Calculo 2 De dos variables_9na Edición - Ron Larson

780 CAPÍTULO 11 Vectores y la geometría del espacio
780 Chapter 11 Vectors and the Geometry of Space
780 Chapter 11 Vectors and the Geometry of Space
Ejercicios
11.2 11.2 Exercises Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
En los ejercicios 1 y 2, aproximar las coordenadas de los puntos.
In Exercises 1 and 2, approximate the coordinates of the points.
In 1. Exercises 1 zand 2, approximate 2. the coordinates z of the points.
B
1. 2.
z
5
1. 2.
z B
B
5 B
5
3
−2
4 3 2
A
4
y
−2
−2
−2
−2
4 5
3
5
4
2
4
3
−2 −3−4
3
−2 −3−4
2
−2
2
−2 −3−4
−3−4
1
2
y
A −2
x
x
1
1
2
y
A
y
A
x
2
x
En los ejercicios 3 a 6, representar los puntos en el mismo sistema
In Exercises de coordenadas
3–6, plot tridimensional.
the points on the same three-dimensional
In Exercises 3–6, plot the points the same three-dimensional
coordinate system.
coordinate 3. a) 2, 1,
system.
3 b) 1, 2, 1
3. (a) 2, 1, 3
(b)
4. 3 1, 2, 1
3. (a) 3, 2, 2, 1, 3
5 (b)
1, 2, 4, 2 2, 1
4. (a) 3, 2, 5
(b)
5. 5, 2
5, 4. (a) (b) 3 3, 2, 5
3 2, 4, 2, 2
2, 4, 2 2
5. (a) 5, 2, 2
(b)
5, 2, 2
6. 5. (a) 0, 5, 4, 2, 5 2
(b)
4, 5, 0, 2, 5 2
6. (a) 0, 4, 5
(b)
4, 0, 5
6. (a) 0, 4, 5
(b)
4, 0, 5
En los ejercicios 7 a 10, hallar las coordenadas del punto.
In Exercises 7–10, find the coordinates of the point.
In Exercises 7–10, find the coordinates of the point.
7. El punto se localiza tres unidades detrás del plano yz, cuatro
7. The point is located three units behind the yz-
plane, four units
7. unidades The point a is la located derecha three del units plano behind xz y cinco the yz-
unidades plane, four arriba units
del
to the right of the xz-
plane, and five units above the xy-
plane.
plano to the xy. right of the xz-
plane, and five units above the xy-
plane.
8. The point is located seven units in front of the yz-
plane, two
8. El The punto point se is localiza located seven siete unidades units in front delante of the del yz-
plano plane, yz, two
dos
units to the left of the xz-
plane, and one unit below the xy-
plane.
unidades units to the a left la izquierda of the xz-
plane, del plano and one xz y unit una below unidad the debajo xy-
plane.
del
9. plano The point xy. is located on the x-
axis, 12 units in front of the
9. The point is located on the x-
axis, 12 units in front of the
yz-
plane.
9. El yz-
punto plane.
se localiza en el eje x, 12 unidades delante del plano yz.
10. The point is located in the yz-
plane, three units to the right of
10. El The punto point se is localiza located en in el the plano yz-
plane, yz, tres three unidades units a to la the derecha right del of
the xz-
plane, and two units above the xy-
plane.
plano the xz-
xz plane, y dos and unidades two units arriba above del the plano xy-
plane. xy.
11. Think About It What is the z-
coordinate of any point in the
11. Para Think pensar About It ¿Cuál What es is la the coordenada z-
coordinate z de of todo any point punto in en the
el
xy-
plane?
plano xy-
plane?
xy?
12. Think About It What is the x-
coordinate of any point in the
12. Para Think pensar About It ¿Cuál What es is la the coordenada x-
coordinate x de of todo any point punto in en the
el
yz-
plane?
plano yz-
plane?
yz?
In Exercises 13–24, determine the location of a point
x, y, z
En In Exercises los ejercicios 13–24, a determine 24, determinar the location la localización of a point
de un x, punto y, z
that satisfies the condition(s).
(x, that y, satisfies z) que satisfaga the condition(s).
la(s) condición(es).
13. z 6
14.
y 2
13. z 6
14.
y 2
15. x 3
16.
z 15. x 3
16.
z 5 5 2
2
17. y < 0
18. x > 0
17. y < 0
18.
x > 0
19. 19. y y 3
20.
3
20.
x x > > 4
4
21. xy > 0, z 3 22. xy < 0,
z 4
21. xy > 0,
z 3
22.
xy < 0,
z 4
23. xyz < 0
24.
xyz > 0
23. xyz < 0
24.
xyz > 0
In Exercises 25–28, find the distance between the points.
En In Exercises los ejercicios 25–28, a find 28, the hallar distance la distancia between entre the los points.
puntos.
25.
0, 0, 0,
4, 2, 7
25.
0, 0, 0,
4, 2, 7
26.
2, 3, 2,
2, 5, 2
26.
2, 3, 2,
2, 5, 2
27.
1, 2, 4,
6, 2, 2
27.
1, 2, 4,
6, 2, 2
28.
2, 2, 3,
4, 5, 6
28.
2, 2, 3,
4, 5, 6
En los ejercicios 29 a 32, hallar las longitudes de los lados del
triángulo In Exercises con 29–32, los vértices find que the lengths se indican, of the y determinar sides of the si triangle
el triángulo
with es the un indicated triángulo vertices, rectángulo, and determine un triángulo whether isósceles, the o triangle
ningu-
In Exercises 29–32, find the lengths of the sides of the with the indicated vertices, and determine whether the triangle
na is a de right ambas triangle, cosas. an isosceles triangle, or neither.
is a right triangle, an isosceles triangle, or neither.
