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FORMULARIO DE GEOMETRÍA
ANALÍTICA
Coordenadas Cartesianas y Polares
01 x = ρ ⋅ Cos(θ) 02 y = ρ ⋅ Sen(θ)
03 ρ = √x2 + y2 04 θ = ∢Tan (
y
x
)
1) ACCESORIOS
05 Punto Medio: PM = (
x1 + x2
2
,
y1 + y2
2
)
Distancias
06 D. . = √(x2 − x1)2 + (y2 − y1)2
07 D. | =
|y1 − m ⋅ x1 − b|
√m2 + 1
=
|D ⋅ x + E ⋅ y + F|
√D2 + E2
08 D|| =
|b2 − b1|
√m2 + 1
09 Traslación de Ejes:
x = X + h
y = Y + k
10 Rotación de Ejes:
x = X ⋅ Cos(θ) − Y ⋅ Sen(θ)
y = X ⋅ Sen(θ) − Y ⋅ Cos(θ)
(
x
y) = (
Cos(θ) −Sen(θ)
Sen(θ) −Cos(θ)
) ⋅ (
X
Y
)
2) TRIGONOMETRÍA ANALÍTICA
11 Centroide: G = (
x1 + x2 + x3
3
,
y1 + y2 + y3
3
)
12 Área: A =
1
2
⋅ |
x1 y1 1
x2 y2 1
x3 y3 1
|
3) GEOMETRÍA ANALÍTICA 2D
13 Cónicas: A ⋅ x2
+ B ⋅ x ⋅ y + C ⋅ y2
+ D ⋅ x + E ⋅ y + F = 0
Recta
14 Ec. General: D ⋅ x + E ⋅ y + F = 0
2
15 Pendiente: m =
y2 − y1
x2 − x1
m = Tan(θ)
θ = ∢Tan(m)
16 Ec. Cartesiana:
y − y1
x − x1
=
y2 − y1
x2 − x1
17 Ec. Punto Pendiente: y − y1 = m ⋅ (x − x1)
18 Ec. Pendiente Ordenada: y = m ⋅ x + b
19 Ec. Pendiente Abscisa: y = m ⋅ (x − a)
20 Ec. Abscisa Ordenada:
x
a
+
y
b
= 1
21 Ángulo entre Rectas: θ = ∢Tan (
m2 − m1
1 + m1 ⋅ m2
)
Circunferencia
22
Ec. General:
x2
+ y2
+ D ⋅ x + E ⋅ y + F = 0
23
Ec. Canónica:
(x − h)2
+ (y − k)2
= R2
24 P = 2 ⋅ π ⋅ R
25 A = π ⋅ R2
26 e = 0
Parábola
27
Ec. Generales:
a, b] x2
+ D ⋅ x + E ⋅ y + F = 0
c, d] y2
+ D ⋅ x + E ⋅ y + F = 0
a]
28
Ec. Canónicas:
a, b](x − h)2
= ±4 ⋅ p ⋅ (y − k)
c, d](y − k)2
= ±4 ⋅ p ⋅ (x − h)
29 e = 1
30 AF = 4 ⋅ p
31
Focos:
a] (h, k + p)
b] (h, k − p)
c] (h + p, k)
d] (h − p, k)
32
Directrices:
a] y = k − p
b] y = k + p
c] x = h − p
d] x = h + p
3
a] b]
c] d]
Elipse
33 Ec. General:
A ⋅ x2
+ C ⋅ y2
+ D ⋅ x + E ⋅ y + F = 0
A ⋅ C > 0
A ≠ C
34
Ec. Canónicas:
a]
(x − h)2
a2
+
(y − k)2
b2
= 1
b]
(y − k)2
a2
+
(x − h)2
b2
= 1
a]
a > b
a > c
35 AF =
2 ⋅ b2
a
36 0 < e =
c
a
< 1
37
a2
= b2
+ c2
4
38 P = π ⋅ (a + b) b]
39 A = π ⋅ a ⋅ b
40
Focos:
a] (h ± c, k)
b] (h, k ± c)
41
Ápside = 2 ⋅ a
Ápside = Afelio + Perihelio
42 Afelio = a + c
43 Perihelio = a − c
44 Distancia Media al Sol = a
Hipérbola
45 Ec. General:
A ⋅ x2
+ C ⋅ y2
+ D ⋅ x + E ⋅ y + F = 0
A ⋅ C < 0
46
Ec. Canónicas:
a]
(x − h)2
a2
−
(y − k)2
b2
= 1
b]
(y − k)2
a2
−
(x − h)2
b2
= 1
a]
c > a
c > b
47
Ec. de Asíntotas:
a]
(x − h)
a
±
(y − k)
b
= 0
b]
(y − k)
a
±
(x − h)
b
= 0
48 AF =
2 ⋅ b2
a
49 e =
c
a
> 1
5
50
c2
= a2
+ b2
b]
c > a
c > b
51
Focos:
a](h ± c, k)
b](h, k ± c)
4) GEOMETRÍA ANALÍTICA 3D
Vectores
52 A
⃗
⃗ = (ax, ay, az) 53 A
⃗
⃗ = ax ⋅ î + ay ⋅ ĵ + az ⋅ k
̂
|A
⃗
⃗ | = √ax
2 + ay
2 + az
2 {
î = (1,0,0)
ĵ = (0,1,0)
k
̂ = (0,0,1)
54
Cosenos Directores.
