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Zusammenfassung Mathematik Analytische Geometrie
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Vollständige Zusammenfassung der Inhalte der Analytischen Geometrie, Mathematik
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MatheAnalytische Geometrie
Abstand von Punkten im Raum Skalarprodukt
d (A ; B) =
(a .
-
b ,
12 + (az be) -
+
(ay by)2
-
ob =
a,b, + 22bz +
23bs
2 15 E)205 = 0
Vektoren
·
Ortsvektor : == ( Rechenregeln :
·
Richtungsvektor : Ä = - +Ö : k L . er
Vertreter eines Vektors : gleich lang parallel gleich
, ,
orientiert (1) 05 505 =
Gegenvektoren :
gleich lang parallel , ,
andersherum orientiert (2) (k 2(0 (5) = k .
(2 05)
(3) 2o(5 2) + = 205 + 30
Parameterdarstellung Gerade (4) k . 01 .
5 = (k 1) · .
(205)
g
:= 0 +
s
.
AB (5) = /äl(äl bzw .
0 = ä
Parameterdarstellung Ebene Winkel zwischen Vektoren
E: 0 AB AC
-c) 05
+
s
.
+ t .
d =
COS
Normalenform Ebene
Einok = mo Lagebeziehung von Geraden
identisch :
Richtungsvektoren Kollinear
Koordinatenform Ebene ·
Stützpunkt von
g liegt auf h
Ein ,
X, + mzXz +
nzXz
= no echt parallel :
Richtungsvektoren Kollinear
·
Stützpunkt von
g liegt nicht auf h
Normalenform in Parameterform schneiden :
Richtungsvektoren linear unabhängig
·
Normalenform in Koordinatenform ·
LGS zu
guh hat eine Lösung
·
3 Punkte auf der Ebene bestimmen windschief : Richtungsvektoren linear unabhängig
·
daraus Parameterform ·
LGS zu
guh hat keine Lösung
Parameterform in Normalenform Lagebeziehung Gerade -
Ebene
E: 0x +
r
-
Xy +
s
.
Xz g in E (Koordinatenform) einsetzen
noXy = 0 R, =
Schnittpunkt -
r =
Zahl
= nz =
noxz = 0 Ry = mit ce
parallel
-
false
E. (0 = (g)0 of Gerade in Ebene -
true / r = c
wa
ausrechnen
g ~ E(Parameterform) gleichsetzen
Das Vektorenprodukt Schnittpunkt - r = Zahl s= Zahl t = Zahl
① parallel
-
false
1) @ = n mit +5 1+ Gerade in Ebene
> n =
c ... S
=
c .... t= 2,
1 -....
-
2)
2
A A = (üv)
....
-
" A= /0) Lagebeziehung Ebene-Ebene
-
4) vo = -
(8) = und =
Kollinear/ linear unabhängig ? (Normalenformen
5) V = (Qi)0i Schnittgerade - und r
linear unabhängig
parallel und Kollinear : kein gemeinsamer Punkt
~
= -
identisch -
in und -
Kollinear ; gemeinsamer Punkt
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