seventh edition vector mechanics for engineers: staticskurtcebece/sta201-bolum2.pdf ·...
TRANSCRIPT
![Page 1: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/1.jpg)
VECTOR MECHANICS FOR ENGINEERS: STATICS
Seventh Edition
Ferdinand P. BeerE. Russell Johnston, Jr.
Ders Notu:Hayri ACARİstanbul Teknik Üniveristesi
Tel: 285 31 46 / 116E-mail: [email protected]: http://atlas.cc.itu.edu.tr/~acarh
© 2002 The McGraw-Hill Companies, Inc. All rights reserved.
2. MADDESEL NOKTALARINSTATİĞİ
![Page 2: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/2.jpg)
DÜZLEM KUVVETLER SİSTEMİ
• KUVVET VEKTÖREL BİR BÜYÜKLÜKTÜR
θ
y
x
F
F
• UYGULAMA NOKTASI
• ŞİDDETİ
• YÖNÜ
BİLİNMESİ GEREKENLER
![Page 3: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/3.jpg)
F3
F2
F1
θ1
θ2
θ3
![Page 4: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/4.jpg)
VEKTÖR ÖZELLİKLERİ
NEGATİF VEKTÖR:
TOPLAMA:
![Page 5: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/5.jpg)
F1
F2
F3
- F1 + F2 + F3 = F4
F4
F1
F2
F3
= F4
TESİR ÇİZGİSİ AYNI OLAN VEKTÖRLERİN TOPLANMASI:
![Page 6: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/6.jpg)
F1
F2 F3
x
y
z
y
x
F1
F2F3F4
F5
AYNI NOKTAYA TESİR EDEN VEKTÖRLERİN TOPLANMASI:
![Page 7: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/7.jpg)
y
x
F1
F2F3
F4
F5
y
x
F
FFFFFFrrrrrr
=++++ 54321
![Page 8: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/8.jpg)
PARALEL KENAR YASASI:
toplamaçıkarma
![Page 9: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/9.jpg)
COSİNÜS TEOREMİ:
P
Q
αP
Qβ
γ
180 - α
R2 = P2 + Q2 – 2PQCos(180-α)
R2 = P2 + Q2 + 2PQCos(α)
SİNÜS TEOREMİ:αβγ sinsinsin
RQP==
α
![Page 10: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/10.jpg)
![Page 11: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/11.jpg)
İKİDEN FAZLA VEKTÖRÜN TOPLANMASI
![Page 12: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/12.jpg)
VEKTÖRLERİN TOPLANMASI
![Page 13: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/13.jpg)
Şekildeki somuna A noktasından iki kuvvet etkimektedir. Bu kuvvetlerin bileşkesini bulunuz.
Örnek Problem 2.1
![Page 14: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/14.jpg)
• Grafik çözüm - paralelkenar yöntemi:
• Trigonometrik çözüm – cos ve sin teoremleri:
°== 35N 98 αR
°== 35N 98 αR
![Page 15: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/15.jpg)
α) α = 45o olduğu durumda her iki halattaki kuvveti,
b) 2 nolu halattaki kuvvetin minimum olması için α açısını bulunuz.
