MOISES VILLENA
R
3
2 2 .1 RECTAS EN R 3 2 .2 PLANOS 2 .3 P OSI OSI CI ONES RELATI VAS 2 .4 SUPERFICIES 2.4.1 SUPERFICIES CILINDRICAS 2.4.2 SUPERFICIES DE REVOLUCIÓN 2.4.3 CUADRICAS 2 .5 COORDENADAS OORDENADAS CILÍ NDRI CA. 2 .6 COORDENADAS ESFÉRICAS.
Se persigue que el estudiante: • Encuentre ecuaciones de Rectas y Planos. • Grafique Rectas y Planos. • Encuentre distancias. • Grafique Superficies Cilíndricas, de Revolución y Cuádricas.
13
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MOISES VILLENA
R
3
2.1 RECTAS EN R 3 2.1.1 DEFINICIÓN →
Sea P0 un punto de R y sea S un vector de R 3 . Una Recta l se define como el conjunto de puntos P de R 3 que 3
→
⎯ ⎯→
contiene a P0 y tal que los vectores V = P0 P son paralelos →
a S. Es decir: → → → ⎯ ⎯→ ⎧ ⎫ l = ⎨ P( x, y, z ) / P ∈ l y S // V donde V = P P ⎬ ⎩ ⎭ 0
0
→
Al Vector S se lo llama VECTOR DIRECTRIZ de la recta. 2.1.2 ECUACIÓN →
Sea P0 ( x0 , y0 , z 0 ) y sea el vector S = (a, b, c ) . z
l
P( x, y , z )
•
→
S = (a, b, c ) →
V
•
P 0 ( x 0 , y 0 , z 0 )
y
x →
El vector S es paralelo al vector entonces: →
→
V = k S
Reemplazando resulta:
→
→
V = P0 P = ( x − x0 , y − y0 , z − z 0 ) ,
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MOISES VILLENA
R
3
⎧( x − x ) = ka ⎪( y − y ) = kb ⎨ ⎪( z − z ) = kc ⎩ 0
0
0
Entonces tenemos:
x − x0 a
=
y − y0 b
=
Ecuación de la recta definida por un punto P0 ( x 0 , y 0 , z 0 ) y → un vector paralelo S = (a, b, c )
z − z 0 c
En ocasiones anteriores ya se ha mencionado que dos puntos definen una recta, observe la figura: z
l
• P ( x , y , z ) 2 2 2 2 P( x, y , z )
•
→
→
V
S
• y
P 1 ( x1 , y1 , z1 )
x
Ahora tenemos que, P0 = P1 ( x1 , y1 , z1 ) y el vector directriz sería: →
S
→
= PP 1
2
⎛ ⎞ ⎜ = x − x , y − y , z − z ⎟ , ⎜ ⎟ ⎝ ⎠ 2 1 2 1 2 1 1 2 3 123 1 2 3
a
b
c
Entonces, se tiene: x − x1 x2
− x
1
=
y − y1 y 2
− y
1
=
z − z1 z 2
− z
1
Ecuación de la recta definida por dos puntos
También se la llama ECUACIÓN CANÓNICA O ECUACIÓN SIMÉTRICA.
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MOISES VILLENA
R
Si consideramos:
x − x0
=
a
Tenemos:
y − y 0 b
⎧ x = x + at ⎪ ⎨ y = y + bt ⎪ z = z + ct ⎩
z − z 0
=
c
3
= t
Ecuaciones Parámetricas
0
0
0
De lo anterior:
( x, y, z ) = ( x + at , y + bt , z + ct ) ( x, y, z ) = ( x , y , z ) + t (a, b, c ) 0
0
0 0 0 1 4 24 3
0
123 ⎯ ⎯→
⎯ ⎯→
V
S
0
Se puede expresar de la siguiente manera: →
→
→
Ecuación Vectorial
V = V 0 + t S
Hallar las Ecuaciones paramétricas de la recta que contiene al punto
P(1,−1 − 1)
y
→
es paralela al vector S = (1,0,2) . SOLUCIÓN: De a cuerdo a lo definido:
⎧ x = x 0 + at = 1 + t ⎪ = + = −1 ⎨ y y 0 bt ⎪ z = z + ct = 1 + 2t 0 ⎩
1.
Halle ecuaciones paramétri paramétricas cas de la recta que contiene los puntos (2, 1, 3)
Grafíquela
y
(1, 2, -1).
⎧ x = 1 + t ⎪ Resp. l : ⎨ y = 2 − t ⎪ z = −1 − 4t ⎩ 2.
