Calculo 2 De dos variables_9na Edición - Ron Larson

En los ejercicios 41 a 46, hallar una ecuación del plano que pasa
In
por In el Exercises punto y es 41–46, perpendicular find an an al equation of
vector o of of recta the dado.
plane passing
In through
Exercises the point
41–46, perpendicular
find an equation to
to to the given
of the vector
plane or or line.
passing
through the point perpendicular to the given vector or line.
Punto
a
Point
Perpendicular to
to
to
Point
1,
1,
3,
3, Perpendicular j
to
41.
1, 3, 41.
0,
1, 3,
0,
1,
1, 7
n
j
42.
0, 1, 42.
3,
0,
3,
2,
2, 1, 4
n
2i
k
43.
3, 2, 2i 2i 3j
3j 3j 43.
0,
3,
0,
0,
2,
0, 2
n 2i 3i
3j
3i
2k
k
44.
0, 0, 3i 2k
44. 0, 1,
0,
1,
4,
0
4, x
n 3i 2t,
2k
45.
1, 4, 2t, y t,
t, t, z 2t
2t
2t
45. 1, 4, 0
x
x 1 2t, y z
5 t, z 3 2t
3,
3, 2,
2, y 46. 3, 2, x 1
z 3
46. 3, 2, 2
y 2
4
3
In
En In Exercises los ejercicios 47–58, a find an
58, an hallar equation of
una ecuación
of of the plane.
In Exercises 47–58, find an equation of the plane. del plano.
47. 47. El The plano plane que passes pasa through 0,
por (0, 0,
0, 0, 0,
0),
0, 0, (2, ,
2,
0,
2, 2, 0,
3)
0, 0, y ,
(3, and 3,
–1, 5).
3, 3, 1,
1, 1, .
47.
48. The
El plane
plano passes
que through
pasa 3,
0, 0,
por (3, –1,
3,
1,
0
2),
1, ,
(2, ,
2, 2,
0,
1,
2,
1,
3
5)
1, ,
y ,
and
(1,2, 1,
3,
–2).
1,
2,
1, 5
2, .
48. The plane passes through 3, 1, 2, 1, and
1, 2, 2.
48.
49. The
El plane
plano passes
que through
pasa 1,
3,
por (1, 2,
1,
2,
1,
3),
2, (3, ,
2 3,
, 2,
2,
3,
2,
1,
1)
2, y ,
5
(1, , and 1, 1,
2,
2,
1,
2,
2).
2, 2.
49. The plane passes through 1, 2, 3, 2, and
1, 2, .
49.
50. The
El plane
plano passes
que through
pasa por 1, 2,
el punto 3 , 3,
(1, 1,
2,
2, 3)
1,
2,
1
y
2, , and
es paralelo is
is 1, 2, 2
al to
.
50. The plane passes through the point 1, 2, and is parallel
plano yz.
to
to
50. The the
yz-
plane.
passes through the point 1, 2, 3 and is parallel to
El plano que pasa por el punto (1, 2, 3) y es paralelo al plano xy.
the yz-
plane.
51. The plane passes through the point 1,
1, 1, 2,
2, 2, and is
is is parallel to
to
to
52. 51. El The the plano xy-
plane. contiene passes el through eje y y forma the point un ángulo 1, 2, 3de and 6is con parallel eje to x
positivo.
the
xy-plane.
52. The plane contains the y-
y-
y-
axis and makes an an angle of
of of with
53. 52. El The the plano positive
plane contiene contains x-
x-
x-
axis. las the rectas y-axis dadas and por makes an angle of 6 with
x the positive
1
x 2
y
3 y 1 z 2
2
y x-axis.
53. The plane contains 4 z
the lines given by
by
53. The plane contains the lines given by
x 4 1 .
x y z y z
x
54. El plano 1
and x
pasa por el punto (2, 2, 1) 2 y
y contiene 1 z
la recta .
2
y 4 z and
dada por
2
3 4 1 .
54. x
The
2 y plane 4
1 passes z. through the point 2,
2, 2, 2,
2, 2, and contains the
54. The line given
plane by
by
passes through the point 2, 2, 1 and contains the
line given by
x
55. El y
plano pasa z. por los puntos (2, 2, 1) y (1, 1, 1) y es perpendicular
x
y 4
z.
al plano z. 2x 3y z 3.
z.
2 1
56. 55. El The plano plane pasa passes por los through puntos the (3, points 2, 1) y 2,
2, 2, (3, 2,
2, 2, 1, 5) and y es 1,
1, 1, perpendicular
The and is
1,
1, 1, 55.
is is
plane al perpendicular plano passes 6x through to to to 7y the the 2z plane points 2x 10.
