Calculus A New Horizon

Howard Anton

Chapter 13

Three-Dimentional Space; Vectors - all with Video Answers

Educators

Section 1

Rectangular Coordinates in 3-Space; Spheres; Cylindrical Surfaces

Problem 1

In each part, find the coordinates of the eight corners of the box.
(FIGURE CANNOT COPY)

Carson Merrill

Carson Merrill

Numerade Educator

Problem 2

A cube of side 4 has its geometric center at the origin and its faces parallel to the coordinate planes. Sketch the cube and give the coordinates of the corners.

Linh Vu

Numerade Educator

Problem 3

Suppose that a box has its faces parallel to the coordinate planes and the points (4,2,-2) and (-6,1,1) are endpoints of a diagonal. Sketch the box and give the coordinates of the remaining six corners.

Linda Hand

Numerade Educator

Problem 4

Suppose that a box has its faces parallel to the coordinate planes and the points $\left(x_{1}, y_{1}, z_{1}\right)$ and $\left(x_{2}, y_{2}, z_{2}\right)$ are endpoints of a diagonal.
(a) Find the coordinates of the remaining six corners.
(b) Show that the midpoint of the line segment joining
$$ \left(x_{1}, y_{1}, z_{1}\right) \text { and }\left(x_{2}, y_{2}, z_{2}\right) \text { is } $$
$\left(\frac{1}{2}\left(x_{1}+x_{2}\right), \frac{1}{2}\left(y_{1}+y_{2}\right), \frac{1}{2}\left(z_{1}+z_{2}\right)\right)$

Carson Merrill

Numerade Educator

Problem 5

Find the center and radius of the sphere that has (1,-2,4) and (3,4,-12) as endpoints of a diameter. [Sec Exercise 4.]

Carson Merrill

Numerade Educator

Problem 6

Show that $(4,5,2),(1,7,3),$ and (2,4,5) are vertices of an equilateral triangle.

Linh Vu

Numerade Educator

Problem 7

(a) Show that $(2,1,6),(4,7,9),$ and (8,5,-6) are the vertices of a right triangle.
(b) Which vertex is at the $90^{\circ}$ angle?
(c) Find the area of the triangle.

Linh Vu

Numerade Educator

Problem 8

Find the distance from the point (-5.2,-3) to the
(a) xy-plane
(b) $x z$ -plane
(c) yz-plane
(d) $x$ -axis
(e) $y$ -axis
(f) $z$ -axis.

Madi Sousa

Numerade Educator

Problem 9

In each part, find the standard equation of the sphere that satisfies the stated conditions.
(a) Center (1,0,-1)$;$ diameter $=8$
(b) Center (-1,3,2) and passing through the origin.
(c) A diameter has endpoints (-1,2,1) and (0,2,3)

Carson Merrill

Numerade Educator

Problem 10

Find equations of two spheres that are centered at the origin and are tangent to the sphere of radius 1 centered at (3,-2,4)

Carson Merrill

Numerade Educator

Problem 11

In each part, find an equation of the sphere with center (2,-1,-3) and satisfying the given condition.
(a) Tangent to the $x y$ -plane
(b) Tangent to the $x$ z-plane
(c) Tangent to the $y z$ -plane

Linh Vu

Numerade Educator

Problem 12

(a) Find an equation of the sphere that is inscribed in the cube that is centered at the point (-2,1,3) and has sides of length 1 that are parallel to the coordinate planes.
(b) Find an equation of the sphere that is circumscribed about the cube in part (a).

Carson Merrill

Numerade Educator

Problem 13

Describe the surface whose equation is given.
$$x^{2}+y^{2}+z^{2}+10 x+4 y+2 z-19=0$$

Linh Vu

Numerade Educator

Problem 14

Describe the surface whose equation is given.
$$x^{2}+y^{2}+z^{2}-y=0$$

Linh Vu

Numerade Educator

Problem 15

Describe the surface whose equation is given.
$$2 x^{2}+2 y^{2}+2 z^{2}-2 x-3 y+5 z-2=0$$

Linh Vu

Numerade Educator

Problem 16

Describe the surface whose equation is given.
$$x^{2}+y^{2}+z^{2}+2 x-2 y+2 z+3=0$$

Linh Vu

Numerade Educator

Problem 17

Describe the surface whose equation is given.
$$x^{2}+y^{2}+z^{2}-3 x+4 y-8 z+25=0$$

Linh Vu

Numerade Educator

Problem 18

Describe the surface whose equation is given.
$$x^{2}+y^{2}+z^{2}-2 x-6 y-8 z+1=0$$

Linh Vu

Numerade Educator

Problem 19

In each part, sketch the portion of the surface that lies in the first octant.
(a) $y=x$
(b) $y=z$
(c) $x=z$

Madi Sousa

Numerade Educator

Problem 20

In each part, sketch the graph of the equation in 3 -space.
(a) $x=1$
(b) $y=1$
(c) $z=1$