Halle ecuaciones paramétri paramétricas cas de la recta que contiene los puntos (2, 1, 0)
y
Grafíquela. ¿Qué conclusió n puede emitir? ¿Cuál sería la ecuación del eje z?
(2,1, 5).
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MOISES VILLENA
R
5.
Halle ecuaciones paramétri paramétricas cas de la recta que contiene los puntos (2, 3, 5)
3
y
(2,2, 0).
Halle ecuaciones paramétri paramétricas cas de la recta que contiene los puntos (0, 2, 2)
y
(2,2, 0).
Halle ecuaciones paramétri paramétricas cas de la recta que contiene los puntos (2, 0, 2)
y
(0,2, 2).
Grafíquela. ¿Qué concl usión puede emitir? 6.
7.
Grafíquela. ¿Qué concl usión puede emitir? Grafíquela. ¿Qué concl usión puede emitir?
8.
Halle ecuaciones ecuaciones paramétricas étricas de la recta que contiene el punto (-1, -6, 2) y es paralel paralela a al vector (4, 1, -3). Grafíquela
⎧ x = −1 + 4t ⎪ Resp. l : ⎨ y = −6 + t ⎪ z = 2 − 3t ⎩ 9.
Halle ecuaciones paramétricas de la recta que pasa por el origen y es perpendicular a la recta 1 1 1 ( x − 10) = y = z . cuya ecuación es: 4 3 2
Resp.
⎧ x = t ⎪ l : ⎨ y = 2t ⎪ z = −5t ⎩
2.2 PLANOS 2.2.1 DEFINICIÓN →
Sea P0 un punto de R y sea n un vector de R 3 . Un Plano π se define como el conjunto de puntos P de R 3 tales que 3
→
→
n es perpendicular al vector V que se define entre P0 y P .
Es decir: → → → → ⎧ π = ⎨ P( x, y, z ) / n• V = 0 donde V = P P y P ∈ R ⎫ ⎬ ⎩ ⎭ 3
0
0
2.2.2 ECUACIÓN →
Sean n = (a, b, c ) y P0 ( x0 , y0 , z 0 ) . Observe la figura:
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MOISES VILLENA
R
3
z
→
n = (a, b, c )
π
P 0 ( x 0 , y1 , z1 )
→
V P( x, y, z ) y
x
Entonces →
→
n• V = 0
(a, b, c ) • ( x − x
0
, y − y 0 , z − z 0 ) = 0
Por tanto, tenemos: a( x − x 0 ) + b( y − y 0 ) + c( z
Ecuación de un plano definida por UN PUNTO Y UN VECTOR PERPENDICULAR.
− z )= 0 0
Si se simplifica, tenemos: a( x − x0 ) + b( y − y0 ) + c( z − z 0 ) = 0 ax + by + cz + (− ax0
− by − cz ) = 0 0
0
Considerando d = −ax0 − by0 − cz0 , tenemos: ax + by + cz + d = 0
ECUACIÓN GENERAL de un plano.
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MOISES VILLENA
R
Hallar la ecuación del plano que contiene a los puntos P3 (2,−1,0) SOLUCIÓN:
P1 (1,2,3) , P2 (−1,0,1)
3
y
Con los tres puntos dados se forman dos vectores (no importa el orden) para de ahí obtener un vector perpendicular al plano buscado. →
→
→
n = V 1 × V 2
P2 (− 1,0,1) →
V 2 P3 (2,−1,0) →
P1 (1,2,3)
V 1
EE En este caso: →
V 1 →
V 2
→
= P1P3 = (2 − 1,−1 − 2,0 − 3) = (1,−3,−3) →
= P1 P2 = (− 1 − 1,0 − 2,1 − 3) = (− 2,−2,−2)
Entonces →
n
→
i
→
= V 1× V 2 =
1
−2 →
n
j
k
− 3 − 3 = (6 − 6 )i − (− 2 − 6) j + (− 2 − 6)k −2 −2
= 0i + 8 j − 8k {
{
{
a
b
c
Podemos tomar P0 ( x0 , y 0 , z 0 ) = P1 (1,2,3) (puede ser cualquier otro punto del plano) Finalmente, empleando la ecuación: a( x − x0 ) + b( y − y0 ) + c( z − z 0 ) = 0
Resulta:
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MOISES VILLENA
R
3
Si el plano tiene intersección A, B, C con los ejes coordenados entonces tenemos tres puntos que pertenecen al plano y se puede determinar su ecuación como en el ejemplo anterior. Observe bserve la figura: z
P2 (0, B,0)
π →
→
V 2 = P1 P2 P3 (0,0, C ) →
→
V 1 = P1 P3 P 1 ( A,0,0) x →
→
En este caso tomamos: V 1 = (− A, B,0 ) y V 2 Entonces: →
n
→
i
→
= V 1× V 2 = − A − A
j
k
B
0
0
C
= (− A,0, C )
= ( BC )i − (− AC ) j + ( AB )k
Si tomamos P0 ( x0 , y 0 , z 0 ) = P1 ( A,0,0) y reemplazando en la ecuación a( x − x0 ) + b( y − y0 ) + c( z − z 0 ) = 0
Resulta: BC ( x − A) + AC ( y − 0) + AB( z − 0) = 0 BCx − ABC + ACy + ABz BCx + ACy + ABz
=0
= ABC
Dividiendo para ABC BCx ABC
+
ACy
+
ABz
ABC ABC x y z A
+
B
+
C
=
=1
ABC ABC
y
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MOISES VILLENA
R
1.
