Chuyên đề Hình học Euclide
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Hình học Euclide
Đề tài thuyết trình:
“ Thể tích m – chiều của m – hộp,
m – đơn hình trong không gian
Euclide n – chiều.”
Nhóm thực hiện: Nhóm 3 – Toán 2A.
1. Một số khái niệm liên quan đến m – hộp và m – đơn hình trong không
gian Euclide n – chiều E
n
(V
n
):
a. Tâm tỷ cự:
m
{ }
m
A A A
m!"
m
a a a K∈
!
#
m
i
i
a
=
≠
∑
$%&%'(
)%
n
G E∈
!*
#
m
i i
i
a GA
=
=
∑
uuur
+,-%-./0 %1%2
34
{ }
m
A A A
567!"
m
a a a K∈
89 *
{ }
: ::
m m
A a A a A a
G G=
b. Tập lồi:
*
;<+'%=*
1>%?@%AB'%=?@.!*
+C*
{ } { }
* D E < # AB C A B ABC A B= ∈ + < ∪
F6?@ GH%4
'%=?@
+C*
[ ]
{ }
{ }
:
* D # *
n
A B
AB M E M G
λ λ
λ
−
= ∈ ∃ ∈ =
;<&*
I%%J%
%%&K60?@%
J%L'%=?@M % %J
$4N%& %&J(%&%'%&>O)%M
%PJ%
40%&5J0 >&4J
89 *PEJ<
F(>& %&>O)%%%)%QN%&5J
c. Định nghĩa hình hộp m – chiều trong không gian Euclide E
n
(V
n
)
;<+AB*
EF
<R m%S
%(K%9
{ }
m
u u u
TLU7A>VR
{ }
m
u u u
%/7I%WX*
D :#
m
n
i i i
i
M E IM t u t
=
∈ = ≤ ≤
∑
uuur
8Y#%L#
8Y%L '%=
8Y%L L>L
8YZ%LZ "L%AB%%0[
%
;<+A\*%
EF
< %%&
d. Định nghĩa m – đơn hình trong không gian Euclide n – chiều E
n
(V
n
):
+C*
EF
<m+1
{ }
#
m
A A A
@&5
{ }
#
m
A A A
m SL6]
#
m
A A A
+C*+SLm57
#
m
A A A
9
( )
#
m
S A A A
%/7I%W*
[ ]
{ }
# #
#
: ::
#
D # *
m m
m
n
m i
A t A t A t
i
M E t t t t M G
=
∈ ∃ ∈ = =
∑
8Y#%L#SL
8Y%LSL '%=
8Y%LSL %7
8YZ%LZSL "%5
;<+A\*SL%
EF
< %%&
2. M ột số vấn đề về ma trận và định thức Gram:
P%S^
n
E
V
%S
{ }
m
u u u
_O%%%'>V7%9.64%S%-*
( )
E <
m
m
m
m m m m
m
u u u u u u
u u u u u u
Gr u u u
u u u u u u
=
I%%-0 %,4%S
{ }
m
u u u
,0
( )
i i ni
a a a
%04%S
:
i
u i m∀ =
%%S!V%3`
$4
n
E
V
_O%%*
E <
m
m
n n nm
n m
a a a
a a a
A
a a a
×
=
+A\*
I%
( )
m
Gr u u u
%"U5
( )
t
m
Gr u u u A A=
Z
( )
% #
m
Gr u u u ≥
a)bYc]UQ( ]%S
{ }
m
u u u
d%%(K%9
e
{ }
m
u u u
%3 ]
( )
m
Gr u u u
%.fO
{ }
m
u u u
%3` ]
( )
m
Gr u u u
%SA
P5*
I%
( )
m