2x
2, 3y 3y
2, 1z and 3.
3.
3.
1, 1, 1
57. El and plano is perpendicular pasa por los puntos
to the plane
(1, 2,
2x 1) 3,
3y
3, y 2,
2, (2, z 5, 6)
3.
56. The plane passes through the points 3, and y 3,
3,
es 3, 1,
1,
parale-
1, 56. lo The and al is
is
eje is
plane perpendicular x. passes through to
to to the plane
the points 6x 6x 7y 7y
3, 2, 2z 1
2z
and
10.
3, 1, 5
58. El and plano is perpendicular pasa por los puntos
to the plane
(4, 2,
6x
1) y 1,
1, (3,
7y
2,
2, 5,
2z 7) y
10.
57. The plane passes through the points 1, 2, and es paralelo 2,
2, 2, 5,
5, 5, 57. al The and eje is
is is
plane z.
parallel
passes to
to to the
through x-
x-
x-
axis.
the points 1, 2, 1 and 2, 5, 6
58. and The is parallel plane to the
passes x-axis.
through the points 4,
4, 4, 2,
2, 2, and 3,
3, 3, 5,
5, 5, En 58. los The and ejercicios is
is is
plane parallel
passes 59 to
to to y the 60, through z-
z-
z- representar axis.
the points gráficamente 4, 2, 1 and la recta 3, y 5, hallar
los and puntos is parallel de intersección to the z-axis.
(si los hay) de la recta con los planos
7
xy, In In xz Exercises y yz. 59 59 and 60, sketch graph of
of of the line and find the
In points
Exercises (if (if any)
59 where
and 60, the line
sketch intersects
a graph the
of xy
xy
the -,
-, -, xz
xz
line -,
-, -, and
and yz
yz
yz
find -planes.
the
59. points x (if 1 any) 2t, where y 2 the line 3t, intersects z 4 the xyt
-, xz-, and yz-planes.
59. x 2
2t,
2t, y 60. y 1 z 3 3t,
3t, z t
59.
x
x 3
1 2t, y z
2 2
3t, z 4 t
60. y x 2
z 3
60. y 1
3
2
En In los ejercicios a 64, hallar una ecuación del plano que con-
In Exercises 61– 64, find an equation of
of of the plane that
tiene In todos los puntos equidistantes de los puntos dados
contains
Exercises all all the 61– points
64, that
find are equidistant
equation from
of the the given
plane points.
that
contains all the points that are equidistant from the given points.
61. 2,
2, 2, 2,
2, 2, ,
0,
0, 0, 2,
2, 2, 62. 1,
1, 1, 0,
0, 0, ,
2,
2, 2, 0,
0, 0, 1)
1)
1)
61. 2, 63. 3,
2,
3, 3, 1,
0
1, 1, ,
,
0, 6,
2, 2
6, 6, 2,
2, 2, 62. 1, 64. 5,
0,
5, 5, 1,
2 ,
1, 1, 3 3, ,
2, 0,
2,
1)
2, 2, 1,
1, 1, 63. 3, 1, 2 , 6, 2, 4 64. 5, 1, 3, 2, 1, 6
CAS
CAS
SECCIÓN 11.5 Rectas y planos en el espacio 809
11.5 Lines and Planes in
in in Space 809
11.5 Lines and Planes in Space 809
En los ejercicios 65 a 70, determinar si los planos son paralelos,
In ortogonales, In Exercises o 65–70, ninguna determine las dos whether cosas. Si the no planes son ni are paralelos parallel, ni
In ortogonales, orthogonal,
Exercises or 65–70,
hallar or neither.
determine If
el ángulo If If de they
whether intersección.
are neither
planes parallel
are parallel,
nor
orthogonal, orthogonal, find
or the
neither. angle of
of of
If intersection.
they are neither parallel nor
65. orthogonal, 5x 3y find z the angle of intersection. 66. 3x y 4z 65. 5x 5x 3y 4y 3y z
7z 66. 3x 3x y
9x 4z
3y 4z 12z 65. x
5x
67. 4y
3y 7z
z
3y 4
66. 3x
6z 9x
y 68. 3x 3y
4z
2y 3
4y 7z 9x 3y
z 12z x
x 3y
4y
5x 6z
7z
y 1
z 3x
9x 2y
3y 4y z
12z
2z 4
67. 3y 6z 68.
3x 2y 67. 5x
x 3y
69. y 5y z
6z
4
68.
x
3x 70.
4y
2y
2x 2z
z
z 7
5x 4y 2z x
5x 5y
y
5x z
z
25y 4
5z 3
2x
x 4y z
4x 2z 0
69. 5y 70.