Şekildeki mavna iki römorkör ile çekilmektedir. Römorkörlerin uyguladığı kuvvetin bileşkesi 5000 N ise
Örnek Problem 2.2
![Page 16: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/16.jpg)
• Grafik çözüm – Paralelkenar yöntemi
NTNT 26003700 21 ==
• Trigonometrik çözüm – Sinüs kuralı
°=
°=
° 105sin5000
30sin45sin21 NTT
NTNT 25903660 21 ==
5000 N
5000 N
(a)
![Page 17: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/17.jpg)
• 2. halattaki minimum kuvvet için üçgen kuralı kullanılarak değişik α açıları için çözüm yapılır:
(b)
• 2. halattaki minimum kuvvet T1 ve T2birbirine dik olduğu zaman oluşur:
( ) °= 30sin50002 NT NT 25002 =
( ) °= 30cos50001 NT NT 43301 =
°−°= 3090α °= 60α
5000 N
5000 N
![Page 18: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/18.jpg)
Fxθ
Fy
Fx = F cos θ
Fy = F sin θ
y
x
F
Fx
θ
Fy
y
x
F
BİR KUVVETİN DİK BİLEŞENLERİNE AYRILMASI
F = (Fx, Fy) = Fx i + Fy j
F = (Fx2 + Fy
2)1/2
Tan θ = Fy / Fx
![Page 19: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/19.jpg)
F1 = F1x i + F1y j
F2 = F2x i + F2y j
Fn = Fnx i + Fny j
R = (ΣFix i)0n+ (ΣFiy j)0
n = 0
Rx = (ΣFix)
F3
F2
F1
F1 cos θ1
F2 cos θ2
F3 cos θ3
F1 sin θ1
F2 sin θ2
F3 sin θ3
Ry = (ΣFiy)
![Page 20: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/20.jpg)
F3
F2
F1
F1 cos θ1
F2 cos θ2
F3 cos θ3
F1 sin θ1
F2 sin θ2
F3 sin θ3
F1 cos θ1F3 cos θ3
x
Fx = F1 cos θ1 + F2 cosθ2 + F3 cosθ3
![Page 21: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/21.jpg)
F3
F2
F1
F1 cos θ1
F2 cos θ2
F3 cos θ3
F1 sin θ1
F2 sin θ2
F3 sin θ3
F1 sin θ1
F2 sin θ2
F3 sin θ3
y
Fy = F1 sinθ1 + F2 sinθ2 + F3 sinθ3
F = (Fx2 + Fy
2 )1/2
θ = tan -1 (Fy / Fx )
![Page 22: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/22.jpg)
MADDESEL NOKTANIN DENGESİ
ΣM = 0 (moment)
Σ F = 0 (kuvvet)
F1 = F1x i + F1y j
F2 = F2x i + F2y j
Fn = Fnx i + Fny j
R = (ΣFix i)+ (ΣFiy j) = 0
(ΣFix) = 0
(ΣFiy) = 0
![Page 23: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/23.jpg)
Şekildeki somuna A noktasında dört kuvvet etkimektedir. Bileşke kuvveti bulunuz.
Örnek Problem 2.3
![Page 24: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/24.jpg)
Her kuvvetin dik bileşenleri hesaplanır:
9.256.961000.11001102.754.27800.759.129150
4
3
2
1
−+−+−++
−−
FFFF
bilybilxşidkuvvet
r
r
r
r
1.199+=xR 3.14+=yR
22 3.141.199 +=R N6.199=R
• Bileşkenin şiddeti ve yönü:
N1.199N3.14tan =α °= 1.4α
![Page 25: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/25.jpg)
Parçacıkların Dengesi• Parçacığa etkiyen kuvvetlerin bileşkesi sıfır ise parçacık dengededir.
• Parçacığa iki kuvvet etkiyor :- eşit şiddetli - aynı tesir çizgisi - zıt yönlü
• Parçacık ikiden fazla kuvvet etkisi altında: - grafik çözüm zorlaşır.- cebirsel çözüm uygundur.
000
==
==
∑∑∑
yx FFFRrr
• Newton’un 1. Kanunu: Bileşke kuvvet sıfır ise, parçacık başlangıçtaki hareketini korur. Dengede ise dengede kalır, belirli bir hızı varsa aynı hızda devam eder.
![Page 26: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/26.jpg)
Serbest Cisim Diyagramı
Uzay Diyagramı: Problemin fiziksel durumunu gösteren resim
Serbest Cisim Diyagramı: Seçilen elemana etkiyen kuvvetleri gösteren çizim.
![Page 27: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/27.jpg)
Bir gemiye araç yüklemesi sırasında 3500 N’luk bir araba kablolar ile kaldırılmaktadır. AC kablosundaki kuvveti bulunuz.