{
Cos(αx) =
ax
|A
⃗
⃗ |
Cos(αy) =
ay
|A
⃗
⃗ |
Cos(αz) =
az
|A
⃗
⃗ |
55
Vector Unitario.
A
̂ =
A
⃗
⃗
|A
⃗
⃗ |
A
̂ =
(ax, ay, az)
|A
⃗
⃗ |
A
̂ = (
ax
|A
⃗
⃗ |
,
ay
|A
⃗
⃗ |
,
az
|A
⃗
⃗ |
)
56 Cos2(αx) + Cos2
(αy) + Cos2(αz) = 1
57
Producto PUNTO.
A
⃗
⃗ ⊡ B
⃗
⃗ = (ax, ay, az) ⊡ (bx, by, bz) = {
ax ⋅ bx + ay ⋅ by + az ⋅ bz
|A
⃗
⃗ | ⋅ |B
⃗
⃗ | ⋅ Cos(γ)
6
58
Producto CRUZ.
A
⃗
⃗ ⊠ B
⃗
⃗ = (ax, ay, az) ⊠ (bx, by, bz) = [
î ĵ k
̂
ax ay az
bx by bz
]
59 |A
⃗
⃗ ⊠ B
⃗
⃗ | = |A
⃗
⃗ | ⋅ |B
⃗
⃗ | ⋅ Sen(γ)
60 Area =
|A
⃗
⃗ ⊠ B
⃗
⃗ |
2
=
|A
⃗
⃗ | ⋅ |B
⃗
⃗ | ⋅ Sen(γ)
2
61
Producto Mixto.
Volumen = (A
⃗
⃗ ⊠ B
⃗
⃗ ) ⊡ C
⃗
Volumen = [
ax ay az
bx by bz
cx cy cz
]
62
|A
⃗
⃗ ⊠ B
⃗
⃗ |
A
⃗
⃗ ⊡ B
⃗
⃗
= Tan(γ)
63
Proyección (o componente) de A
⃗
⃗ sobre B
⃗
⃗ .
A
⃗
⃗ B = (A
⃗
⃗ ⊡ B
̂) ⋅ B
̂
64
Paralelismo:
Si: A
⃗
⃗ ∥ B
⃗
⃗ ⇒ A
⃗
⃗ = k ⋅ B
⃗
⃗ , k ∈ ℝ
65
Perpendicularidad:
Si: A
⃗
⃗ ⊥ B
⃗
⃗ ⇒ A
⃗
⃗ ⊡ B
⃗
⃗ = 0
Plano
66 Vector en el Plano:
V
⃗
⃗ = (x − x1, y − y1, z − z1) ∈ Plano
P1 = (x1, y1, z1) ∈ Plano
67
Vector perpendicular al
Plano:
N
⃗⃗ = (a, b, c)
68
Ecuación “Punto-Normal”.
a ⋅ (x − x1) + b ⋅ (y − y1) + c ⋅ (z − z1) = 0
N
⃗⃗ ⊡ V
⃗
⃗ = 0
69
Ecuación “General”.
a ⋅ x + b ⋅ y + c ⋅ z + d = 0
d = −a ⋅ x1 − b ⋅ y1 − c ⋅ z1
Recta
70
Ecuación “Cartesiana”.
x − x1
x2 − x1
=
y − y1
y2 − y1
=
z − z1
z2 − z1
= t
P1 = (x1, y1, z1) ∈ Recta
P2 = (x2, y2, z2) ∈ Recta
71
Ec. “Punto-Directriz”.
x − x1
𝒶
=
y − y1
𝒷
=
z − z1
𝒸
= t
72
Directriz.
