Geometrie_vectoriala FOARTE BUN
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GEOMETRIE VECTORIALĂ I 1. În Oxyz , M (x, y, z) este determinat de vectorul de poziţ versori k j i k z j y i x OM = ⋅ + ⋅ + ⋅ = , , y M (x,y,z) j i x k z 2. rodus scalar ) , ( 2 1 v v ) , ( unde cos 2 1 2 1 2 1 v v v v v v = < ⋅ ⋅ = ⋅ α α ) 0 0 ( 0 2 1 2 1 2 1 = = = ⋅ ⇔ ⊥ v sau v v v v v ( ) ( ) = + + = = + + = 3 2 1 3 2 1 2 3 2 1 3 2 1 1 , , , , Daca b b b k b j b i b v a a a k a j a i a v 3 3 2 2 1 1 2 1 b a b a b a v v + + = ⋅ analitic
Transcript of Geometrie_vectoriala FOARTE BUN
GEOMETRIE VECTORIALA
GEOMETRIE VECTORIAL
I 1. n Oxyz , M (x, y, z) este determinat de vectorul de poziie
y
M (x,y,z)
x
z
2. Produs scalar
analitic
Proprieti. Fie
3. Produs vectorial
Proprieti. Fie
coliniari
4. Produsul mixt a trei vectori
5. Aplicaii.
Distana de la un punct la o dreapt. Fie ABC,
Aria ABC.
Aria ABCD (paralelogram)
Volumul paralelipipedului oblic.
d
A
P
hp C D
O B
Volumul tetraedrului.
Determinarea nlimii tetraedrului din A.
6. Preliminarii.
Vector director al unei drepte.
d
A B
Parametrii directori ai unei drepte.
parametrii directori ai dreptei d Cosinusurile directoare ale direciei d.
Vectori directori ai unui plan.
Vectori necoliniarivectori directori,
Vector normal la un plan.
B
A
P
II ECUAIILE PLANULUI.
1. Ec. unui plan care trece printr-un punct i are un vector normal dat.
2. Ecuaia general a planului n spaiu tridimensional.
3. Ecuaia normal a planului care trece printr-un punct. M0 (x0,y0,z0)
4. Ecuaia planului care trece prin 3 puncte necoliniare.
Condiie de necoliniaritate a 3 puncte.
5. Poziiile relative ale planelor.
EMBED Equation.3
6. Ecuaia planului care trece prin origine.
Ecuaia planului || cu Ox ( a = 0).
Ecuaia planului || cu Oy ( b = 0).
Ecuaia planului || cu Oz ( c = 0).
7. Ecuaia planelor || cu planele de coordonate.
8. Ecuaia planului prin tieturi.
III ECUAIILE DREPTEI N SPAIU
1. Ecuaia vectorial a dreptei (d).
2. Ecuaii parametrice ale dreptei.
3. Ecuaia dreptei sub form canonic.
4. Ecuaia vectorial a dreptei determinat de 2 puncte.
Fie M1, M2
5. Ecuaii parametrice ale dreptei determinate de 2 puncte:
6. Ecuaii canonice ale dreptei determinate de 2 puncte.
7. Ecuaia dreptei sub form explicit.
8.
IV 1. Poziiile relative a dou drepte.
a) Drepte concurente.
b) Drepte paralele.
c) Drepte oarecare (nesituate n acelai plan).
2. Unghiul format de dreptele i :
cu i
3. Dreapta intersecteaz planul.
Observaii!
Scriem d :
Dreapta sistemul ; este verificat ptr.
4. Unghiul format de o dreapt cu un plan.
Fie ; =>
5. Distana de la un punct la un plan
6. Unghiul format de
7. Aria cu vrfurile M1, M2, M3
unde
8. Volumul tetraedrului (M0, M1, M2, M3).
EMBED Equation.3
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