Calculus

James Stewart

Chapter 12

Vectors and the Geometry of Space - all with Video Answers

Educators

Section 1

12.1 Exercises

Problem 1

Suppose you start at the origin, move along the $x$-axis a distance of 4 units in the positive direction, and then move downward a distance of 3 units. What are the coordinates of your position?

Howard Francis

Howard Francis

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Problem 2

Sketch the points $(0,5,2),(4,0,-1),(2,4,6)$, and $(1,-1,2)$ on a single set of coordinate axes.

Howard Francis

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Problem 3

Which of the points $A(-4,0,-1), B(3,1,-5)$, and $C(2,4,6)$ is closest to the $y z$-plane? Which point lies in the $x z$-plane?

Howard Francis

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Problem 4

What are the projections of the point $(2,3,5)$ on the $x y$-, $y z$-, and $x z$-planes? Draw a rectangular box with the origin and $(2,3,5)$ as opposite vertices and with its faces parallel to the coordinate planes. Label all vertices of the box. Find the length of the diagonal of the box.

Nicholas Barvinok

Nicholas Barvinok

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Problem 5

Describe and sketch the surface in $\mathbb{R}^3$ represented by the equation $x+y=2$.

Lucas Gagne

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Problem 6

(a) What does the equation $x=4$ represent in $\mathbb{R}^2$ ? What does it represent in $\mathbb{R}^3$ ? Illustrate with sketches.
(b) What does the equation $y=3$ represent in $\mathbb{R}^3$ ? What does $z=5$ represent? What does the pair of equations $y=3$,
$z=5$ represent? In other words, describe the set of points $(x, y, z)$ such that $y=3$ and $z=5$. Illustrate with a sketch.

Howard Francis

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Problem 7

Find the lengths of the sides of the triangle $P Q R$. Is it a right triangle? Is it an isosceles triangle?
$P(3,-2,-3), \quad Q(7,0,1), \quad R(1,2,1)$

Sarah Parrigin

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Problem 8

Find the lengths of the sides of the triangle $P Q R$. Is it a right triangle? Is it an isosceles triangle?
$P(2,-1,0), \quad Q(4,1,1), \quad R(4,-5,4)$

Sarah Parrigin

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Problem 9

Determine whether the points lie on straight line.
(a) $A(2,4,2), \quad B(3,7,-2), \quad C(1,3,3)$
(b) $D(0,-5,5), \quad E(1,-2,4), \quad F(3,4,2)$

Howard Francis

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Problem 10

Find the distance from $(4,-2,6)$ to each of the following.
(a) The $x y$-plane
(b) The $y z$-plane
(c) The $x z$-plane
(d) The $x$-axis
(c) The $y$-axis
(f) The $z$-axis

Howard Francis

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Problem 11

Find an equation of the sphere with center $(-3,2,5)$ and radius 4 . What is the intersection of this sphere with the $y z$-plane?

Dorcas Attuabea Addo

Dorcas Attuabea Addo

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Problem 12

Find an equation of the sphere with center $(2,-6,4)$ and radius 5. Describe its intersection with each of the coordinate planes.

Howard Francis

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Problem 13

Find an equation of the sphere that passes through the point $(4,3,-1)$ and has center $(3,8,1)$.

Alice Trott

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Problem 14

Find an equation of the sphere that passes through the origin and whose center is $(1,2,3)$.

Howard Francis

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Problem 15

Show that the equation represents a sphere, and find its center and radius.
$x^2+y^2+z^2-2 x-4 y+8 z=15$

Howard Francis

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Problem 16

Show that the equation represents a sphere, and find its center and radius.
$x^2+y^2+z^2+8 x-6 y+2 z+17=0$

Howard Francis

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Problem 17

Show that the equation represents a sphere, and find its center and radius.
$2 x^2+2 y^2+2 z^2=8 x-24 z+1$

Howard Francis

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Problem 18

Show that the equation represents a sphere, and find its center and radius.
$3 x^2+3 y^2+3 z^2=10+6 y+12 z$

A M

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Problem 19

(a) Prove that the midpoint of the line segment from $P_1\left(x_1, y_1, z_1\right)$ to $P_2\left(x_2, y_2, z_2\right)$ is

$$
\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right)
$$

(b) Find the lengths of the medians of the triangle with vertices $A(1,2,3), B(-2,0,5)$, and $C(4,1,5)$.

Howard Francis

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Problem 20

Find an equation of a sphere if one of its diameters has endpoints $(2,1,4)$ and $(4,3,10)$.