29.
0, 0, 4, 2, 6, 7, 6, 4, 8
29.
0, 0, 4, 2, 6, 7, 6, 4, 8
30.
3, 4, 1, 0, 6, 2, 3, 5, 6
30.
3, 4, 1, 0, 6, 2, 3, 5, 6
31.
1, 0, 2, 1, 5, 2, 3, 1, 1
31.
1, 0, 2, 1, 5, 2, 3, 1, 1
32.
4, 1, 1, 2, 0, 4, 3, 5, 1
32.
4, 1, 1, 2, 0, 4, 3, 5, 1
33. Para
Think pensar
About It El triángulo
The triangle del ejercicio
in Exercise 29 se
29 traslada
is translated
cinco
33. Think unidades About hacia It arriba The a triangle lo largo in del Exercise eje z. Determinar 29 is translated
five units upward along the z-
axis. Determine the coordinates las coordenadas
units del upward triángulo along trasladado. the z-
axis. Determine the coordinates of
the translated triangle.
the Para translated pensartriangle.
34. Think About It El triángulo
of
five The triangle del
in ejercicio
Exercise 30
30 se
is traslada
translated
tres
34. Think unidades About a la derecha It The a triangle largo in del Exercise eje y. Determinar 30 is translated
three units to the right along the
y-
axis. Determine the las
coordi-
coordenadas
units del to triángulo the right trasladado. along the
y-
axis. Determine the coordinates
of the translated triangle.
nates of the translated triangle.
En los ejercicios 35 y 36, hallar las coordenadas del punto medio
three In Exercises 35 and 36, find the coordinates of the midpoint of
In del Exercises segmento 35 de and recta 36, que find une the los coordinates puntos. of the midpoint of
the line segment joining the points.
the 35.
line
5, 9,
segment
7, 2,
joining
3, 3
the points.
36. 4, 0, 6, 8, 8, 20
35. 5, 9, 7, 2, 3, 3
36.
4, 0, 6, 8, 8, 20
35. 5, 9, 7, 2, 3, 3
36.
4, 0, 6, 8, 8, 20
En los ejercicios 37 a 40, hallar la ecuación estándar de la esfera.
In Exercises 37– 40, find the standard equation of the sphere.
In 37.
Exercises
Centro:
37–
0, 2,
40,
5
find the standard
38. Centro:
equation
4,
of
1,
the
1
sphere.
37. Center: 0, 2, 5
38. Center:
4, 1, 1
37. Center: Radio: 0, 2 2, 5
38. Center:
Radio: 4, 5
1, 1
Radius: 2 Radius: 5
39. Radius: Puntos terminales 2 de un diámetro: Radius: (2, 0, 0), 5
(0, 6, 0)
39. Endpoints of a diameter:
2, 0, 0, 0, 6, 0
39. 40. Endpoints Centro: (3, of a 2, diameter:
4), tangente 2, 0, al 0, plano 0, yz 6, 0
40. Center: 3, 2, 4,
tangent to the yz-
plane
40. En
Center:
los ejercicios
3, 2,
41
4,
a
tangent
44, completar
to the yz-plane
el cuadrado para dar la
ecuación In Exercises de la 41– esfera 44, complete en forma the canónica square to o write estándar. the equation Hallar of
el
In Exercises 41– 44, complete the square to write the equation of
centro the sphere y el radio. in standard form. Find the center and radius.
the sphere in standard form. Find the center and radius.
41.
x 41.
x 2 y 2 y 2 z 2 z 2 2x 6y 8z 1 0
2 2x 6y 8z 1 0
42.
x 42.
x 2 y 2 y 2 z 2 z 2 9x 2y 10z 19 0
2 9x 2y 10z 19 0
43.
9x 43.
9x 2 9y 2 9y 2 9z 2 9z 2 6x 18y 1 0
2 6x 18y 1 0
44.
4x 44.
4x 2 4y 2 4y 2 4z 2 4z 2 24x 4y 8z 23 0
2 24x 4y 8z 23 0
En In Exercises los ejercicios 45–48, a describe 48, describir the solid el sólido satisfying que satisface the condition.
la condición.
In Exercises 45–48, describe the solid satisfying the condition.
45. x 45. x 2 y
45. 2 2 z y
2 2 36 46. x z
2 36 46.
x 2 y
≤ 36 46. x 2 2 z 2 y
y 2 2 2 z
z 2 > 4
2 > 4
47.
x > 4
47.
x 2 y
47.
2 2 z y
2 2 z
2 < 4x 6y 8z 13
< 4x 6y 8z 13
48.
x < 6y 8z 13
48.
x 2 y
48. x 2 2 z 2 y
y 2 2 2 z
z 2 > 4x 6y 8z 13
2 > 4x 6y 8z 13
> 4x 6y 8z 13
In Exercises 49–52, (a) find the component form of the vector v,
In En Exercises los ejercicios 49–52, a (a) 52, find a) encontrar the component las componentes form of the del vector vector v,
(b) write the vector using standard unit vector notation, and (c) v,
(b) escribir write the el vector utilizando using standard la notación unit vector del vector notation, unitario and estándar
y c) the dibujar vector el with vector its con initial su punto point at inicial the origin.
en el origen.
(c)
sketch the vector with its initial point at the origin.
sketch 49. 50.
49. z
50.
z
6
4
2 (2, 4, 3)
(4, 0, 3)
6
4
2
v (0, 5, 1)
(4, 2, 1)
v
y
6
4
2 4 6
y
6
6
x
x