Calculo 2 De dos variables_9na Edición - Ron Larson

810 Chapter 11 Vectors and the Geometry of Space
810 CAPÍTULO 11 Vectores y la geometría del espacio
810 810 Chapter Chapter 11 11 Vectors Vectors and and the the Geometry Geometry of Space of Space
r 11 Vectors and the Geometry of Space
In Exercises 93–96, find the point(s) of intersection (if any) of WRITING ABOUT CONCEPTS (continued)
the plane and the line. Also determine whether the line lies in
In En los ejercicios a 96, hallar el o los puntos de intersección (si 115. Describe a method for determining when two planes
the In Exercises plane. Exercises 93–96, 93–96, find find the the point(s) point(s) of intersection of intersection (if (if any) any) of of
6, find the point(s) los hay) of del intersection plano y la recta. (if any) Investigar of además si la recta se halla
WRITING Desarrollo WRITING ABOUT ABOUT de conceptos CONCEPTS (continued) (continuación)
(continued)
the the plane plane and and the the line. line. Also Also determine determine whether WRITING whether the the line ABOUT line lies lies in CONCEPTS in a 1 (continued) x b 1 y c 1 z d 1 0 and
line. Also determine en el plano. whether the line lies 1in
y 32 z 1
115. 115.
115. Describe a method for determining
115. Describe Describe
when two planes
93.
x 1 a 1 x b 1 y c 1 z d 1 0 and
1 y 32 z 1 2 y 32 z 1
a
Describir a method a method
2 x b 2 y
un
c
método for for determining determining
2 z d
para
2 0
determinar when when
cuándo two two planes planes
the the
93. plane. plane.
dos planos
2x 2y z 12, x
2 1 2
a 1 xa 1 xb 2x 2y z 12,
are a 1 x(a) 1 yb 1 yc bparallel 1 y 1 zc 1 zd c 1 zand 1
d 1 0 and 0 and
1 1 y y
1
32 32z z
2
1 1
d 1 (b) 0perpendicular. y a 2 x b 2 y Explain c 2 z d 2 your 0
93. 93. 2x 2x 2y 2yz z12,
x12,
x 1x
y z 3
a
12, x 94. 2x 3y 5,
a 2 x b 2 y c 2 z d 2 0
2 1 2 x 1
94.
y are (a) parallel and (b) perpendicular. Explain your
x 1 y z 3
4 2 z 3
reasoning.
2 xa 2 x b 2 yb 2 y c 2 zc 2 z d 2
d 2 0 0
2 2 1 1 2 2
4 2 6
2x 3y 5,
116.
are
son are
Let
(a)
a) (a)
L and
parallel
paralelos parallel
L be nonparallel
and
y and b)
(b)
perpendiculares. (b) perpendicular. Explicar
lines that do not
Explain Explain el
intersect.
your
razonamiento.
your
x x 1 1 y y z z
6
3 3
1 2
Is
1
94. 94. 2x 2x 3y 3y 5, 5,
reasoning. reasoning.
5, 95. 2x 3y 10, 4 4 2
reasoning.
4 2 6
x 1
95.
y 2
1 6 z6
3
it possible to find a nonzero vector v such that v is
2x 3y 10,
3
116. Let L 1 and L 2 be nonparallel lines that do not intersect. Is
x 1 y 1
3 2 z 3
116. 116. Let
Sean
perpendicular Let L
L 1 y L 2 rectas paralelas que no se cortan. ¿Es posible
hallar un vector v distinto de cero tal que v sea perpen-
1 and L 1 and L 2 be Lto 2 nonparallel be both nonparallel L 1 and lines L 2 ? lines Explain that that do do not your not intersect. reasoning. intersect. Is Is
x x 1 1 y y 1 1
95. 95. 2x 2x 3y 3y 10, 10, x 4 y z z 32
3
it possible to find a nonzero 117.
it
Find
possible it possible
vector an vequation to find find
such that of
a
vthe nonzero a nonzero
is plane
vector vector
with x-intercept
v such v such that that
a, 0,
v
0
is v is
0,
96. 5x z 3y 3 17, 3 3
3 2
x 4
96.
y 2
1
2
,
2 3
perpendicular to both L 1 and L 2 ? Explain your reasoning.
x 4 y 1 z 2 2 3 z 5 2
perpendicular a ambos
y-intercept 0,
to
b,
both to Lboth 1 y
0 , and
L
L 2 ? Explicar el razonamiento.
