Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.

Géométrie dans un espace affine euclidien

459 views

Published on

www.touscours.net, Groupes,Permutations,Anneaux,Arithmétique dans Z,Corps commutatif,Les polynômes formels à une indéterminée à coefficients dans un corps K,Fonctions polynomiales,racines,Espaces vectoriels,K-algèbres,Espaces vectoriels de type fini,Matrices,Déterminants,Fractions rationnelles,Produit scalaire sur un R-ev,Espace vectoriel euclidien,R-ev euclidien orienté de dimension 2,R-ev euclidien orienté de dimension 3,Espaces affines,Géométrie dans un espace affine euclidien

Published in: Education
  • Be the first to comment

Géométrie dans un espace affine euclidien

  1. 1. ! " # ε $ % & ' # "(= ) % * • ="( " % +# "("( →×εε , $ + , =⇔= -"( , "("( = , "("("( +≤ • . ε " ε " { }∈= ""("( • . &" / ε " { }&&&& """("( ∈∈= 0 1 # " # "2 3 , % 4/ . &" / &" " # 5"( & # 5"( & % ' 3 6 , # ε∈ -≠ " # { }+ ∈+ λλ "% % +# / , # % touscours.net
  2. 2. & 7 8 . , ε % . ε∈ % + 8 = # 8 9 # 3 3 ⊥ ( : ,$, 3 ⊥ ∈ ) % ; + 8 # 3 3 "("( = % < 8 % ∈ " &&& += % ≥ " = = % 4 ∈ " ( ∈ = % > + = • +#= ε∈ $ { }-?∈ " # { }⊥∈= "ε • @ ' 3 A + # ⋅ →ε " ) ε∈ { }-?∈ ∈ " # ⋅ " # { }=⋅∈= "ε % < "( 3 ∈ % 4 " 3 & % = % " ε∈ -=⋅⇔ ⋅=⋅⇔ =⋅⇔∈ #= $ % • B= / . ε∈" " % . { } { }=∈==∈= ""("(" εε . C"D % &((( && && ⋅=+⋅−=−=− -=⋅⇔= touscours.net
  3. 3. * { }-" =⋅∈= ε " E #= $ % < # #= % • # $ = % . = $ % . ε∈ F ="( ) 8 % %λ += % & %λ=⋅ % ⋅ == & %λ G + . εε → % ε ⇔ " : ,$, "(("(("" =∈∀ ε + ε / ε ε $ ( ( : ,$, • . (∈ϕ " " ε∈" " #" # "((:::":( ==== ϕ • . 3 % < 3 # % . +=ϕ % H 3 ϕ ( : ,$, 3 (∈ϕ . ∈ " ε∈ " += % = ===== ::::(( ϕϕ ( I ! ε = ε " : ,$, $ ( % ! ε = ε " : ,$, $ (?( % (J ε " ε " ( , (ε " # (ε ε , % 4/ + = (J ε % touscours.net
  4. 4. 0 4 " = $ , " " = $ ( ) % 4 " 3 ⊥ = " 3 = $ F = $ " 3 # ( . =( " =( " " = $ " 3 − " % + / ( = $ = % 4 / ε " / / 3 " # / #= #= % (! + & + 4 . " +=ϕ ( " (ε∈ (∈ϕ % < 3 ϕ " # % • ϕ # " # # % " % ' 3 " (ε % • / ϕ # θ ( π& % ' % . ε∈ " % . ε∈ % (:(( ϕ+=+= % :(J( :(:(/ =−⇔ =−⇔=⇔ ϕ ϕϕ < { } { }-("JK ( ==∈=− ϕϕ ϕ 8 " 8 ( / :(J( − −=⇔ ϕ % < / Ω % ($ :(J( − −+=Ω ϕ H (((" : Ω+Ω=Ω+Ω=∈∀ ϕε " (: Ω=Ω ϕ % touscours.net
  5. 5. 7 < 3 Ω # θ % J " " # 3 Ω Ω (∈θρ % ' 3 3 (:= $ =ΩΩ Ω=Ω C&D:5"( : πθ (ε 4 1 ∅ ; ε J ; ' { }Ω ' # Ω & / . / " & / & % . &= " εJ& = % . &≠ &LL " =& " ) &&= . Ω θ & Ω # / # θ& )θ # 5"( & (J 8 # J " L " / L Ω 3 / / L % " / / % ; / " M -" " &" * / % 4 . (J ε∈ % . " 3 # - & / % . / 3 3 " (ε∈ " - & / " − = * / % touscours.net
