1. Produse de vectori: Produsul scalar: < v 1 , v 2 >= x 1 x 2 + y 1 y 2 + z 1 z 2 , k vk = p x 2 1 + y 2 1 + z 2 2 , cos( \ v 1 , v 2 )= < v 1 , v 2 > k v1k·k v2k . Ortogonalitate: < v 1 , v 2 >= 0. Dacˇa Δ - dreaptˇa de vector director u, k uk = 1, atunci Pr Δ v 1 =(k vk cos( d v 1 , u)) u,¸ si pr Δ v 1 = k vk cos( d v 1 , u). Produsul vectorial: v 1 × v 2 = fl fl fl fl fl fl i j k x 1 y 1 z 1 x 2 y 2 z 2 fl fl fl fl fl fl ¸ si k v 1 × v 2 k = k v 1 k·k v 2 k sin( \ v 1 , v 2 ). Avem: σ triunghi = 1 2 k v 1 × v 2 k, σ paralelogr = k v 1 × v 2 k. Coliniaritate: v 1 × v 2 = 0. Produsul mixt: ( v 1 ; v 2 ; v 3 )=< v 1 , v 2 × v 3 >= fl fl fl fl fl fl x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 fl fl fl fl fl fl . Avem: V paralelipiped =( v 1 ; v 2 ; v 3 ), V tetraedru = 1 6 ( v 1 ; v 2 ; v 3 ). Coplanaritate: ( v 1 ; v 2 ; v 3 ) = 0. 2. Mi¸ scˇ ari: Translat ¸ie de vector r = x 0 i + y 0 j + z 0 k : x 0 = x - x 0 y 0 = y - y0 z 0 = z - z 0 sau x = x 0 + x 0 y = y 0 + y0 z = z 0 + z 0 . Rotat ¸ie de unghi θ ˆ ın jurul originii: x 0 y 0 ¶ = cos θ sin θ - sin θ cos θ ¶ · x y ¶ sau x y ¶ = cos θ - sin θ sin θ cos θ ¶ · x 0 y 0 ¶ . 3. Dreapta: Un punct ¸ si direct ¸ia: x = x 0 + tl y = y 0 + tm z = z 0 + tn sau x - x 0 l = y - y 0 m = z - z 0 n . Douˇ a puncte: l = x 1 - x 0 , m = y 1 - y 0 , n = z 1 - z 0 ¸ si se ˆ ınlocuie¸ ste mai sus. 4. Planul: Un punct ¸ si douˇa direct ¸ii: x = x 0 + t 1 l 1 + t 2 l 2 y = y 0 + t 1 m 1 + t 2 m 2 z = z 0 + t 1 n 1 + t 2 n 2 sau fl fl fl fl fl fl x - x 0 y - y 0 z - z 0 l 1 m 1 n 1 l 2 m 2 n 2 fl fl fl fl fl fl = 0. Trei puncte: l 1 = x 1 - x 0 , m 1 = y 1 - y 0 , n 1 = z 1 - z 0 , l 2 = x 2 - x 0 , m 2 = y 2 - y 0 , n 2 = z 2 - z 0 ¸ si se ˆ ınlocuie¸ ste mai sus. Prin tˇaieturi: x a + y b + z c = 1. Dat de un punct ¸ si de normala A i + B j + C k:(x - x 0 )A +(y - y 0 )B +(z - z 0 )C =0. 5. Intersect ¸ii, proiect ¸ii, unghiuri: d(M,D)= k M 0 M × v D k k v D k ,M 0 ∈ D, d(M,α)= |Ax M +By M +Cz M +D| √ A 2 + B 2 + C 2 , d(D 1 ,D 2 )= |( M 1 M 2 ; v D 1 ; v D 2 )| k v D 1 × v D 2 k ,M 1 ∈ D 1 ,M 2 ∈ D 2 Perpendiculara comunˇ a: direct ¸ie v = v D1 × v D2 , ec. - la intersect ¸ia planelor (M 1 , v, v D1 )¸ si (M 2 , v, v D2 ), M 1 ∈ D 1 ,M 2 ∈ D 2 . 6. Sfera: (x - a) 2 +(y - b) 2 +(z - c) 2 = R 2 sau x 2 + y 2 + z 2 + mx + ny + pz + q = 0. Cerc=sferˇa ∩ plan. Planul tangent - prin dedublare: x 2 → xx 0 , x → x+x0 2 . 