2x y 69. 8z 10
5x
x 5y z 5z
1
70.
4x
2x y
z 8z
1
5x 25y 5z 4x 8z 10
10
5x 25y 5z 3
4x y 8z 10
In En In Exercises los ejercicios 71–78, a sketch 78, marcar graph toda of
of of intersección the plane and y dibujar label any la
gráfica In intercepts.
Exercises del plano. 71–78, sketch a graph of the plane and label any
intercepts.
71. 4x 4x 2y 2y 6z 6z 12
12
72. 3x 3x 6y 6y 2z 2z 71. 2x
4x y
2y 3z
6z 12
72.
2x
3x y
6y z
2z 6
73. 2x 3z 74.
2x 73. x
2x z
y 3z 4
74.
2x
2x y
y z 4
75. 76.
2x 75. x
x z 6
76.
z
2x y 8
77. 78.
77. x 5
78. z 8
In En In Exercises los ejercicios 79–82, a use 82, usar computer un sistema algebra algebraico system to
to to por graph compu-
the
In tadora plane.
Exercises 79–82, use a computer algebra system to graph the
para representar gráficamente el plano.
plane.
79. 2x 2x y
z 80. x
3z 3z 79. 2x 5x
5x
y
4y
z
6z
6
8
80.
x 3z 3
81. 4y 6z 82.
2.1x 4.7y z 3
81. 5x 4y 6z 8 82. 2.1x 4.7y z 3
In In En Exercises los ejercicios 83–86, a determine if
86, determinar
if if any of
si
of of algunos the planes de los are planos parallel
or In son
or Exercises
paralelos identical.
83–86, determine if any of the planes are parallel
or identical. o idénticos.
83. 1 :
6y 17
1 :
2x y 3z 1 15x 6y 24z 17 84.
1 2x 3z 83. P 1 2 :
: 15x 5x
6y 2y
24z 8z 17 84.
P 1 2 :
:
3x
2x 5y
y 3z 2z
8
2 5x 2y 8z 2 3x 5y 2z P 3 : 2 :
6x
5x 4y
2y 4z
8z 6
P 2 3 :
:
8x
3x 4y
5y 2z 6
3 6x 4y 4z 3 8x 4y 12z P 4 : 3 :
3x
6x 2y
4y 2z
4z 9
P 3 4 :
: 8x 4x
4y 2y
12z 6z
5
11
4 3x 2y 2z 4 4x 2y 6z 11
P 1 : 4 :
3x
3x 2y
2y 5z
2z 10
4 P 4 : 4x 2y 6z 11
85.
1 3x 2y 5z 10
85.
P 1 2 :
: 3x 6x
2y 4y
5z 10
2 6x 4y 10z P 3 : 2 :
3x
6x 2y
4y 5z
10z 5
3 3x 2y 5z P 4 : 3 :
3x 2y 5z 8
4 75x 50y 125z 250
P 1 : 4 : 75x 50y 125z
86.
250
27
1 60x 90y 30z 27
86.
P 1 2 :
:
6x
60x 9y
90y 3z
30z 27
2 6x 9y 3z P 3 : 2 : 6x 9y 3z 2
3 20x 30y 10z P 4 : 3 :
20x 12x 30y 18y 6z
10z 6z 9
4 P 4 : 12x 18y 6z 5
In In Exercises 87– 90, describe the family of
of of planes represented by
by
En In the
Exercises los equation, ejercicios 87– where
90, 87 cdescribe a is
is 90, is any describir the real family number. a la of familia planes de represented planos representada
the equation, por la where ecuación, c is any donde real c number. es cualquier número real.
87. x y z c
88. x y c
87. 89. cy
by
x y cy z
z c
88.
90. x
x cz
y c
cz cz 89. cy z 0
90.
x cz 0
In In Exercises 91 91 and 92, (a) find the angle between the two
In En planes,
Exercises
los ejercicios and (b)
91 find
and 91 y set
92, of
92, of of
(a) a) parametric
find the angle
encontrar equations
between el ángulo for entre the
the of
los line
two
dos of
of
planes, of
planos intersection
and (b)
y b) hallar of of the
find un planes.
a set of parametric equations for the line of
conjunto de ecuaciones paramétricas de la
intersection of the planes.
recta de intersección de los planos.
91. 3x 3x 2y 2y z 92. 6x 6x 3y 3y z 91.
91. x
3x 3x 4y 2y 4y 2y 2z z 2z 7
92. 6x 92. 6xx 3y 3y y 5z z 5
5z x 4y 2z 0
x y 5z 5
4y 2z x 5z