  6. 6. > * + . (?(J εε∈ % +=ϕ % (?(∈ϕ % ϕ / " (N= % / % " ε∈ " " (((: ϕ+=+== (: ϕ= / % J " / (?(J εε & # / % . ε∈ " (:= % < 3 % " ϕϕ =J % = / : = % H " : # ⊥ ∈∈ +=: % == : % ( ( < = % ϕϕ == J+ % H / ( " / &+= % / LL: &+ ( " = " ) / @ A $ / % (:= : ( touscours.net
  7. 7. O ! / % < = / % 4 / 1 ∅ ; ε εJ { }Ω ' # Ω ' / % J ∅ ' / ( : ,$, ) / { }-?(∈ G = 3 & . ""(= " % ! 3 =+ " ) -"-("( ≠ % ! $ " − % # $ . - - - " =+ % < . ∈ % && -- && --- - (( "( + −+ = + −+− = ⋅ = % &&&& =+ =+ θ θ +# θ " # # 5"( & % & & & & && && & & ++ + = ⋅ =θ 43 . " & & & % +#3 - & (( & (( -- & & & & = + −−+ + −− % touscours.net
  8. 8. + . εε → % ⇔ "(%("(("""P =∈∀∈∃ + ε % " . " "(%("(("""PQ =∈∀∈∃ + ε % % . εε → " -> % < 3 ⇔ # # % ⇔ # ϕ% ) (∈ϕ ⇔ % " . % . ψ % (∈= ψϕ % • . ϕ ( " 3 " 3 % • " & ( • . " " % +# ε , "(( ε % & 4 / < ) ""-( % R 3 • ; ) ∈ % !! + • .= ( / $ #/ !! touscours.net
  9. 9. • B ∈α !! %α % B -! ∈α %( -- !!!! −+α ( ---- %( α+=+ • ' # ∈θ !"! θ% ' -! # ∈θ ( - % - !!"!! −+ θ + + / = !! +% " ) ×∈ P"( % • . ×∈ P"( % # θ α % %" " ) P∈α ∈θ % !! +% !"! θ% ( α+ ( 4 %α ( # # # " % • J " % 4 # # ( : ,$, # " = !! +% 3 # = !! +% % 4 . !! +% % -! / !!! =−⇔=+⇔ --- (% . = -≠ " #= / " !! + % . = -= " # % . ≠ " / -! % !! += %( " !! += -- % % (( -- !!!! −=− % # θ α % %" ) P∈α ∈θ " 3 " " # -! α -! # θ % J 3 !! +% " ) ×∈ P"( % touscours.net
  10. 10. - 4 " # !! +% " # !! +% " % J " " !! " " " # +% " = " !! +% % / 4 1 ∅ ; J { }-! . $ " : ,$, ( # -! # θ # -! α ( % % ≠θ α " (' . C"D C:":D / ( / % / 3 C"D C:":D ( " / " " " % . ∈βα" % . βα +!! % % =+ =+ : : % % !! !! βα βα " : ,$, −= −=− !! !!!! % %( : :: αβ α % < 3 P∈α " ∈β % =:: % . " $ :: # ::"( !! !! − − = :: α " !! !! :::: = − − =α C&D::"( "(::"( ((( :: π α !!!! = −= −−−= touscours.net
  11. 11. . ε % . ""(= ε % . ε∈ & "( ∈θρ % < 3 "( θρ = 3 (% θρ= " ) (θ %% θθ + " : ,$, 3 C&D(5"( πθθ = % J 3 # 8 = " • + = / "-( θρ = = % • ! ≠ / = ∈++−=∈+= "&"(""&"( ππαρπαρ " ) α # "( % 43 " / • + #3 *=ρ ( $ # ε∈ 3 # = "( θρ *=ρ % " # # ε∈ 3 # / ∈θ 3 (%* θ= " # = *% • #3 θρ = " + ∈θ ( ∈θ π& π &Lπ &L*π J " ε *" # """(= % (+ * + 4 . (ε∈ " ϕ % touscours.net
  12. 12. & (∈ϕ % # " #/ "( ω # θ % • . C&D- πθ = " J=ϕ " % • . C&D- πθ ≠ , . # / # " 3 % ε∈ " ( 3 (: ϕ= % . " 8 % ∈ " ∈ " ϕ " ( =: % (: ϕ= % < 3 #/ "( ω # θ ω θ # / % J " = % , . # / . ε∈ " % : = % ϕϕ =J % # #/ "( ω # θ ) " θ # ϕ " : = ω θ H : : ⊥ ∈∈ +=: % 4 = % . $ % ( / ϕ ⊥ : ,$, ( M " $ " $ 3 ϕ # # % < / L % / % # #/ ":( ω # θ ) # % touscours.net