1 and L
z 1 and
-intercept
L 2 ? LExplain 2 ? Explain
0, 0,
your your
c . (Assume
reasoning. reasoning.
a,
5x 3y 17, x x 4 4 y y 1 1 z z
5
2 2
Hallar una ecuación del plano con intersección en x (a, 0),
96. 96. 5x 5x 3y 3y 17, 17,
117. 117. Find b, Find and an can equation are equation nonzero.) of of the the plane plane with with x-intercept
x-intercept
a, 0, 00, , 0 ,
7, In Exercises 97–100, find 2 the 2 distance 3 3between 117. 5 5 Find the point an equation and the of the plane ywith intersección y-intercept x-intercept
0, b,
en 0, 0 b, a, y
, 0 and
(0, , 0 and b,
z,
0) z-intercept e intersección
0, 0, 0, c 0, . c (Assume
en . (Assume z (0, 0,
a,
c). a,
2 plane. En 3los ejercicios 5 97 a 100, hallar la distancia del y-intercept punto al 0, plano. b, 0 , and z-intercept b,
(Suponer
and b, and 0, c are 0, c c are que
nonzero.) . (Assume nonzero.) a, b y c son a, distintos de cero.)
In Exercises In Exercises 97–100, 97–100, find find the the distance distance between between the b, the and point point c are and nonzero.) and the the
0, find the distance plane. 97. plane. 0, between 0, 0 the point and the 98. 0, 0, 0
CAPSTONE
2x 3y z 12
5x y z 9
118. Para Match discusión the equation or set of equations with the description
97. 97. 0, 0, 0, 00, 0
98. 98. 0, 0, 0, 00, 0
CAPSTONE
99. 98. 2, 8, 0, 40, 0
100. 1, CAPSTONE
3, 1
it represents.
Encontrar la correspondencia entre la ecuación o conjunto
2x 2x 3y 3yz z12
12
5x 5xy yz z9
9
118. 118. Match Match
12
2x 5x y y z z 5 9
3x 118. 4y Match 5z the 6 equation or set of equations (a) Set
the the
with of
equation equation
the parametric
or set set
description equations
of equations of equations
of a
with with
line
the the description description
de ecuaciones que cumple con la descripción indicada.
99. 99. 2, 8, 2, 48, 4
100. 100. 1, 3, 1, 3, 1 1
it represents. it represents.
100. 1, 3, 1
it represents.
(b) a) Conjunto Set of symmetric de ecuaciones equations paramétricas of a line de una recta
In En Exercises los 2x 2x ejercicios y y101–104, z z5
a 5verify 104, verificar that the que 3xtwo los 4y planes dos 4y planos son paralelos,
y
5
3x
hallar
4y
la distancia
5z 6
(a)
5zare Set
6
of
parallel, 6
(a) (a) Set Set of parametric of parametric equations equations of a of line a line
parametric equations (c) of a Standard line equation of a plane in space
and find the distance between entre the ellos. planes.
(b)
b) (b) Conjunto
Set Set of symmetric of de symmetric ecuaciones
equations equations simétricas
of a of line a de line una recta
In In Exercises Exercises 101–104, 101–104, verify verify that that the the two two planes planes (b) are Set are parallel, of symmetric parallel, equations (d) of a General line form of an equation of a plane in space
104, verify that 101. the xx two 3y 3y planes 4z 4z are 10 10 parallel, 102.
4x 4x 4y 4y (c) 9z 9z Standard 7
(c)
c) (c) Ecuación
Standard Standard estándar
equation equation de
of
un
a of plane
plano a plane in
en
space in el space espacio
and and find find the the distance distance between between the the planes. planes.