D
⃗⃗ = (x2 − x1, y2 − y1, z2 − z1)
D
⃗⃗ = (𝒶, 𝒷, 𝒸)
7
73
Ec. “Paramétricas”.
{
x = x1 + 𝒶 ⋅ t
y = y1 + 𝒷 ⋅ t
z = z1 + 𝒸 ⋅ t
(x, y, z) = (x1, y1, z1) + (𝒶, 𝒷, 𝒸) ⋅ t
Distancias
74
Distancia Punto (x1,y1,z1) a Punto (x2,y2,z2):
δ. . = √(x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2
75
Distancia Punto (x0,y0,z0) a Plano:
δ. □ =
|a ⋅ x0 + b ⋅ y0 + c ⋅ z0 + d|
√a2 + b2 + c2
δ. □ = (P0 − P1) ⊡ N
̂ = (x0 − x1, y0 − y1, z0 − z1) ⊡
(a, b, c)
√a2 + b2 + c2
P0 = (x0, y0, z0) ∉ Plano
P1 = (x1, y1, z1) ∈ Plano
76
Distancia Punto (x0,y0,z0) a Recta:
δ. | = √|P0 − P1|2 − [(P0 − P1) ⊡ D
̂]
2
δ. | = √|P0 − P1|2 − [(x0 − x1, y0 − y1, z0 − z1) ⊡
(𝒶, 𝒷, 𝒸)
√𝒶2 + 𝒷2 + 𝒸2
]
2
|P0 − P1| = √(x0 − x1)2 + (y0 − y1)2 + (z0 − z1)2
P0 = (x0, y0, z0) ∉ Recta
P1 = (x1, y1, z1) ∈ Recta
77
Distancia entre dos Rectas Alabeadas:
δ||a = (P2 − P1) ⊡
D1
⃗⃗⃗⃗ ⊠ D2
⃗⃗⃗⃗
|D1
⃗⃗⃗⃗ ⊠ D2
⃗⃗⃗⃗ |
δ||a = (x2 − x1, y2 − y1, z2 − z1) ⊡
D1
⃗⃗⃗⃗ ⊠ D2
⃗⃗⃗⃗
|D1
⃗⃗⃗⃗ ⊠ D2
⃗⃗⃗⃗ |
P1 = (x1, y1, z1) ∈ Recta #1
P2 = (x2, y2, z2) ∈ Recta #2
8
78
Distancia entre dos Rectas Paralelas:
δ||p = √|P1 − P2|2 − [(P1 − P2) ⊡ D
̂]
2
δ||p = √|P1 − P2|2 − [(x1 − x2, y1 − y2, z1 − z2) ⊡
(𝒶, 𝒷, 𝒸)
√𝒶2 + 𝒷2 + 𝒸2
]
2
|P1 − P2| = √(x1 − x2)2 + (y1 − y2)2 + (z1 − z2)2
P1 = (x1, y1, z1) ∈ Recta #1
P2 = (x2, y2, z2) ∈ Recta #2
La Directriz Unitaria “D
̂” puede ser de la Recta#1 o de la Recta#2.
79
Distancia entre dos Planos Paralelos.
δ□□ = (P2 − P1) ⊡ N
̂ = (x2 − x1, y2 − y1, z2 − z1) ⊡
(a, b, c)
√a2 + b2 + c2
P1 = (x1, y1, z1) ∈ Plano #1
P2 = (x2, y2, z2) ∈ Plano #2
La Normal “N
̂” puede ser de la Recta#1 o de la Recta#2.