7. Conice: H(x, y)= a 11 x 2 +a 22 y 2 +2a 12 xy +2a 13 x+2a 23 y +a 33 =0. Invariant ¸i: Δ= fl fl fl fl fl fl a 11 a 12 a 13 a 12 a 22 a 23 a 13 a 23 a 33 fl fl fl fl fl fl , δ = fl fl fl fl a 11 a 12 a 12 a 22 fl fl fl fl , I = a 11 + a 22 . 8. Cuadrice: Σ 2 : (a 11 x 2 + a 22 y 2 + a 33 z 2 +2a 12 xy +2a 13 xz +2a 23 yz) + (2a 14 x +2a 24 y +2a 34 z)+ a 44 = 0, D = a 11 a 12 a 13 a 14 a12 a22 a23 a24 a 13 a 23 a 33 a 34 a 14 a 24 a 34 a 44 , A = a 11 a 12 a 13 a12 a22 a23 a 13 a 23 a 33 . Invariant ¸i: Δ = det D, δ = det A, ρ = rangD, r = rangA, p=nr. de pˇatrate pozitive. 9. Suprafet ¸e cilindrice, conice, de rotat ¸ie: Ec. supr. cilindrice care se sprijinˇa pe curba Γ: ‰ f (x, y, z)=0 g(x, y, z)=0 ¸ si are generatoareaparalelˇacu dreapta D : ‰ P 1 (x, y, z)=0 P 2 (x, y, z)=0 se obt ¸ineeliminˆand λ, μ din sist. a). Ec. supr. cilindrice tangente la supr. F (x, y, z)=0¸ si cu generatoarea paralelˇa cu D : ‰ P 1 (x, y, z)=0 P 2 (x, y, z)=0 se obt ¸ineeliminˆand λ, μ din sist. b). Conul de vˆarf V (sist. c)) care se sprijinˇa pe curba Γ: ‰ f (x, y, z)=0 g(x, y, z)=0 se obt ¸ineeliminˆand λ, μ din sist. d). Conul devˆarf V (sist. c)) tangent la supr. F (x, y, z) = 0 se obt ¸ineeliminˆand λ, μ din sist. e). Suprafat ¸a de rotat ¸ie generatˇ a de rotirea curbei Γ: f (x, y, z)=0,g(x, y, z)=0ˆ ın jurul dreptei D : x - x 0 l = y - y 0 m = z - z 0 n se obt ¸ineeliminˆand λ, μ din sist. f). 1
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fl k fl triunghi kfl paralelogr = 0. : ( ) = fl V ) = 0. fl paralelipiped …fliacob/An1/2012-2013... · 2010-03-02 · y = y0 +tm z = z0 +tn sau x ¡0 l = y0 m = z0 n: Dou•a
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1. Produse de vectori: Produsul scalar: < v1, v2 >= x1x2 + y1y2 + z1z2, ‖v‖ =√
x21 + y2
1 + z22 , cos(v1, v2) = <v1,v2>
‖v1‖·‖v2‖ .
Ortogonalitate: < v1, v2 >= 0. Daca ∆ - dreapta de vector director u, ‖u‖ = 1, atunci Pr∆v1 = (‖v‖ cos(v1, u))u, si
Muchii: tangenta (TM0(Γ)), cu vect. dir. t, normala principala (Dn(Γ)), cu vect. dir.n, binormala (Db(Γ)), cu vect. dir. b. Vectori directori: t = xi + yj + zk
∣∣t0
, b =
Ai + Bj + Ck∣∣t0
, n = li + mj + nk∣∣t0
, unde: A =∣∣∣∣y zy z
∣∣∣∣t0
, B = −∣∣∣∣x zx z
∣∣∣∣t0
, C =∣∣∣∣x yx y
∣∣∣∣t0
,
l =∣∣∣∣y zB C
∣∣∣∣t0
, m = −∣∣∣∣x zA C
∣∣∣∣t0
, n =∣∣∣∣x yA B
∣∣∣∣t0
. Versori: t0 =
t
‖t‖ , b0
=b
‖b‖ , n0 =n‖n‖ .
Relatii: b× t = n, t× n = b, n× b = t. Plane: planul normal (PN (Γ) = (M0, n, b)), planulosculator (Po(Γ) = (M0, n, t)), planul rectificant (Pr(Γ) = (M0, t, b)).