7 equation of a plane i) in space x 6 2 y 1 3 z 1
nce between the planes.
xx 3y 3y 4z 4z 6
6
4x 4x 4y 4y(d) 9z 9z General 18
(d)
d) (d) Forma
General General general
form form of
de
an of la
equation an ecuación equation of
de
a of un
plane a plano plane in
en
space in el space espacio
18
101. 101. x x 3y 3y 4z 4z 10 10 102. 102. 4x 4x 4y 9z 9z 7 7 form of an equation ii) of a 2xplane 7yin space 5z 10 0
10 103. 102. 3x 4x 6y 4y 6y 7z 9z 7z 17
1 104.
2x 2x 4z 4z i)
4
i) i) x x 6 62 2 y y 1 1 3 3z 1z 1
4
x x 3y 3y 4z 4z 6 6
4x 4x 4y 4y 9z 9z x 186182 y 1 3 iii) z 1x 4 7t, y 3 t, z 3 3t
6
6x 6x4x 12y 12y 4y14z 14z 9z 25
18 25 2x 2x 4z 4zii)
10
ii) ii) 2x 2x 7y 7y 5z 5z 10 10 0 0
10
103. 103. 3x 3x 6y 6y 7z 7z 1 1 104. 104. 2x 2x 4z 4 2x 4 7y 5z 10 0 iv) 2(x 1) (y 3) 4(z 5) 0
7z 1 104. 2x 4z 4
iii) iii) x x 4 47t, 7t, y y 3 3 t, zt, z 3 33t
3t
In Exercises 6x 6x 12y105–108, 14z 14zfind 25 25 the distance 2x 2x between 4z 4z iii) 10 the x10point 4 and 7t, y 3 t, z 3 3t
14z 25 En los ejercicios 2x 4z 105 10a 108, hallar la distancia del punto a la recta iv) iv) 2(x 2(x 1) 1) (y (y 3) 3) 4(z 4(z 5) 5) 0 0
the line given by the set of parametric equations. iv) 2(x 1) (y 3) 4(z 5) 0
In
dada In Exercises Exercises por medio
105–108, 105–108, del conjunto
find find the the de
distance
ecuaciones distance between between paramétricas.
the the point point and and 119. Describe and find an equation for the surface generated by all
108, find the the 105. distance the 1, 5, between 2; x 4t point 2, and y 3, z t 1
119. Describir y hallar una ecuación para la superficie generada por
105. line line
1, given given
5, 2; by by the the
x set set
4t of parametric of parametric
2, y 3, equations. equations.
z t 1
points x, y, z that are four units from the point 3, 2, 5 .
he set of parametric
106. 1,
equations.
2, 4 ; x 2t, y t 3, 2t 2
119. 119. todos los puntos (x, y, z) que están a cuatro unidades del punto
106. 1, 2, 4; x 2t, y t 3, 119. z Describe 2t 2 and find an equation 120.
Describe Describe
for Describe
and and
3, the 2, surface and
find find
5. generated find
an
an
equation an equation
equation
for for
by all for
the the
the
surface surface
surface
generated generated
generated
by by all all
by
105. 105. 1, 5, 1, 5, 2; 2; x x 4t 4t 2, 2, y y 3, 3, z z t t1
1
points points
x 4t 2, 107. y 3, 2, z1, 3 ; t x 1 1 t, y 2 t, z
107. 2, 1, 3; x 1 t, y 2 t, z
points 2t
2t
x, y, z that are four units all from points
x, y, x, zy, the point x,
that z that
y, 3, z
are are
that
four four
2, 5 . are
units units
four
from from
units
the the point point
from
3, 3,
the
2, 52, plane
. 5 .
