Summary
Zusammenfassung Hesse‘sche Normalenform
- Course
- Institution
Hesse‘sche Normalenform erklärt, mit Rechenbeispielen.
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jasmin-hirsch
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Menschengemachten_ .Eine Ebenen
gleichung der Form E :
(F -
Ä) -
= 0 mit dem Normalen einheitsvektor
heißt Hesse '
sehe Normalenform .
Satz : Für den Abstand d eines Punktes R ( r
,
/ r
,
/ r
}
) von der Ebene E
gilt :
d =/ IT F ) ü -
-
.
Ist a. × , azx + b eine Koordinaten
gleichung der Ebene E so
gilt
:
+
asx
=
,
} ,
b Ann b
91h +
azr , -19 r +
azr , -19 r -
-
, } , }
D= Oder : D=
< <
a.
z
/ ,
a. +
a. +
Bsp .
:
1. Bestimmen sie den Abstand d. Punktes R / 11612 ) von der Ebene E. :
Ü -
| ! ) f- § )
'
=
0
und Ez
:
×, -
2×2+4×3=1
Es :
1. Normalen einheitsvektor bestimmen :
=
(E) ;
in / =
FF =3
i. =
} -
E) =
( ;) ÷
Normalen einheitsvektor mit
2. + Punkt in Ebenen
gleichung Betrag einsetzen :
I KH ! ) -
E;) / =D
I I !) I ! ) -
.EE/t--fE).fE)l--l-z .
} + 4.
f) + z .
3) 1=9
Ez :
1 .
Alles auf eine Seite
bringen
:
Ez
:
1×1 -
2×2+4 ×
}
-
1 =
0
2.
Betrag des Normalen Vektors ausrechnen .
/ /
'
42
'
=
1 + ( 2) -
+
=
21
3. Punkt in Ebenen
gleichung einsetzen und durch tnt teilen ( mit Betrag )
:
1. 1- 2 .
6+4 '
2- 1
D= = 0,873
21
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