Dibuje los planos cuyas ecuaciones son: a) 4 x + 2 y + 6 z = 12 d) x + 2 y b)
3 x + 6 y + 2 z
=4 e) 2 x + y − z = 6 f) x − 3z = 3
=6
3.
=0
y + z = 5 Encuentre Encuentre la ecuación del plano que contienen contienen al punto (-5,7,-2) (-5,7,-2) y que es paralelo al plano "xz" Resp. y = 7 Encuentre Encuentre la ecuación del plano que contienen contienen al punto (-5,7,-2) (-5,7,-2) y que es perpendicular perpendicular al eje "x" Resp. x = −5 c)
2.
g) x + y + z
3
4.
Encuentre Encuentre la ecuación del plano que contienen al punto (-5,7,-2) (-5,7,-2) y que es paralelo tanto al eje "x" como al de "y" Resp. z = −2
5.
Encuentre Encuentre la ecuación del plano que contienen contienen al punto (-5,7,-2) (-5,7,-2) y que es paralelo al plano 3 x − 4 y + z = 7
3 x − 4 y + z
Resp. 6.
Hallar la ecuación del plano paralelo al plano x + 3 y − 2 z + 14 = 0 y tal que la suma de sus intersecciones con los ejes coordenados sea igual a 5. Resp. x + 3 y − 2 z
7.
= −45
=6
Hallar la ecuación del plano que es paralelo al plano 3 x + 8 y − 5 z + 16 = 0 y que intercepta a los ejes coordenados en los puntos A, B y C, de tal manera que A + B + C = 31. Resp. 3 x + 8 y − 5 z = 120
2. 3. POSICIONES RELATIVAS 2.3.1 ENTRE UN PUNTO
P0
Y UNA RECTA
l
2.3.1.1 EL PUNTO PERTENECE A LA RECTA: P0 ∈ l l:
x − x1 a
=
y − y1 b
=
z − z1 c
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MOISES VILLENA
R
3
2.3.1.2 EL PUNTO NO PERTENECE A LA RECTA: P0 ∉ l l:
x − x1
=
y − y1
a
=
b
z − z1 c
P0 ( x 0 , y 0 , z 0 )
Si un punto no pertenece a una recta entonces las coordenadas del punto no satisfacen la ecuación de la recta, es decir: x − x z −z y0 − y1 z − z x0 − x1 y − y ≠ 0 1 ≠ 0 1 o 0 1≠ 0 1 o a
a
b
c
b
c
2.3.1.2.1 Distancia del punto a la recta Si escogemos un punto P cualquiera de la recta y definimos un vector →
V entre este punto P y el punto P0 .
P0 ( x 0 , y 0 , z 0 ) d = h →
V
θ
→
S
l:
x − x1 a
=
y − y1 b
=
z − z1 c
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MOISES VILLENA
R
Observe que senθ =
3
→
h
entonces h = V senθ
→
V →
→
→
Reemplazando resulta V × S = S h Finalmente: →
→
V × S h = d (P0 , l ) =
→
S
recta.
2.3.1.2.2 Ecuación del plano que contiene al punto y a la →
Un vector normal al plano será el resultante del producto cruz de V con →
S →
→
→
n = V × S
P0 ( x 0 , y 0 , z 0 ) →
V
π
→
S
P( x, y, z )
Como punto del plano tenemos para escoger entre P0 y cualquier punto de la recta.
