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Zusammenfassung Lernzettel Mathematik Leistungskurs Niedersachsen $8.10
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Zusammenfassung Lernzettel Mathematik Leistungskurs Niedersachsen

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In diesem Dokument fasse ich alles zusammen, was man für das Abitur in Niedersachsen benötigt wenn man den Mathematik Leistungskurs belegt. Diese Themen umfassen sowohl Analysis, Analytische Geometrie und Stochastik

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  • May 21, 2022
  • 13
  • 2021/2022
  • Summary
  • Secondary school
  • Gymnasium
  • 1
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Lineare Algebra
Dreidimensionales Koordinatensystem
z Punkte:
P (x, y, z)
Als erstes x-Richtung, dann y-Richtung und als
letztes z-Richtung.
Vektoren:
y Beschreibt Richtung und Bewegung im Raum
ð Vektoren haben keinen Start und Endpunkt
x



Punkte in Vektoren umschreiben
𝑥
,,,,,⃗ 𝑦
𝐴(𝑥, 𝑦, 𝑧) => 𝑂𝐴(Null – Vektor) = 8 9
𝑧

Distanz zwischen zwei Punkten
𝑥
,,,,,⃗ 𝑦
𝐴(𝑥, 𝑦, 𝑧) => 𝑂𝐴 = 8 9
𝑧
𝑥! 𝑥"
,,,,,,⃗ = 𝑂𝐵
𝐴𝐵 ,,,,,⃗ − ,,,,,⃗
𝑂𝐴 => ,,,,,⃗
𝐴𝐵 = <𝑦! = − <𝑦" =
𝑧! 𝑧"
𝑥
,,,,,⃗
𝐵(𝑥, 𝑦, 𝑧) => 𝑂𝐵 = 8 𝑦9
𝑧


Abstand zwischen zwei Punkten Vektorkreis
𝑥
,,,,,⃗
𝐴𝐵 = 8𝑦9 => 𝑑 = ?𝑥 ! + 𝑦 ! + 𝑧 !
𝑧
ð norm() im CAS

Addition von Vektoren
𝑥" 𝑥! 𝑥# 𝑥" + 𝑥! = 𝑥#
𝑦 𝑦 𝑦
< " = + < ! = = < # = => 𝑦" + 𝑦! = 𝑦#
𝑧" 𝑧! 𝑧# 𝑧" + 𝑧! = 𝑧# 0
𝑎⃗ + 𝑏,⃗ + 𝑐⃗ + 𝑑⃗ + 𝑒⃗ + 𝑓⃗ + 𝑔⃗ + ℎ,⃗ = <0=
Subtraktion von Vektoren 0
𝑥" 𝑥! 𝑥# 𝑥" − 𝑥! = 𝑥#
<𝑦" = − <𝑦! = = <𝑦# = => 𝑦" − 𝑦! = 𝑦#
𝑧" 𝑧! 𝑧# 𝑧" − 𝑧! = 𝑧#




1

, Skalare Multiplikation
𝑥" 𝑥! 𝑟 ∙ 𝑥" = 𝑥!
𝑟 ∙ <𝑦" = = <𝑦! = => 𝑟 ∙ 𝑦" = 𝑦!
𝑧" 𝑧! 𝑟 ∙ 𝑧" = 𝑧!

Skalarprodukt
𝑎⃗ ∗ 𝑏,⃗ => (𝑥" ∙ 𝑥! ) + (𝑦" ∙ 𝑦! ) + (𝑧" ∙ 𝑧! )
ð dotP() im CAS


Winkel zwischen Vektoren
1. Startpunkte aneinanderlegen
2. In Formel einsetzen
"⃗
!"⃗∗%
ð cos(∝) = |!"⃗|∙(%"⃗
(


Verbindungsvektor
𝑥! 𝑥"
,,,,,,⃗
𝐴𝐵 = 𝐴⃗ − 𝐵
,⃗ => ,,,,,⃗
𝐴𝐵 = <𝑦! = − <𝑦" =
𝑧! 𝑧"

Lineare Abhängigkeit
𝑥" 𝑥! 𝑘 ∙ 𝑥" = 𝑥!
,

𝑎⃗ ∙ 𝑘 = 𝑏 => <𝑦" = ∙ 𝑘 = <𝑦! = 𝑘 ∙ 𝑦" = 𝑦!
𝑧" 𝑧! 𝑘 ∙ 𝑧" = 𝑧!
Nach k auflösen, wenn k nicht gleich ist auf allen Ebenen dann sind die beiden nicht skalar
abhängig.

𝑎⃗ = 𝑘 ∙ 𝑏,⃗ + 𝑙 ∙ 𝑐⃗
Gleiches Prinzip, wenn k und l nicht gleich für beide Vektoren, dann lassen sich die beiden
nicht durcheinander ausdrücken.

Geradengleichung
𝑔: 𝑥⃗ = ,,,,,⃗
𝑂𝐴(𝑍𝑢𝑣𝑒𝑘𝑡𝑜𝑟) + 𝑘 ∙ ,,,,,⃗
𝐴𝐵(𝑅𝑖𝑐ℎ𝑡𝑢𝑛𝑔𝑠𝑣𝑒𝑘𝑡𝑜𝑟)

Streckengleichung
𝑔: 𝑥⃗ = )))))⃗ )))))⃗(𝑅𝑖𝑐ℎ𝑡𝑢𝑛𝑔𝑠𝑣𝑒𝑘𝑡𝑜𝑟); 0 ≤ 𝑘 ≤ 0 (𝐵𝑒𝑔𝑟𝑒𝑛𝑧𝑢𝑛𝑔)
𝑂𝐴(𝑍𝑢𝑣𝑒𝑘𝑡𝑜𝑟) + 𝑘 ∙ 𝐴𝐵




2

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