80
Distancia entre Recta y Plano Paralelos:
δ|□ =
|a ⋅ x0 + b ⋅ y0 + c ⋅ z0 + d|
√a2 + b2 + c2
δ|□ = (P0 − P1) ⊡ N
̂ = (x0 − x1, y0 − y1, z0 − z1) ⊡
(a, b, c)
√a2 + b2 + c2
P0 = (x0, y0, z0) ∈ Recta
P1 = (x1, y1, z1) ∈ Plano
Elaboró: MCI José A. Guasco.
https://www.slideshare.net/AntonioGuasco1/

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Formulario Geometria Analitica

  • 1. 1 FORMULARIO DE GEOMETRÍA ANALÍTICA Coordenadas Cartesianas y Polares 01 x = ρ ⋅ Cos(θ) 02 y = ρ ⋅ Sen(θ) 03 ρ = √x2 + y2 04 θ = ∢Tan ( y x ) 1) ACCESORIOS 05 Punto Medio: PM = ( x1 + x2 2 , y1 + y2 2 ) Distancias 06 D. . = √(x2 − x1)2 + (y2 − y1)2 07 D. | = |y1 − m ⋅ x1 − b| √m2 + 1 = |D ⋅ x + E ⋅ y + F| √D2 + E2 08 D|| = |b2 − b1| √m2 + 1 09 Traslación de Ejes: x = X + h y = Y + k 10 Rotación de Ejes: x = X ⋅ Cos(θ) − Y ⋅ Sen(θ) y = X ⋅ Sen(θ) − Y ⋅ Cos(θ) ( x y) = ( Cos(θ) −Sen(θ) Sen(θ) −Cos(θ) ) ⋅ ( X Y ) 2) TRIGONOMETRÍA ANALÍTICA 11 Centroide: G = ( x1 + x2 + x3 3 , y1 + y2 + y3 3 ) 12 Área: A = 1 2 ⋅ | x1 y1 1 x2 y2 1 x3 y3 1 | 3) GEOMETRÍA ANALÍTICA 2D 13 Cónicas: A ⋅ x2 + B ⋅ x ⋅ y + C ⋅ y2 + D ⋅ x + E ⋅ y + F = 0 Recta 14 Ec. General: D ⋅ x + E ⋅ y + F = 0
  • 2. 2 15 Pendiente: m = y2 − y1 x2 − x1 m = Tan(θ) θ = ∢Tan(m) 16 Ec. Cartesiana: y − y1 x − x1 = y2 − y1 x2 − x1 17 Ec. Punto Pendiente: y − y1 = m ⋅ (x − x1) 18 Ec. Pendiente Ordenada: y = m ⋅ x + b 19 Ec. Pendiente Abscisa: y = m ⋅ (x − a) 20 Ec. Abscisa Ordenada: x a + y b = 1 21 Ángulo entre Rectas: θ = ∢Tan ( m2 − m1 1 + m1 ⋅ m2 ) Circunferencia 22 Ec. General: x2 + y2 + D ⋅ x + E ⋅ y + F = 0 23 Ec. Canónica: (x − h)2 + (y − k)2 = R2 24 P = 2 ⋅ π ⋅ R 25 A = π ⋅ R2 26 e = 0 Parábola 27 Ec. Generales: a, b] x2 + D ⋅ x + E ⋅ y + F = 0 c, d] y2 + D ⋅ x + E ⋅ y + F = 0 a] 28 Ec. Canónicas: a, b](x − h)2 = ±4 ⋅ p ⋅ (y − k) c, d](y − k)2 = ±4 ⋅ p ⋅ (x − h) 29 e = 1 30 AF = 4 ⋅ p 31 Focos: a] (h, k + p) b] (h, k − p) c] (h + p, k) d] (h − p, k) 32 Directrices: a] y = k − p b] y = k + p c] x = h − p d] x = h + p