Gr u u u
%"U5
T-L%$%9)%4%9$.6 *
:
i j j i
u u u u i j= ∀
( )
t
m
Gr u u u A A=
$*
( )
E <
n n n
i i i
m
n n n
m
i i i
m
m m m m
m
a a a a a
u u u u u u
u u u u u u a a a a a
Gr u u u
u u u u u u
= = =
= = =
= =
∑ ∑ ∑
∑ ∑ ∑
n n n
i i i
n m
n
m m nm
a a a a a
a a a a a a
a a a a a a
a a a
= = =
=
∑ ∑ ∑
E <
m
t
n n nm
A A dpcm
a a a
=
ZP%*
$*
( )
( )
% % % E < #
t
m
Gr u u u A A A= = ≥
a)bYcUQ(
%?Y#
_O%.S%L
# # #E<
m
m
m m
n n nm m
x
a a a
x
a a a
x u x u x u A X
a a a x
+ + + = ⇔ = ⇔ =
a%?Y#-E<$7%G%.fJ(
{ }
m
u u u
d%%(K%9
PQ*
CK%S
{ }
m
u u u
d%%(K%9%5%&%'%S
i
u
>%A
%(K%9g7%Sh'*
j m
i i i i i m m j j
j i
u b u b u b u b u b u b u
=
− − + +
≠
= + + + + + + =
÷
∑
( )
i m
i m i i i i m
m m i m m
j m
j j m
j i
j j
j i
u u u u u u
Gr u u u u u u u u u u
u u u u u u
u u u b u u u
b u
=
≠
≠
=
÷
=
∑
j m j m j m j m
j j j j j j m
j i j i j i
j m
m m j j m m
j i
u b u b u b u u
u u u b u u u
= = = =
≠ ≠ ≠
=
≠
÷ ÷ ÷ ÷
÷
∑ ∑ ∑ ∑
∑
%)(%%5 >%A%(K%947%h'-
( )
% #
i m
Gr u u u u =
eP%*
T%S
{ }
m
u u u
%3%5
#:
i j
u u i j= ∀ ≠
J(
( )
m
Gr u u u
%.fO
( )
%
m m
Gr u u u u u u=
T%S
{ }
m
u u u
%3`%5
-Yi
#- i
i j ij
u u
δ
= =
≠
J(
( )
m
Gr u u u
%SA
( )
%
m
Gr u u u =
PQ*
CK
( )
m
Gr u u u
%.fO%5 7
#:
i j
u u i j= ∀ ≠
5
i j
u u
%%36J(%S
{ }
m
u u u
%3
CK
( )
m
Gr u u u
%SA%5
#: :
i j i i
u u i j u u i m= ∀ ≠ = ∀ =
J(%S
{ }
m
u u u
%3`
3. Thể tích của m – hộp trong không gian Euclide n – chiều E
n
( V
n
):
a. Định nghĩa:
AB%%94%
9
( )
( )
m
V H I u u u
>Mg('*
Y*
( )
( )
V H I u u=
j*
( )
( )
( )
( )
m m m
V H I u u u V H I u u u h
−
=
F6
m
h
Q7%k]
m m m
S u IS=
uuur
KE<=
gR6%S].S
m
u u u
α
−
=
ur
PH\*
87%%956Y$ '%=h56Y
$ %94L>L
b. Định lý:
@L.S%%94gR3%%S
{ }
m
u u u
>MA%5,4%S
{ }
m
u u u
5 *
( )
( )
( )
( )
%
m m
V H I u u u Gr u u u=
(
( )
( )
( )
%
m m
V H I u u u Gr u u u=
P5*
@Mg('%%$*
Y*
( )
( )
V H I u u=
Y
( )
%Gr u
j*,Q!lH6E<%5*
( )
( )
( )
( )