Linh Vu

Numerade Educator

Problem 21

In each part, sketch the graph of the equation in 3 -space.
(a) $x^{2}+y^{2}=25$
(b) $y^{2}+z^{2}=25$
(c) $x^{2}+z^{2}=25$

Linh Vu

Numerade Educator

Problem 22

In each part, sketch the graph of the equation in 3 -space.
(a) $y=x^{2}$
(b) $z=x^{2}$
(c) $y=z^{2}$

Linh Vu

Numerade Educator

Problem 23

In each part, write an equation for the surface.
(a) The plane that contains the $x$ -axis and the point (0,1,2)
(b) The plane that contains the $y$ -axis and the point (1,0,2)
(c) The right circular cylinder that has radius 1 and is centered on the line parallel to the $z$ -axis that passes through the point (1,1,0)
(d) The right circular cylinder that has radius 1 and is centeredon the line parallel to the $y$ -axis that passes through the point (1,0,1)X

Carson Merrill

Numerade Educator

Problem 24

Find equations for the following right circular cylinders. Each cylinder has radius $a$ and is "tangent" to two coordinate planes.
(FIGURE CANNOT COPY)

Linh Vu

Numerade Educator

Problem 25

Sketch the surface in 3 -space.
$$y=\sin x$$

Linh Vu

Numerade Educator

Problem 26

Sketch the surface in 3 -space.
$$y=e^{x}$$

Linh Vu

Numerade Educator

Problem 27

Sketch the surface in 3 -space.
$$z=1-y^{2}$$

Linh Vu

Numerade Educator

Problem 28

Sketch the surface in 3 -space.
$$z=\cos x$$

Linh Vu

Numerade Educator

Problem 29

Sketch the surface in 3 -space.
$$2 x+z=3$$

Linh Vu

Numerade Educator

Problem 30

Sketch the surface in 3 -space.
$$2 x+3 y=6$$

Linh Vu

Numerade Educator

Problem 31

Sketch the surface in 3 -space.
$$4 x^{2}+9 z^{2}=36$$

Linh Vu

Numerade Educator

Problem 32

Sketch the surface in 3 -space.
$$z=\sqrt{3-x}$$

Linh Vu

Numerade Educator

Problem 33

Sketch the surface in 3 -space.
$$y^{2}-4 z^{2}=4$$

Linh Vu

Numerade Educator

Problem 34

Sketch the surface in 3 -space.
$$y z=1$$

Linh Vu

Numerade Educator

Problem 35

Use a graphing utility to generate the curve $y=x^{3} /\left(1+x^{2}\right)$ in the $x y$ -plane, and then use the graph to help sketch the surface $z=y^{3} /\left(1+y^{2}\right)$ in 3 -space.

Linh Vu

Numerade Educator

Problem 36

Use a graphing utility to generate the curve $y=x /\left(1+x^{4}\right)$ in the $x y$ -plane, and then use the graph to help sketch the surface $z=y /\left(1+y^{4}\right)$ in 3 -space.

Linh Vu

Numerade Educator

Problem 37

If a bug walks on the sphere
$$ x^{2}+y^{2}+z^{2}+2 x-2 y-4 z-3=0 $$
how close and how far can it get from the origin?

Carson Merrill

Numerade Educator

Problem 38

Describe the set of all points in 3 -space whose coordinates satisfy the inequality $x^{2}+y^{2}+z^{2}-2 x+8 z \leq 8$

Linh Vu

Numerade Educator

Problem 39

Describe the set of all points in 3 -space whose coordinates satisfy the inequality $y^{2}+z^{2}+6 y-4 z>3$

Linh Vu

Numerade Educator

Problem 40

The distance between a point $P(x, y, z)$ and the point $A(1,-2,0)$ is twice the distance between $P$ and the point $B(0,1,1) .$ Show that the set of all such points is a sphere. and find the center and radius of the sphere.

Carson Merrill

Numerade Educator

Problem 41

As shown in the accompanying figure, a bowling ball of radius $R$ is placed inside a box just large enough to hold it.and it is secured for shipping by packing a Styrofoam sphere into each comer of the box. Find the radius of the largest Styrofoam sphere that can be used. [Hint: Take the origin of a Cartesian coordinate system at a corner of the box with the coordinate axes along the edges.]
(FIGURE CANNOT COPY)

Carson Merrill

Numerade Educator

Problem 42

Consider the equation
$$ x^{2}+y^{2}+z^{2}+G x+H y+I z+J=0 $$
and let $K=G^{2}+H^{2}+l^{2}-4 J$
(a) Prove that the equation represents a sphere if $K>0,$ a point if $K=0,$ and has no graph if $K<0$
(b) In the case where $K>0,$ find the center and radius of the sphere.

Carson Merrill

Numerade Educator

Problem 43

Show that for all values of $\theta$ and $\phi$, the point
$$ (a \sin \phi \cos \theta, a \sin \phi \sin \theta, a \cos \phi) $$
lies on the sphere $x^{2}+y^{2}+z^{2}=a^{2}$

Linh Vu

Numerade Educator