Carson Merrill

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Problem 21

Find equations of the spheres with center $(2,-3,6)$ that touch (a) the $x y$-plane, (b) the $y z$-plane, (c) the $x z$-plane.

Howard Francis

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Problem 22

Find an equation of the largest sphere with center $(5,4,9)$ that is contained in the first octant.

Howard Francis

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Problem 23

Describe in words the region of $\mathbb{R}^3$ represented by the equations or inequalities.
$x=5$

Carson Merrill

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Problem 24

Describe in words the region of $\mathbb{R}^3$ represented by the equations or inequalities.
$y=-2$

Howard Francis

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Problem 25

Describe in words the region of $\mathbb{R}^3$ represented by the equations or inequalities.
$y<8$

Howard Francis

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Problem 26

Describe in words the region of $\mathbb{R}^3$ represented by the equations or inequalities.
$x \geqslant-3$

Carson Merrill

Numerade Educator

Problem 27

Describe in words the region of $\mathbb{R}^3$ represented by the equations or inequalities.
$0 \leqslant z \leqslant 6$

Carson Merrill

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Problem 28

Describe in words the region of $\mathbb{R}^3$ represented by the equations or inequalities.
$z^2=1$

Howard Francis

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Problem 29

Describe in words the region of $\mathbb{R}^3$ represented by the equations or inequalities.
$x^2+y^2=4, \quad z=-1$

Steven Clarke

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Problem 30

Describe in words the region of $\mathbb{R}^3$ represented by the equations or inequalities.
$y^2+z^2=16$

Nicholas Barvinok

Nicholas Barvinok

Numerade Educator

Problem 31

Describe in words the region of $\mathbb{R}^3$ represented by the equations or inequalities.
$x^2+y^2+z^2 \leqslant 3$

Howard Francis

Numerade Educator

Problem 32

Describe in words the region of $\mathbb{R}^3$ represented by the equations or inequalities.
$x=z$

Howard Francis

Numerade Educator

Problem 33

Describe in words the region of $\mathbb{R}^3$ represented by the equations or inequalities.
$x^2+z^2 \leqslant 9$

Howard Francis

Numerade Educator

Problem 34

Describe in words the region of $\mathbb{R}^3$ represented by the equations or inequalities.
$x^2+y^2+z^2>2 z$

William Semus

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Problem 35

Write inequalities to describe the region.
The region between the $y z$-plane and the vertical plane $x=5$

Pawan Yadav

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Problem 36

Write inequalities to describe the region.
The solid cylinder that lies on or below the plane $z=8$ and on or above the disk in the $x y$-plane with center the origin and radius 2

Carson Merrill

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Problem 37

Write inequalities to describe the region.
The region consisting of all points between (but not on) the spheres of radius $r$ and $R$ centered at the origin, where $r<R$

Carson Merrill

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Problem 38

Write inequalities to describe the region.
The solid upper hemisphere of the sphere of radius 2 centered at the origin

Carson Merrill

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Problem 39

The figure shows a line $L_1$ in space and a second line $L_2$, which is the projection of $L_1$ on the $x y$-plane. (In other words, the points on $L_2$ are directly beneath, or above, the points on $L_1$.)
(a) Find the coordinates of the point $P$ on the line $L_1$.
(b) Locate on the diagram the points $A, B$, and $C$, where the line $L_1$ intersects the $x y$-plane, the $y z$-plane, and the $x z$-plane, respectively.

Lucas Gagne

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Problem 40

Consider the points $P$ such that the distance from $P$ to $A(-1,5,3)$ is twice the distance from $P$ to $B(6,2,-2)$. Show that the set of all such points is a sphere, and find its center and radius.

James Leslie

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Problem 41

Find an equation of the set of all points equidistant from the points $A(-1,5,3)$ and $B(6,2,-2)$. Describe the set.

Howard Francis

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Problem 42

Find the volume of the solid that lies inside both of the spheres

$$
x^2+y^2+z^2+4 x-2 y+4 z+5=0
$$

and

$$
x^2+y^2+z^2=4
$$

Carson Merrill

Numerade Educator

Problem 43

Find the distance between the spheres $x^2+y^2+z^2=4$ and $x^2+y^2+z^2=4 x+4 y+4 z-11$.

Brittany Knowlton

Brittany Knowlton

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Problem 44

Describe and sketch a solid with the following properties. When illuminated by rays parallel to the $z$-axis, its shadow is a circular disk. If the rays are parallel to the $y$-axis, its shadow is a square. If the rays are parallel to the $x$-axis, its shadow is an isosceles triangle.

Howard Francis

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