%
m m
V H I u u u Gr u u u
− −
=
PG5H6%%(*
+m%
m
u u u
α
−
=
ur
m m
u x h= +
r uur
6
:
m m i
x u u u h u i m
α
−
∈ = ⊥ ∀ = −
r ur uur
%$*
m m
h h=
uur
_O%
( )
( )
( )
( )
( ) ( ) ( ) ( )
% %
m m
m
m
m m m m
Gr u u u Gr u u x h
u u u u u x h
u u u u u x h
x h u x h u x h x h
= + =
+
+
= =
+ + + +
r uur
r uur
r uur
r uur r uur r uur r uur
m m
m m m m m m
u u u h u u u h u u u x
u u u x
h u h h x u x h h u h
x u x x
= + + +
uur uur r
r
uur uur uur r r uur uur urr r r
( ) ( )
% %
m m m
x
Gr u u u x Gr u u u h M N
− −
= + + +
u r
r uur
$*
#:
m i m i
h u h u i m⊥ ⇒ = ∀ = −
uur uur
m
x u u u
α
−
∈ =
r ur
!(
#
m
h x =
uur r
J(
#
#
#
m
m
u u
u u u h
M
x u
x u x h
= = =
uur
r
r r uur
#
# #
m m
u u u x
u u u x
N
h u h x
= = =
r
r
uur uur r
( )
% #
m
Gr u u u x
−
=
r
L
{ }
m
u u u x
−
r
d%%(K%9
F(
( )
( ) ( )
% % %
#
#
m
m m m m
m m m
m
m m m
m
u u u h
Gr u u u Gr u u x h Gr u u u h
h u h h
u u u u
u u u u
h
−
−
− − −
= + = =
=
uur
r uur uur
uur uur uur
u
( )
%
m
m m m
m m m
m
u u u u
h h Gr u u u
u u u u
h
−
−
− − −
= =
uur
ur uur
F(
( ) ( ) ( )
( )
( )
( )
( )
( )
% %
E<
m m m m m
m m
Gr u u u h Gr u u u h V H I u u u
V H I u u u u
− −
−
= =
=
Hệ quả:
CK
{ }
m
u u u
%3%L
( )
( )
m m
V H I u u u u u u=
CKY ? %47%%04
{ }
n
u u u
%%S!V%3` $%L%%94L>M7
%A%(%"4A%5?%5 *
( )
( )
%
n
V H I u u u A=
Z P
( )
* →
ur
n n
f f E E
%O>K[nn
{ }
n
I u u
%o(\%
%L
( ) ( ) ( )
{ }
n
f I f u f u
ur ur
p % %%94$*
( ) ( ) ( )
( )
( )
( )
( )
%
n n
V H f I f u f u f V H I u u=
ur ur ur
P5*
+1( %9)%./!(%kA%5,
F 1(p O5%[g7%%5%9%%94
hình hộp chữ nhật%Z!S)*
V abc=
6>G
./% >9%.64LN%
%5%9%%94 A%5,%$*
( )
( )
( ) ( )
% % % E <
n n
V H I u u u Gr u u u A A dpcm= = =
Z,0
E < *
n
α α α α
%S!V%3`%
0
( )
i i ni
a a a
%04%S
:
i
u i n∀ =
%
E <
α
_O%
n
n
n n nn
a a a
a a a
A
a a a
=
,0
( )
i i ni
b b b
%04
E <f
α
ur
%
E <
α
%$*
n
n
n n nn
b b b
b b b
B
b b b
=
%[S!V%k
E <
α
!