  • 3. 3 a] b] c] d] Elipse 33 Ec. General: A ⋅ x2 + C ⋅ y2 + D ⋅ x + E ⋅ y + F = 0 A ⋅ C > 0 A ≠ C 34 Ec. Canónicas: a] (x − h)2 a2 + (y − k)2 b2 = 1 b] (y − k)2 a2 + (x − h)2 b2 = 1 a] a > b a > c 35 AF = 2 ⋅ b2 a 36 0 < e = c a < 1 37 a2 = b2 + c2
  • 4. 4 38 P = π ⋅ (a + b) b] 39 A = π ⋅ a ⋅ b 40 Focos: a] (h ± c, k) b] (h, k ± c) 41 Ápside = 2 ⋅ a Ápside = Afelio + Perihelio 42 Afelio = a + c 43 Perihelio = a − c 44 Distancia Media al Sol = a Hipérbola 45 Ec. General: A ⋅ x2 + C ⋅ y2 + D ⋅ x + E ⋅ y + F = 0 A ⋅ C < 0 46 Ec. Canónicas: a] (x − h)2 a2 − (y − k)2 b2 = 1 b] (y − k)2 a2 − (x − h)2 b2 = 1 a] c > a c > b 47 Ec. de Asíntotas: a] (x − h) a ± (y − k) b = 0 b] (y − k) a ± (x − h) b = 0 48 AF = 2 ⋅ b2 a 49 e = c a > 1
  • 5. 5 50 c2 = a2 + b2 b] c > a c > b 51 Focos: a](h ± c, k) b](h, k ± c) 4) GEOMETRÍA ANALÍTICA 3D Vectores 52 A ⃗ ⃗ = (ax, ay, az) 53 A ⃗ ⃗ = ax ⋅ î + ay ⋅ ĵ + az ⋅ k ̂ |A ⃗ ⃗ | = √ax 2 + ay 2 + az 2 { î = (1,0,0) ĵ = (0,1,0) k ̂ = (0,0,1) 54 Cosenos Directores. { Cos(αx) = ax |A ⃗ ⃗ | Cos(αy) = ay |A ⃗ ⃗ | Cos(αz) = az |A ⃗ ⃗ | 55 Vector Unitario. A ̂ = A ⃗ ⃗ |A ⃗ ⃗ | A ̂ = (ax, ay, az) |A ⃗ ⃗ | A ̂ = ( ax |A ⃗ ⃗ | , ay |A ⃗ ⃗ | , az |A ⃗ ⃗ | ) 56 Cos2(αx) + Cos2 (αy) + Cos2(αz) = 1 57 Producto PUNTO. A ⃗ ⃗ ⊡ B ⃗ ⃗ = (ax, ay, az) ⊡ (bx, by, bz) = { ax ⋅ bx + ay ⋅ by + az ⋅ bz |A ⃗ ⃗ | ⋅ |B ⃗ ⃗ | ⋅ Cos(γ)
  • 6. 6 58 Producto CRUZ. A ⃗ ⃗ ⊠ B ⃗ ⃗ = (ax, ay, az) ⊠ (bx, by, bz) = [ î ĵ k ̂ ax ay az bx by bz ] 59 |A ⃗ ⃗ ⊠ B ⃗ ⃗ | = |A ⃗ ⃗ | ⋅ |B ⃗ ⃗ | ⋅ Sen(γ) 60 Area = |A ⃗ ⃗ ⊠ B ⃗ ⃗ | 2 = |A ⃗ ⃗ | ⋅ |B ⃗ ⃗ | ⋅ Sen(γ) 2 61 Producto Mixto. Volumen = (A ⃗ ⃗ ⊠ B ⃗ ⃗ ) ⊡ C ⃗ Volumen = [ ax ay az bx by bz cx cy cz ] 62 |A ⃗ ⃗ ⊠ B ⃗ ⃗ | A ⃗ ⃗ ⊡ B ⃗ ⃗ = Tan(γ) 63 Proyección (o componente) de A ⃗ ⃗ sobre B ⃗ ⃗ . A ⃗ ⃗ B = (A ⃗ ⃗ ⊡ B ̂) ⋅ B ̂ 64 Paralelismo: Si: A ⃗ ⃗ ∥ B ⃗ ⃗ ⇒ A ⃗ ⃗ = k ⋅ B ⃗ ⃗ , k ∈ ℝ 65 Perpendicularidad: Si: A ⃗ ⃗ ⊥ B ⃗ ⃗ ⇒ A ⃗ ⃗ ⊡ B ⃗ ⃗ = 0 Plano 66 Vector en el Plano: V ⃗ ⃗ = (x − x1, y − y1, z − z1) ∈ Plano P1 = (x1, y1, z1) ∈ Plano 67 Vector perpendicular al Plano: N ⃗⃗ = (a, b, c) 68 Ecuación “Punto-Normal”. a ⋅ (x − x1) + b ⋅ (y − y1) + c ⋅ (z − z1) = 0 N ⃗⃗ ⊡ V ⃗ ⃗ = 0 69 Ecuación “General”. a ⋅ x + b ⋅ y + c ⋅ z + d = 0 d = −a ⋅ x1 − b ⋅ y1 − c ⋅ z1 Recta 70 Ecuación “Cartesiana”. x − x1 x2 − x1 = y − y1 y2 − y1 = z − z1 z2 − z1 = t P1 = (x1, y1, z1) ∈ Recta P2 = (x2, y2, z2) ∈ Recta 71 Ec. “Punto-Directriz”. x − x1 𝒶 = y − y1 𝒷 = z − z1 𝒸 = t 72 Directriz. D ⃗⃗ = (x2 − x1, y2 − y1, z2 − z1) D ⃗⃗ = (𝒶, 𝒷, 𝒸)