  13. 13. * = " ) #/ ":( ω # θ " % ω θ # M $ #( -≠ " #= / % < 3 ( #/ ":( ω " # θ % ( 4 1 ∅ J ; ε J εJ θρ " -≠θ #/ "( ω ) (= ' #/ "( ω # θ ∅ θρ " -≠θ #/ "( ω N #/ "( ω # θ " ) " ∈≠ - 3 " / ' E : : : : !! ) += −= −= &: &: >: !! ( # # # 3 & & > : # − − !! : " : ,$, # ϕ − − -- -- -- ""( % 4 " 3 (:((" ϕε +=+=∈∀ . − − += !! -- -- -- & & > : : : touscours.net
  14. 14. 0 (& J # # π ! ( = $ ! " #/ ! # ( =ϕ (∈ϕ % J≠ϕ " " #/ % (* ' ! = = = ⇔ += −= −= ⇔ -& * & & > !! < # " # % (0 #/ " # . #/ "( ω # -≠θ -≠ % { }$:"ε∈= " # 3 : 3 % $" − − & && &> : % = = ⇔∈ * (N: #/ ! - * % / / , . &LL " =& " ) &&= ( # &= & & , . α =∩ & " & # α& % ' 3 ! 3 / " / & ( # " " " % touscours.net
  15. 15. 7 G = 3 * 43 #! =++ " ) -"-"-(""( ≠# % ! $ # < 3 " ε∈ -=⋅⇔⋅=⋅⇔=⋅⇔∈ <) 3 %λ= & =λ % J / &&&&& !# !# =++ =++ < # " & & & & # % . -& =∧ " &LL % . -& ≠=∧ " &∩ " % 4 ⊥ " (∈ " &⊥ ( &∈ % (( &∩=∈ ( -≠ % # $ . $ % % ="( " %λ= % += % && %-% λλ =+=⋅ " & ⋅ = " &&& # #! ++ −++ = ⋅ = % # $ ="( % < ∧=∧+∧=∧ % =∧ ( ⊥ % ∧ =="( % touscours.net
  16. 16. > = 3 3 = 3 . ε∈ " ( ""( ! % < = = 3 ""( !$ θ * ""(""( !$$! θθ= % " " 3 ∈θ " %%( θθθ += " " 3 ε∈ " % 8 " 3 ""( !$ θ = = 3 $ ⇔+=⇔ !$ %(% θ "( θ$ = % ""( ! S % 4 ε = = 3 • . !∈ " = ""-( !θ " ∈θ 3 3 % • . !∉ " # # / $ = " 5"( %=θ ( "( " ! S % 4 "&"( !$ πθ + "&"( !$ ππθ ++− ∈ 3 3 % ' 3 / $ θ ( !∉ " && $ += " $=θ " $=θ % θ & 3 . ε∈ " ( ""( ! % < = 3 ""( ϕθ$ * ϕθ$= " ϕθ$= " ϕ$! = % 4 . ε∈ " ""( ! % < 8 % 8 % 3 ""( ϕθ$ = 3 $ 3 ( "( ϕθ$ = % $ ""( θ$! = % touscours.net
  17. 17. O , . = " ( 3 -== θθ $$ " 3 -=$ % ' 3 " ""( ϕθ$ 3 -=$ = 3 % , . ≠ " !∈ " ( 3 θ$! = -=θ % ' 3 " ""( ϕθ$ " : ,$, = "&"( ϕπ! "&"( ϕππ! +− ( ∈ϕ 3 3 = 3 % , 4 " !∉ "( αρ = % ""( " ( 3 +=−== === C&D C&D ππαϕρθθ παϕρθθ $!$ $!$ ' 3 " ""( ϕθ$ = 3 % J 3 ε = 3 % ' 3 . ""( ϕθ$ = 3 " 8 & & $ = 4 &&&&&&& & ( $$$ =++= θϕϕθ ' # = 3 !∉ % . ε∈ " 3 !∉ 8 % 8 " ""( ! % $ = % . [ ]πθ "-∈ # "( θ$! =⋅= % " & && !% += " θθ &&&& & ( $$% =−= " -≥$ -≥θ ( [ ]πθ "-∈ " θ$% = . ϕ # "( "( % %%($% ϕϕθ += 4 ""( ϕθ$ = 3 % ϕ θ touscours.net
  18. 18. ' 3 3 !∉ " ->$ ] [πθ "-∈ % ϕθ""$ &&& !$ ++= " = $ ! θ " θ ϕ $ = " θ ϕ $ = % " = 3 3 ] [ [ ]ππϕπθ """-"- −∈∈>$ # # " = 3 :&"&"( :&"&"( :&"&"( :&"&"( ππϕππθ πϕππθ ππϕπθ πϕπθ $ $ $ $ ++++−− +++− +++− ++ 4 3 3 ε " 3 # #/ ! " = 3 ""( ϕθ$ -≥$ " [ ]πθ "-∈ C"D ππϕ −∈ % H $ = " 3 3 ( / % touscours.net

×