1
PHẦN I : MỞ ĐẦU

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&
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 7)8(96 ,2
0::3$%;3
#0<4=$!>?!@9A$!>
B&
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-$EE5!"6344,!3F
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9
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M
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K
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H
4,
J
%
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K
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)*+,/E5(8)
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2+>&
T!4UV()*+,9
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%!X YZ[S"25()
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()*+,3O
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I/ TÓM TẮT LÝ THUYẾT LIÊN QUAN
1/ Phương trình tham số của đường thẳng trong mặt phẳng
)*+,"2D0<45
  
- ` .M x y
 EOC
- ` .u a b
r
 Y


x x at
y y bt
= +


= +

Y
# #
a b+ ≠
2/ Phương trình tham số của đường thẳng trong không gian
a)*+,/3"2D0<45
   
- ` ` .M x y z
 EOC
- ` ` .u a b c
r
 Y


x x a t
d
y y b t
= +


= +

# # #
#
# # #
Y
x x a t
d
y y b t
= +


= +

+c)Y
   # # #
   # # #
x a t x a t
y b t y b t
+ = +


+ = +




= +


# # #
# # # #
# # #
Y
x x a t
d y y b t
z z c t
= +


= +


= +


+c)Y
   # # #
   # # #
   # # #
x a t x a t
y b t y b t
z c t z c t
+ = +




T!4d

 52+>Dd<45e$%*+,/"Y



#
#
#
C M C
C M C
C M C
x x x
y y y
z z z

= −


= −


= −


6/ Các bài toán liên quan :
Bài toán 1 : Tìm hình chiếu của một điểm M trên một đường thẳng d :
Cách 1 :


 9
#
bE"9

Y


#
g
x t
y t
= +


= +

39
#

#
#

l #
x t
y t
= +


= −


EOC

-`.u =
ur
E
   
&     CM u t t t
= ⇔ − + + + = ⇔ =
uuuur ur
7@,e-#`g.
m"TY
T∈9
#

⇒
# #
- `l # .N t t+ −
# #
- # `# # .CN t t
= − + −
uuur
9
#
EOC
#
-` #.u
= −
uur
E




= − = − =


7@,d

-`l.
d
#
 52+>Dd<49
#
4,T 45dd
#
[E
#
#
## q
# g
o o
n n
# #
o o
C N C
C N C
x x x
y y y

= − = − =

g #
x t
y t
z t
= +


= +


= −

39
#
Y
#
#
#

l #
g
x t
y t
z t
= +


= −



-  ` # .CM t t t
= − + + −
uuuur
9

EOC

-`` #.u = −
ur
E
    
&    l  CM u t t t t
= ⇔ − + + + + = ⇔ =
uuuur ur
7@,e-#`g`g.
m"5TY
T∈9
#

⇒
# # #
- `l # `g .N t t t+ − +
# # #
- # `# # ` .CN t t t
= − + −
uuur
9
#
EOC
#

= − = − =


= − = − =


= − = − =


7@,d

-`l`g.
d
#
 52+>Dd<49
#
4,T 45dd
#
[E
#
#
#
# l g 
# l # #
# p g o
C N C
C N C
C N C
x x x
y y y

= +


= +

# # #
#
# # #
Y
x x a t
d
y y b t
= +


= +

B,"Cr3'0Y
GV : Trương Tử Trang THPT Vĩnh Thạnh
C
M
N
A B
6
&s
 #
3d d
 0&
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3d d
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tdrE7d 

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ae 45d'4,'O
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&f()E
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GV : Trương Tử Trang THPT Vĩnh Thạnh
C
A B
M
N
C
M
N
A B
d
1
d
2