  • 7. 7 73 Ec. “Paramétricas”. { x = x1 + 𝒶 ⋅ t y = y1 + 𝒷 ⋅ t z = z1 + 𝒸 ⋅ t (x, y, z) = (x1, y1, z1) + (𝒶, 𝒷, 𝒸) ⋅ t Distancias 74 Distancia Punto (x1,y1,z1) a Punto (x2,y2,z2): δ. . = √(x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2 75 Distancia Punto (x0,y0,z0) a Plano: δ. □ = |a ⋅ x0 + b ⋅ y0 + c ⋅ z0 + d| √a2 + b2 + c2 δ. □ = (P0 − P1) ⊡ N ̂ = (x0 − x1, y0 − y1, z0 − z1) ⊡ (a, b, c) √a2 + b2 + c2 P0 = (x0, y0, z0) ∉ Plano P1 = (x1, y1, z1) ∈ Plano 76 Distancia Punto (x0,y0,z0) a Recta: δ. | = √|P0 − P1|2 − [(P0 − P1) ⊡ D ̂] 2 δ. | = √|P0 − P1|2 − [(x0 − x1, y0 − y1, z0 − z1) ⊡ (𝒶, 𝒷, 𝒸) √𝒶2 + 𝒷2 + 𝒸2 ] 2 |P0 − P1| = √(x0 − x1)2 + (y0 − y1)2 + (z0 − z1)2 P0 = (x0, y0, z0) ∉ Recta P1 = (x1, y1, z1) ∈ Recta 77 Distancia entre dos Rectas Alabeadas: δ||a = (P2 − P1) ⊡ D1 ⃗⃗⃗⃗ ⊠ D2 ⃗⃗⃗⃗ |D1 ⃗⃗⃗⃗ ⊠ D2 ⃗⃗⃗⃗ | δ||a = (x2 − x1, y2 − y1, z2 − z1) ⊡ D1 ⃗⃗⃗⃗ ⊠ D2 ⃗⃗⃗⃗ |D1 ⃗⃗⃗⃗ ⊠ D2 ⃗⃗⃗⃗ | P1 = (x1, y1, z1) ∈ Recta #1 P2 = (x2, y2, z2) ∈ Recta #2
  • 8. 8 78 Distancia entre dos Rectas Paralelas: δ||p = √|P1 − P2|2 − [(P1 − P2) ⊡ D ̂] 2 δ||p = √|P1 − P2|2 − [(x1 − x2, y1 − y2, z1 − z2) ⊡ (𝒶, 𝒷, 𝒸) √𝒶2 + 𝒷2 + 𝒸2 ] 2 |P1 − P2| = √(x1 − x2)2 + (y1 − y2)2 + (z1 − z2)2 P1 = (x1, y1, z1) ∈ Recta #1 P2 = (x2, y2, z2) ∈ Recta #2 La Directriz Unitaria “D ̂” puede ser de la Recta#1 o de la Recta#2. 79 Distancia entre dos Planos Paralelos. δ□□ = (P2 − P1) ⊡ N ̂ = (x2 − x1, y2 − y1, z2 − z1) ⊡ (a, b, c) √a2 + b2 + c2 P1 = (x1, y1, z1) ∈ Plano #1 P2 = (x2, y2, z2) ∈ Plano #2 La Normal “N ̂” puede ser de la Recta#1 o de la Recta#2. 80 Distancia entre Recta y Plano Paralelos: δ|□ = |a ⋅ x0 + b ⋅ y0 + c ⋅ z0 + d| √a2 + b2 + c2 δ|□ = (P0 − P1) ⊡ N ̂ = (x0 − x1, y0 − y1, z0 − z1) ⊡ (a, b, c) √a2 + b2 + c2 P0 = (x0, y0, z0) ∈ Recta P1 = (x1, y1, z1) ∈ Plano Elaboró: MCI José A. Guasco. https://www.slideshare.net/AntonioGuasco1/