106. 106. 1, 1, 2, 42, ; 4 ; x x 2t, 2t, y yt t3, 3, z z 2t 2 2
120. 120. Describe 4xDescribir Describe 3y and y zand hallar find 10. find an una an equation ecuación equation for para for the la the surface superficie surface generated generated generada by por by
x 2t, y 108. t 3, 4, z 1, 52t;
x2
3, y 1 3t,
108. 4, 1, 5; x 3, y 1 3t,
120. z 1
z
Describe t
1 t
and find an equation for todos the los surface puntos generated x, y, z que by
107. 107. 2, 2, 31, ; 3 ; x x 1 1 t, t, y y 2 2 t, t, z z 2t 2t
están a cuatro unidades del plano
x 1 t, y 2 t, z 2t
all points x, y, z that 121.
all all
are four Modeling
points points
units Data
x, y, x, zy, z
from Per
that that the capita
are are
plane consumptions
four four units units from from
(in gallons)
the the plane plane
of
In Exercises 109 and 110, verify that the lines 4xare 3y parallel, z and
4x 4x
10.
different
3y 3y
types
z z
of
10. 10.
108. 108. 4, 4, 1, 51, ; 5 ; x x 3, 3, y y 1 13t,
3t, z z 1 1 t t
milk in the United States from 1999 through
x 3, y find 1En los the 3t, ejercicios distance z 1 between 109 t y 110, them. verificar que las rectas son paralelas 121. 121. Modeling 2005 Modelado Modeling are shown Data matemático Data in Per the Per capita table. capita Los Consumptions consumos consumptions per of (in cápita flavored (in gallons) (en gallons) galones) milk, of of
In y Exercises In hallar Exercises la distancia 109 109 and and 110, entre 110, verify ellas. verify that that the 121. the lines lines Modeling are are parallel, parallel, Dataand
and Per capita consumptions different de diferentes (in tipos gallons) de leche of
plain different reduced-fat types types of milk, of milk in and the in plain the United United en light Estados States and States skim from Unidos from milks 1999 1999 are desde through represented
2005 are by are shown the shown variables in the in the table. x, table. y, en and la Consumptions tabla. z, respectively. El of consumo flavored of flavored (Source: de milk, milk, leche
through 1999
nd 110, verify find 109. find that the L 1 the :
distance x distance lines 2 are between t, between parallel, y them. 3 them. and 2t, z 4 tdifferent types of milk in the United 2005 hasta States 2005 from se 1999 muestran through
etween them. 109.
L
L 1 :
x
x
3t,
2
y
t, y
1
3
6t,
2t,
z 4
z 4
3t
2005 t are shown in the table. Consumptions descremada of y flavored semidescremada, milk,
2 :
plain U.S. plain Department reduced-fat reduced-fat milk, of milk, Agriculture) and and plain plain light leche light and and skim reducida skim milks milks en are grasas are representesented
by by the the variables variables x, y, x, and y, and z, respectively. z, respectively. (Source: (Source:
repre-
y la
109. 109. L 1 L:
L
x 3t, y 1 6t, z 4 3t plain reduced-fat milk, and plain light leche and entera skim milks se representa are represented
by the variables x, y, and
por las variables x, y y z, respectivamente.
z, respectively. (Fuente: U.S. (Source:
t, y 3 110. 2t, zL 2 : x 1 : x 2 2 t, t, y y 3 32t,
2t, z z 4 4 t t
1 : x4 3t
6t, y 2 9t, z 1 12t
110. L 2 L:
L
x 3 6t, y 2 9t, z 1 12t
Department of Agriculture)
y 1 6t, z L4 1 : x 2 : x 3t, 3t, y y 1 16t,
6t, z z 4 43t
3t
x 3t 1 4t,
3 6t, z
AñoU.S. U.S. Department Department
1999 2000of Agriculture) of Agriculture)
2001 2002 2003 2004 2005
2 :
U.S. 8t Department of Agriculture)
110. 110. L 1 L:
L
x 1 4t, y 3 6t, z 8t
6t, y 2 9t, 2 : x 1 : x 3 36t,
6t, y y 2 29t,
9t, z z 1 1 12t 12t
z 1 12t
Año x 1999 1.4 2000 1.4 2001 1.4 2002 1.6 2003 1.6 2004 1.7 2005 1.7
WRITING L 2 : L Año 1999 2000 2001 2002 2003 2004 2005
x 2 : x
ABOUT 1 14t,
4t,
CONCEPTS
y y 3 36t,
6t, z z 8t 8t
4t, y 3 6t, z 8t
Año 1999 2000 2001 2002 2003 2004 2005
111.