Sea
P0 (1 2 3)
y sea l :
x − 1
=
y + 2
=
z −1
. Hallar la distancia de
P0
a
l
y la
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MOISES VILLENA
R
→ →
V × S →
S
=
= (− 14)2 + 4 2 + (− 8)2 = 22
+ 32 + (− 2)2 =
3
276
17
Por lo tanto: →
→
V× S d ( P0 , l ) =
→
=
S →
Por otro lado, un vector normal al plano sería: n
27 6 17
=2
69 17
→ →
= V × S = (− 14,4,−8)
Escogiendo el punto P0 , tenemos: a ( x − x0 ) + b ( y − y 0 ) + c ( z − z 0 )
=0 −14 ( x− 1) + 4 ( y− 2 ) − 8 ( z− 3) = 0 −14 x+ 14 + 4 y− 8 − 8 z+ 24 = 0 Por tanto, la ecuación del plano sería: π : 7 x− 2 y+ 4 z− 15 = 0
2.3.2 POSICIONES RELATIVAS ENTRE UN PUNTO Y UN PLANO π 2.3.2.1 EL PUNTO PERTENECE AL PLANO: P0 ∈ π .
P0
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MOISES VILLENA
R3
En este caso las coordenadas del punto NO satisfacen la ecuación del plano, es decir: ax0 + by 0 + cz 0 + d ≠ 0 . 2.3.2.3
Distancia del punto al plano.
Si tomamos un punto P cualquiera del plano y formamos el vector ⎯ ⎯→
V
⎯ ⎯→
= PP = ( x − x, y − y, z − z ) . Observe la figura anterior. 0
0
0
0
⎯ ⎯→
La distancia del punto al plano será la proyección escalar de V sobre ⎯ ⎯→
n , es decir: ⎯ ⎯→ ⎯ ⎯→
d (P0 , π ) =
V • n →
=
n
Observe que:
( x − x, y − y, z − z ) • (a, b, c ) a +b +c 0
0
0
2
=
ax0
=
ax0
2
2
− ax + by − by + cz − cz a +b +c + by + cz − ax − by − cz a +b +c 0
2
0
0
2
2
0
2
d = − ax − by − cz
2
2
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MOISES VILLENA
R3
2.3.3 POSICIONES RELATIVAS ENTRE DOS RECTAS l Y l . 1
2
2.3.3.1 RECTAS COINCIDENTES l1 :
x − x1
=
a
y − y1 b
=
z − z1 c
⎯⎯→
S1 = ( a, b, c )
l2 :
x − x1´ a´
=
y − y1´ b´
=
z − z1´ c´
⎯⎯→
S2 = ( a´, b´, c´)
Dos rectas son coincidentes si y sólo si: ⎯ ⎯→ ⎯ ⎯→
1. Sus vectores directrices son paralelos: S1 // S 2 ; y, 2. Todos los puntos puntos que pertenecen a una recta también pertenecen a la otra recta; para esto, bastará que un punto de una recta satisfaga la ecuación de la otra recta.
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MOISES VILLENA
R3
2.3.3.2 RECTAS PARALELAS: l1 :
x − x1 a
=
y − y1 b
=
l1 // l 2
z − z1 c
⎯⎯→
S1 = ( a, b, c)
l2 :
x − x1´ a´
=
y − y1´ b´
=
z − z1´ c´
⎯⎯→
S2 = ( a´, b´, c´)
Dos rectas son paralelas si y sólo si: ⎯ ⎯→ ⎯ ⎯→
1. Sus vectores directrices son paralelos: S1 // S 2 ; y, 2. Ningún punto de una recta pertenece a la otra recta; para esto, bastará que un punto de una recta NO satisfaga la ecuación de la otra recta.
Sean
l1 :
x − 1
=
y + 1
=
z−2
y
l2 :
x
=
y − 1
=
z +1
−1 −3 6 9 a) Demuestre que son l1 y l 2 son rectas paralelas. b) Determine la distancia entre l1 y l 2 . c) Encuentre la ecuación del plano qu e contiene a l1 y 2
3
.
