Formulario Geometría Analítica en 2 y 3 dimensiones.

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/