Örnek Problem 2.4
![Page 28: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/28.jpg)
°=
°=
° 58sin3500
2sin120sinNTT ACAB
NTAB 3570=
NTAC 144=
Serbest cisim diyagramı
3500 N
3500 N
![Page 29: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/29.jpg)
Bir botun sürükleme kuvvetinin hesaplanabilmesi için su kanalı kullanılmaktadır. Bot 3 kablo ile desteklenmiştir. AB kablosunda 40 N, AE kablosunda ise 60 N kuvvet oluştuğu biliniyorsa, botun sürüklem kuvvetini ve AC kablosundaki kuvveti bulunuz.
Örnek Problem 2.6
7 m 1.5 m
4 m
4 mAkış
![Page 30: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/30.jpg)
Botun serbest cisim diyagramı:
°=
==
25.60
75.1m 4m 7tan
α
α
°=
==
56.20
375.0m 4m 1.5tan
β
β
• Denge için kuvvet toplamının sıfıra eşit olması gerekir:
0=+++= DAEACAB FTTTRrrrrr
7 m 1.5 m
4 m
4 mAkış
40 N
60 N
![Page 31: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/31.jpg)
( ) ( )( ) ( )
( )
( )( ) jT
iFTR
iFF
jT
jTiT
jTiTT
jijiT
AC
DAC
DD
ACAC
ACACAC
AB
r
r
r
rr
rr
rr
rrr
rr
rrr
609363.084.19
3512.073.340
N 06
9363.03512.0
56.20cos56.20sin
N 84.19N 73.3426.60cosN 4026.60sinN 40
−++
++−=
=
=
−=
+=
°+°=
+−=
°+°−=
(60 N)
(40 N)
(40 N)
42.9 N
40 N
19.66 N
60 N
![Page 32: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/32.jpg)
( )( ) jT
iFTR
AC
DACr
r
r
609363.084.193512.073.34
0
−++
++−=
=
( )( ) 609363.084.1900
3512.073.3400−+==
++−==
∑∑
ACy
DACxTF
FTF
N 66.19N 9.42
+=+=
D
AC
FT
42.9 N
40 N
19.66 N
60 N
![Page 33: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/33.jpg)
B C
AA
A A
RABRAC
WA
A
B
RBC
RBX
RBY
A
C
RCX
RCY
RBC
WCWB
Serbest Cisim Diyagramları
![Page 34: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/34.jpg)
Uzayda Dik Bileşenler
• Vektör OBACdüzlemindedir.
yy FF θcos=
yh FF θsin=
• Yatay bileşenlerin dik bileşenleri
φθ
φ
φθφ
sinsin
sin
cossincos
y
hy
y
hx
F
FF
FFF
=
=
==
• Düşey ve yatay bileşenleri:
![Page 35: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/35.jpg)
• F vektörünün dik eksenlerle yaptığ açı biliniyorsa:
( )
kji
F
kjiF
kFjFiFF
FFFFFF
zyx
zyx
zyx
zzyyxx
rrrr
r
rrr
rrrr
θθθλ
λ
θθθ
θθθ
coscoscos
coscoscos
coscoscos
++=
=
++=
++=
===
• tesir çizgisi üzerindeki birim vektördür.λr
zyx ve θθθ cos cos,cos : doğrultman cosinüsleri
λ (Şiddet=1)
![Page 36: Seventh Edition VECTOR MECHANICS FOR ENGINEERS: STATICSkurtcebece/sta201-bolum2.pdf · 2006-01-03 · VECTOR MECHANICS FOR ENGINEERS: STATICS Seventh Edition Ferdinand P. Beer E](https://reader030.vdocuments.us/reader030/viewer/2022033119/5e500e7381afa8576774af28/html5/thumbnails/36.jpg)
Tesir çizgisi üzerindeki iki nokta ile tanımlanan kuvvetin yönü
( ) ( )222111 ,, ve,, zyxNzyxM
( )
dFdF
dFd
Fd
FdF
FF
kdjdidd
zzdyydxxd
kdjdidd
zz
yy
xx
zyx
zyx
zyx
===
=
++=
−=−=−=
++=
λ
λrr
rrrr
rrrr
1
121212
Noktalar
M ve N noktalarını birleştiren vektör