l2
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MOISES VILLENA
R
→
→
V × S 2 d (l 1 , l 2 ) = d (P0 , l 2 ) =
→
S2
→
→
V × S 2
→
=
i
j
k
1
−2
3
2
3
−1
→
V × S 2 →
S2
=
= (− 7,7,7 )
= (− 7 )2 + 7 2 + 7 2 = 7 6
2
+ 9 2 + (− 3)2 = 3
Por tanto: d (l 1 , l 2 ) = d (P0 , l 2 ) =
3
14
7 3 3 14
d) Las dos rectas paralelas definen un plano que contiene a ambas. →
→
→
n = V × S
π P0 (1,−1,2 ) →
V = (1,−2,3)
3
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MOISES VILLENA
R
2.3.3.2 RECTAS INTERSECANTES. l1
l2
→
S1 →
•
Dos rectas son intersecantes si y sólo si:
S2 P0 ( x 0 , y 0 , z 0 )
3
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MOISES VILLENA
R
t = 1
y
3
k = 1
Entonces:
⎧ x 0 = 1 + 2(1) = 3 ⎪ ⎨ y 0 = −(1) = −1 ⎪ z = −1 + 3(1) = 2 ⎩ 0 Note que igual resultado se obtiene en la segunda condición:
⎧ x 0 = 3(1) = 3 ⎪ ⎨ y 0 = 2 − 3(1) = −1 ⎪ z = 1 + (1) = 2 ⎩ 0 Por tanto, las rectas se intersecan en sólo un punto. b) El ángulo de corte está determinado por el ángulo que forman los vectores directrices; es decir: →
θ = arccos
→
S1 • S 2 →
→
(2,−1,3) • (3,−3,1)
= arccos 2
S1 S 2
= arccos
2
+ (− 1)2 + 3 2
12 14 19
3
2
+ (− 3)2 + 12
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MOISES VILLENA
R
2.3.3.2 RECTAS OBLICUAS O ALABEADAS. Dos rectas son Oblicuas o Alabeadas si y sólo si: 1. Sus vectores vectores directrices directrices NO son paralelos; paralelos; y, 2. Ningún punto punto de una recta pertenece a la otra recta. recta. l1
→
S1
→
S2 l2
En este caso no existirá algún plano que contenga a ambas rectas.
3
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MOISES VILLENA
R
3
2.3.3.2.1 Distancia entre Rectas oblicuas. →
Definamos un vector V , entre un punto cualquiera de una recta con otro punto cualquiera de la otra recta, Observe la figura: →
→
S ×S 1
2
•
l
1
→
S
1
→
S
2
}
d
⎯ ⎯→
V
•
l
2
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MOISES VILLENA
R
3
Por tanto,
⎛ •⎜ ⎝
⎯⎯→
V d ( l1 , l2 ) =
=
d ( l1 , l2 ) =
×
⎯⎯→
⎯⎯→
S1
S2
×
⎞ ⎟ ⎠ = (1, −2, 0 ) • ( −5, 5, 5) 2 ( −5 ) + 52 + 52
⎯⎯ →
S2
(1, −2, 0 ) • 5 ( −1,1,1) 5
=
⎯⎯ →
S1
2 ( −1) + 12 + 12
−1 − 2 + 0 3 3 3
2.3.4 POSICIONES RELATIVAS ENTRE DOS
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MOISES VILLENA
R
Los planos π 1 : 2 x − 3 y + z + 1 = 0 y π 2 : 4 x − 6 y + 2 z + 2 = 0 son coincidentes debido a que: −3 1 1 2 = = = 4 −6 2 2
2.3.4.2 PLANOS PARALELOS: Dos planos son Paralelos si y sólo si:
π 1 // π 2
3
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MOISES VILLENA
R
3
2.3.4.3 PLANOS INTERSECANTES Dos planos son intersecantes si y sólo si sus vectores normales NO son paralelos.
π
1
:
a1 x
+
b1 y
+
c 1 z
+
d 1
=
0
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MOISES VILLENA
R
= −3 − 5 y ⇒ x + y − 3 − 5 y = −2 Haciendo y = t , entonces: ⎧ x = 1 + 4t ⎪ l : ⎨ y = t ⎪ z = −3 − 5t ⎩ z
3
⇒ x = 1 + 4 y
Segundo Método: Un vector directriz de la recta buscada estaría dado por el vector resultante del producto cruz entre los vectores normales de los planos, es decir: i
j
k
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MOISES VILLENA
R
3
2.3.5 POSICIONES RELATIVAS ENTRE UNA RECTA Y UN PLANO. 2.3.5.1 RECTA PERTENECIENTE PERTENECIENTE A UN PLANO. Una recta pertenece a un plano si y sólo si todos los puntos de la recta pertenecen también al plano.
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MOISES VILLENA
R
3
2.3.5.1 RECTA PARALELA A UN PLANO. Una recta es paralela a un plano si y sólo si todos los puntos de la recta NO pertenecen al plano.
→
→
n
S
⎧ x = x0 + a´t ⎪ l : ⎨ y = y0 + b´t ⎪ z = z + c´t 0 ⎩
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MOISES VILLENA
R
6.
Calcule la distancia del plano 2 x + 2 y − z
=6
al punto (2, 2, -4).
3
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⎧ x = t
3
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Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.