Material zum Vorkurs P2 WiSe 2013/14

WiSe 2013/14
04.10.2013
[ ] [ ] [ ] [ ] [ ] [
0
1
4
−3
0
2.) a) p 1 = ; p 2 = ; p 3 = ; p 4 = ; p 5 = ; p 6 =
1
3
−2
−3
5
[ ]
b) Verbindungsvektoren berechnet man mit −−→ P 1 P 2 = p 2 − p 1 = .
1
2
19. Tag – Vektorrechnung in 2D
Lösungen zu Vektorrechnung in 2D I
−5
2
]
P 1
P 2
P 3
P 4
P 5
P 6
P 1 P 2 P 3 P 4 P 5 P 6
[ ]
[ ] [ ] [ ]
−−→ −−→ 1 −−→ 4 −−→ −3 −−→ 0
P1 P 1 = 0 P 1 P 2 = P 1 P 3 = P 1 P 4 = P 1 P 5 =
2
−3
−4
4
[ ]
[ ] [ ] [ ]
−−→ −1 −−→ −−→ 3 −−→ −4 −−→ −1
P2 P 1 = P 2 P 2 = 0 P 2 P 3 = P 2 P 4 = P 2 P 5 =
−2
−5
−6
2
[ ] [ ]
[ ] [ ]
−−→ −4 −−→ −3 −−→ −−→ −7 −−→ −4
P3 P 1 = P 3 P 2 = P 3 P 3 = 0 P 3 P 4 = P 3 P 5 =
3
5
−1
7
[ ]
[ ]
[ ]
[ ]
−−→ 3 −−→ 4 −−→ 7 −−→ −−→ 3
P4 P 1 = P 4 P 2 = P 4 P 3 = P 4 P 4 = 0 P 4 P 5 =
4
6
1
8
[ ] [ ] [ ] [ ]
−−→ 0 −−→ 1 −−→ 4 −−→ −3 −−→
P5 P 1 = P 5 P 2 = P 5 P 3 = P 5 P 4 = P 5 P 5 = 0
−4
−2
−7
−8
[ ] [ ]
[ ] [ ] [ ]
−−→ −−→ −−→ −−→ −−→
P6 P 1 = P 6 P 2 = P 6 P 3 = P 6 P 4 = P 6 P 5 =
5
−1
6
1
9
−4
2
−5
5
3
[
−−→
P 1 P 6 =
[
−−→
P 2 P 6 =
−−→
P 3 P 6 =
[
−−→
P 4 P 6 =
[
−−→
P 5 P 6 =
−−→
P 6 P 6 = 0
[ ] [ ] [ ]
0 2 · 0 0
c) • q 1 = 2p 1 = 2 · = = =⇒ Q 1 = (0, 2)
1 2 · 1 2
[ ] [ ] [ ] [ ] [ ] [ ] [ ]
1 4 3 · 1 2 · 4 3 8 3 + 8
• q 2 = 3p 2 + 2p 3 = 3 · + 2 · = + = + =
3 −2 3 · 3 2 · (−2) 9 −4 9 + (−4)
=
[
]
11
5
• q 3 = −p 2 + 3p 3 − 4p 6
=⇒ Q 2 = (11, 5)
[ ] [ ] [ ]
1 4 −5
= − + 3 · − 4 ·
3 −2 2
=⇒ Q 3 = (31, −17)
[ ] [ ] [ ]
0 4 4
d) • p 1 + p 3 = + = =⇒ R 1 = (4, −1)
1 −2 −1
[ ] ⎛[
] [ ] ⎞ [ ] [ ]
• p 1 + −−→ 0
P 2 P 4 = + ⎝
−3 1
− ⎠
0 − 3 − 1 −4
= =
1 −3 3 1 − 3 − 3 −5
⎛[
] [ ] ⎞ ⎛[
] [ ] ⎞ [
• −−→ P 3 P 6 + −−→ P 1 P 4 = ⎝ − ⎠ + ⎝ − ⎠ =
−5
2
4
−2
−3
−3
0
1
]
−5
1
]
−6
−1
[ ]
−9
4
]
−2
5
]
−5
−3
[
] [
−1 + 3 · 4 − 4 · (−5)
=
=
−3 + 3 · (−2) − 4 · 2
=⇒ R 2 = (−4, −5)
−5 − 4 − 3 − 0
2 − (−2) − 3 − 1
]
=
[
]
−12
0
31
−17
]
=⇒ R 3 = (−12, 0)
Mathematik Vorkurs P2
✞ ☎
✝157 ✆
Dipl. Math. Stefan Podworny