FG($B(A.
, (12;2; 11)
ABC
ABC d
d
u n
u n n
u u
∆


HT 56. :&% =   3 %C !
,
Oxyz
%   $(.
2 5 0
x y z
+ − + =
4 78 
3 1 3
:
2 1 1
x y z
d
+ + −
= =
*
( 2;3; 4)
A
−
7&;78∆_&A$(42%*B
$(4e8B:;* &A∆"%%=%v@ Vg
Giải
J>FoB∩$(⇒
( 1;0;4)
B
−
;
( )
P

 
 

0%*>
1
, (1; 1; 1)
3
P d
u n u
∆
 
= = − −
 
 
  
⇒:∆.
1
4
x t
y t
z t


= − +



= −



3
t
= −
⇔
7 4 16
; ;
3 3 3
M
 



−




 
D@ CJ:SS=
7 4 16
; ;
3 3 3
M
 



−



Giải
J>
,
d
u u
∆
 
K7:GB
∆
H
P
n

::$(
)
2 2 2
( ; ; ), ( 0)
d
u a b c a b c= + + ≠

;B_&%$(A.
P d
n u
⊥
 

⇒
– 0
a b c
+ =

c ac c c+ = ⇔ = = −

 /
0
c
=
.>
1
a b
= =
⇒::MB.
3
1 –
1
x t
y t
z


= +



= −



=









 
GV.Lưu Huy Thưởng 0968.393.899  
BỂ HỌC VÔ BỜ - CHUYÊN CẦN SẼ ĐẾN BẾN<<
HT 58. :&% =  %C !
,
Oxyz
% 78  B.
3 2 1
2 1 1
x y z
− + +
= =
−
   
( ) : 2 0
P x y z
+ + + =
J> %*B$(7&;78
∆
_&%$(4
Be8=%vn 
∆
i_
42

n =

4B:G
(2;1; 1)
d
u
= −


;
∆
_&%$(BA:G
, (2; 3;1)
d P
u u n
∆
 
= = −
 
 
  

J>
( ; ; )
N a b c
; &A
∆
4=
( 1; 3; )
MN a b c

2 0
2 3 11 0
( 1) ( 3) 42
a b c
a b c
a b c



+ + + =



− + − =




− + + + =



⇒S$sH6<H6s(%S$6EH6IHs(
 •S$sH6<H6s(⇒7&;
5 2 5
:
2 3 1
x y z
− + +
∆ = =

' :
1 1 3
x y z
+
∆ = =
7&;78
d
_&%$
α
(V∆′HB
$∆(k%=%v5-i_
6
2

Giải
$α(::
(1;1; 1)
n
= −

4∆:G
( 1; 1;1)
u
∆
= − −

⇒∆⊥$α(
J>
( )
A

u a b c
=

:G$B(⇒
. 0
d
u n a b c
= + − =
 
$(
 
d
u

='7
AB

$<(
 :.
( , ) ( , )
d d d B d
∆ =
⇒
,
6
2
d
d
AB u
u



=



 •
0
a
=
G>
1
b c
= =
⇒
(0;1;1)
d
u =

⇒
0
:
1
x
d y t
z t


=




=



= −



= −





 
GV.Lưu Huy Thưởng 0968.393.899

BỂ HỌC VÔ BỜ - CHUYÊN CẦN SẼ ĐẾN BẾN<E
Dạng 3: Viết phương trình đường thẳng liên quan đến hai đường thẳng khác

HT 60. :&% =   3 %C !
,
Oxyz
 7 &; 78      78 .
1
7 3 9
:
1 2 1

1
∆
.
7 '
3 2 '
9 '
x t
y t
z t


= +



= +



 = −



J> SK7L%*78∆

∆
<
 ⇒
(7 ; 3 2 : 9 ), (3 7 ;1 2 ;1 3 )
M a a a N b b b

 
 
   
⇔ ⇔ ⇒
   
   
=
⊥ =
   
 
 
 
 
   
   

)78∆j78 S
D47&;78
3 1 1
:
2 1 4
x y z
− − −
∆ = =

 
HT 61. :&%=3%C!
,
Oxyz
7&;78B2*

2
2 1 1
:
2 3 5
x y z
d
− + −
= =
−

Giải
:7&;78.
1
1 1
1
5 3
: 7 2
x t
d y t
z t


= −



= − +




1 2
,
A d d B d d
= ∩ = ∩
⇒
1 1 1
(5 3 ; 7 2 ; )
A t t t
− − +
4
2 2 2
(2 2 ; 1 3 ;1 5 )
B t t t
+ − + −

1 1 1
( 3 9;2 2; 3)
MA t t t
= − + − −

4
2 2 2
(2 6; 3 4; 5 2)
MB t t t
= + + − −


1 2 1 2 1 2 2 1 2 1 2
, ( 13 8 13 16; 13 39 ; 13 24 31 48)
MA MB t t t t t t t t t t t




=




⇒
( 1; 3;2), (2; 1;1)
A B
− − −
⇒
(3;2; 1)
AB
= −


)78B2 $6IH6sHE(:G
(3;2; 1)
AB
= −

⇒
4 3
: 5 2
3
x t
d y t
z t

α
(7&;
1 2
2
1 1 2
: 5 3 , : , ( ) : 2 0
1 1 2
x t
x y z
y t x y z
z t
α


= +


− + +

∆ = + ∆ = = − + + =



 =


  7 &; 78  B  2 %
*
1
∆

 
 
⇔ ⇒ −
 
 
= =
 
 
 
− + + = = −
 
 
 

:&lN:G
(0;1;0)
j
=


J>B782@V
2
∆
C
(1 ; 1 ; 2 2 )
B t t t
+ − + − +


( ; 3;2 1); 0 3 (3;0; 5)


HT 63. :&% =   3 %C !
,
Oxyz
%      78   7 &;
( ) : 3 12 3 5 0
P x y z
+ − − =

( ) : 3 4 9 7 0
Q x y z
− + + =
4
1
5 3 1
:
2 4 3
x y z
d
+ − +
= =
−
4
2
3 1 2
:
2 3 4
x y z
d
− + −

4
2
d
:G
2
( 2; 3; 4)
u
= −


J>
1
( 5 2 ;3 4 ; 1 3 )
A d A a a a
= ∆ ∩ ⇒ − + − − +
H
2
(3 2 ; 1 3 ;2 4 )
B d B b b b
= ∆ ∩ ⇒ − − + +

:.
( 2 2 8;3 4 4;4 3 3)
AB b a b a b a
= − − + + − − +


J>
1
[ ; ] (8; 3; 4)

. 3 4 4 3
4 3 3 4
b a k
AB k u b a k
b a k


− − + =



⇔ = ⇔ + − = −



 − + = −




1
1
1
a
b
c


=


3 1 2
:
8 3 4
x y z
+ + −
∆ = =
− −