Desarrollo
Give the parametric
de conceptos
equations and the symmetric equations xy x 1.4 7.31.4 1.4 7.11.4 1.4 7.01.4 1.6 7.01.6 1.6 6.91.6 1.7 6.91.7 1.7 6.91.7
WRITING WRITING x 1.4 1.4 1.4 1.6 1.6 1.7 1.7
OUT CONCEPTSof a line
ABOUT ABOUT
in space. CONCEPTS
Describe what is required to find these
z 6.2 6.1 5.9 5.8 5.6 5.5 5.6
111. 111. 111. Give equations. Dar Give the las the parametric ecuaciones parametric equations paramétricas equations and and the y the
rametric equations and the symmetric equations
ylas symmetric symmetric ecuaciones 7.3
equations equations simétricas
de una recta en el espacio. Describir qué se requiere
y y 7.3 7.3 7.1 7.1 7.0 7.0 7.0 7.0 6.9 6.9 6.9 6.9 6.9 6.9
7.1 7.0 7.0 6.9 6.9 6.9
space. Describe 112.
of
what Give
a of line a line
para
is required the standard
space. in space. Describe Describe
hallar estas
to find equation
what what
ecuaciones.
these of a plane
is required is required
in space.
to find to find
Describe
these these
z z
equations.
what is required to find this equation.
z 6.2 6.1 5.9 5.8 A model
6.2 6.2
5.6for the
6.1 6.1
5.5 data
5.9 5.9
is 5.6 given
5.8 5.8
by 0.92x
5.6 5.6
1.03y
5.5 5.5
z
5.6 5.6
equations.
0.02.
112. 112. 112. Dar la ecuación estándar de un plano en el espacio. Describir
qué
(a) Un Complete modelo para a fourth los datos row está in dado the table por using the model to
andard equation 113.
Give Give
of a Describe
the the standard standard
plane in a space. method
equation equation
se requiere
Describe of finding
of a of
the
plane a plane
line
in
of
space. in space.
intersection
Describe Describe
of
A model A what para hallar esta ecuación. A model for the data is given by 0.92x approximate
for for the the data data
1.03y z for
is
z the
given is given
0.02. given
by by
values
0.92x 0.92x
of x
1.03y 1.03y
and y. Compare
z z0.02.
0.02.
what the
ired to find this equation. two planes.
is required is required to find to find this this equation. equation.
(a) (a) Complete approximations Complete a fourth a fourth with row the row in actual the in the table values table using of using z. the the model model to to
113. 113. 113. Describir un método de hallar la recta de (a) intersección Complete a entre fourth row in the a) table Hacer using un cuarto the model renglón to de la tabla usando el modelo para
method of finding 114.
Describe Describe
the Describe
a
dos
line
planos.
of each
method a method
intersection surface
of finding of finding
of given
the the
by
line line
the
of
equations
intersection of intersection
x
of of
a,
two approximate z for the given values (b)
approximate approximate
According aproximar of x and y. to
z for z for Compare this
the the
con model,
given given
los valores theany values values
increases
of xof and x
dados de in
y.
x y consumption
Compare y. Compare the the
two y. Comparar oflas
y
planes. planes.
b, and z c.
approximations
114. Describir toda superficie dada por las ecuaciones approximations a, with the actual values two aproximaciones types of z. of milk
with with
con will
the los have
actual actual
valores what
values values
effect
of
reales on
z. of z.
de the z. consumption
114. 114. Describe Describe each each surface surface given given by by the the equations equations x xa,
a,
ach surface given by
y
the
b y
equations z x a,
(b) (b) According
(b) According to this model, any increases b)
of According
Según
the third to
in este
type? this to this model, model, any any increases increases consumption in consumption of of
y y b, and b, and z zc.
c.
consumption modelo, cualquier of incremento en el consumo de
z c.
two two types types of milk of milk will will have have what what effect effect on the on the consumption consumption
two types of milk will have what effect dos tipos on the de consumption leche tendrá ¿qué efecto en el consumo del tercer
of the of the third third type? type?
of the third type?
tipo?