E <f
α
ur
p
%4O>K[%(K%9
f
ur
%
E <
α
$*
( ) ( ) ( )
( )
( )
( ) ( ) ( )
( )
% %
n n
V H f I f u f u Gr f u f u f u C= =
ur ur ur ur ur
6P %7%%04
( )
i
f u
ur
%S!V%3`
E <
α
$*
( ) ( ) ( ) ( )
( ) ( ) ( )
n n
n n
n n
n n n nn n
f u a f a f a f
u a a a
u a a a
f u a f a f
u a a a
α α α
α α α
α α α
α α
α α α
= + + +
= + + +
= + + +
= + +
⇒
= + + +
ur ur ur ur
ur ur ur
( )
( ) ( ) ( ) ( )
n n
n n n nn n
a f
f u a f a f a f
α
α α α
+
= + + +
ur
ur ur ur ur
$*
( )
( )
( ) ( )
[ ]
( )
( )
( ) ( ) ( ) ( )
q
q
q q
f
f f f f
f u T u dstt
C A T A B A
α α
α
α
α α α α
→
→
=
⇒ = = =
F(*
( ) ( ) ( )
( )
( )
( ) ( ) ( )
( )
( )
( )
% %
% % % % % E <
n n
n
V H f I f u f u Gr f u f u f u C
B A B A f V H I u u dpcm
= =
= = =
ur ur ur ur ur
ur
F6
( )
( )
%
n
A V H I u u=
E%gQ<
% %B f=
ur
c. Ví dụ minh họa:
F9d*m%=r
6S!V%3-E9%s<R
%So.S
( ) ( )
u a a v b b= =
9%9L
>L U7A>V
( )
I u v
,Q*
,0? %4 %S!V%3-%$*
a b
A
a b
=
a%9L>L U7A>V
( )
I u v
%9>V%5!*
( )
( )
% E < %V H I u v Gr u v A a b a b= = = −
F9d*
e
E
6S!V%3`9%s
# Z
E < E##< E## < E#<A A A A− −
9%%9ZU7
A>V
# # # # Z
E : <A A A A A A A
uuuur uuuuur uuuur
,Q *
kQ%(K%%$*
# # # Z
E <: E# <: E## Z<A A A A A A= − − = − − − = −
uuuur uuuuur uuuur
_O%%,%'>V%S
# # # Z
A A A A A A
uuuur uuuuur uuuur
.
( )
# # # # # # Z
# # # Z # # # # # # Z
# Z # # Z # # Z
A A A A A A A A A A A A
Gr A A A A A A A A A A A A A A A A A A
A A A A A A A A A A
=
uuuur uuuur uuuur uuuuur uuuur uuuur
uuuur uuuuur uuuur uuuuur uuuur uuuuur uuuuur uuuuur uuuur
uuuur uuuur uuuur uuuuur uuuur
# Z
t # t
# Z
t #
A A
= −
−
uuuur
%9ZU7A>V
# # # # Z
E : <A A A A A A A
uuuur uuuuur uuuur
*
( )
Z # # # Z
% Zu
hop
V Gr A A A A A A
−
= =
uuuur uuuuur uuuur
F9dZ *
e
E
6S!V%3`9%s7
# Z e
E###< E###< E< E# #< E# #<A A A A A− −
9%%9e
U7A>V
# # # # Z # e
E : <A A A A A A A A A
uuuur uuuuur uuuur uuuuur
,Q *
kQ%(K%%$*
# # # Z # e
E# #<: E#<: E# #<: E# #<A A A A A A A A= − = = − = −
uuuur uuuuur uuuur uuuuur
_O%%7%%04%%-
# #
#
#
# # #
A
− −
=
−
$*
% #A = − ≠
C-%%-%% $%$%%9e *
e
%
hop
V A
−
= = − =
4. Thể tích của m – đơn hình trong không gian Euclide n – chiều E
n
( V
n
):
a.Định nghĩa:
AB%%94SL
( )
#
m
S A A A
>Mg('
%*
Y*
( )
( )
# #
V S A A A A=
uuuur
j*
( )
( )
( )
( )
# #
m m m
V S A A A V S A A A h
m
−
=
$
m
h
Q7%k
m
A
KE<=g
#
m
A A A
−
PH\*
87%%956Y$ '%=h56Y
$ %94%7
b. Định lý:
%94SL
( )
#
m
S A A A
./%9>M
%5!*
( )
( )
( )
( )
( )
# # # # # # #
%
v v
m m m
V S A A A V H A A A A A Gr A A A A A A
m m
= =
uuuur uuuuur uuuur uuuuur uuuuur
P5*
P5>Mg('%
;<F6Y%$*
( )
( )
( )
( )
# # # #
v
V S A A A A V H A A A= =
uuuur uuuur
EH<
;<F6jQ!lH6E<%5*
( )
( )
( )
( )
( )
# # # #
v
m m
V S A A A V H A A A A A
m
− −
=
−
uuuur uuuuuur
PG5H6%%(*
( )
( )
( )
( )
( )
( )
( )
( )
( )
# # # # #
# # #
v
E <
v
m m m m m
m
V S A A A V S A A A h V H A A A A A h
m m m
V H A A A A A dpcm
m
− −
= =
−
=
uuuur uuuuuur
uuuur uuuuur
c. Ví dụ minh họa:
F9d*
Z
6S!V%3`9%s>
?EZe^<@E#Z<PE^Zue<P5M>%-%=
X(%9%9%7?@P
,Q*
$*
( ) ( )
ee : wuAB AC= − − = −
uuur uuur
$
e e
w u
− −
≠ ≠
−
->?@P
%=
( )
ZZ
w
AB AB AB AC
Gr AB AC
AC AB AC AC
= =
uuur uuur uuur uuur
uuur uuur
uuur uuur uuur uuur
a%9%7?@P *
( )
( )
ZZ
% % uw
w
v v
ABC
S Gr AB AC dvdt
= = =
uuur uuur
F9d*
Z
6S!V%3`9%s>"
?E^Z<@E#<PE^Ze<aEe^Z<TX(%9%%9%5?@Pa
,Q*
kQ%K%%$*
( ) ( ) ( )
e : e Zt : #AB AC AD= − = − − =
uuur uuur uuur
_O%%7%%04%S
: :AB AC AD
uuur uuur uuur
%$*
e
Z
e t #
A
−
= − −
$*
% ex #A
= − ≠
-
: :AB AC AD
uuur uuur uuur
%(K
%9
%94%5?@Pa *
% ex xE <
Zv w
ABCD
V A dvtt= = =
F9dZ*SL
#
E <
m
S P P P
0 SLKQ7
N]
i j
P P
>)%L>M
<P5M%0%1G4SL%
n
E
77]
4SL$
><9Q7%k%0%1G4SLK%]4$
>K%Q7N]4SL$
<9%%94SL>K%Q7N]4SL
$
,Q*
<
,0G %0%1SL
#
E <
m
S P P P
E < #
i j
d d P P i j m= ∀ ≤ ≠ ≤
$
( )
# #
#
# #
m m
i i
i i
GP m GP P P
= =
= ⇒ + + =
∑ ∑
uuur uuur
# #
m
i
i
GP P P
m
=
⇒ = −
+
∑
uuur uuuur
( )
( )
( )
( )
( )
# # # #
! E <
Z
E <
m
i i j
i i j
GP P P P P P P
m
m
md d m
m
m
d m m
m
m m
d
m
m
d
m
π
= ≠
⇒ = +
÷
+
= + −
÷
+
= + −
÷
+
+
=
+
=
+
∑ ∑
uuur uuuur uuuuruuuur
P5%.S%3
$
( )
k
m
GP d k m
m
= =
+
uuur
F(%0%1G4SL%
n
E
77]4SL
$
><
k1<%$
( )
#
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Danh sách sinh viên nhóm 3:
1. Đinh Văn Dương.
2. Trần Tuấn Anh.
3. Cao Văn Hoàng.
4. Lê Khắc Hiếu.
5. Huỳnh Phương Nam.
6. Lư Tấn Cường.
7. Bùi Thị Thanh An.
8. Đỗ Hồng Hiệp.
9. Nguyễn Hoàng Khôi.
10. Võ Thanh Hải.
11. Hà Thị Nguyệt.
12. Trần Huỳnh Thảo Ly.
13. Phạm Nguyễn Trà My.
14. Lương Hoàng Khương.
15. Lưu Quốc Anh.
16. Nguyễn Hữu Lợi.( K33.101.062 )
