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Chapter 12

VECTORS AND THE GEOMETRY OF SPACE

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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?

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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.

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Problem 3

Which of the points $P(6,2,3), Q(-5,-1,4),$ and $R(0,3,8)$ is
closest to the $x z$ -plane? Which point lies in the $y z$ -plane?

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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.

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Problem 5

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

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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.

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Problem 7

$7-8$ Find the lengths of the sides of the triangle $P Q R .$ Is it a right
triangle? Is it an isosceles triangle?
$$P(3,-2,-3), \quad Q(7,0,1), \quad R(1,2,1)$$

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Problem 8

$7-8$ Find the lengths of the sides of the triangle $P Q R .$ Is it a right
triangle? Is it an isosceles triangle?
$$P(2,-1,0), \quad Q(4,1,1), \quad R(4,-5,4)$$

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Problem 9

Determine whether the points lie on straight line.
$$
\begin{array}{ll}{\text { (a) } A(2,4,2),} & {B(3,7,-2), \quad C(1,3,3)} \\ {\text { (b) } D(0,-5,5),} & {E(1,-2,4), \quad F(3,4,2)}\end{array}
$$

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Problem 10

Find the distance from $(3,7,-5)$ to each of the following.
$\begin{array}{ll}{\text { (a) The } x y \text { -plane }} & {\text { (b) The } y z \text { -plane }} \\ {\text { (c) The } x z \text { -plane }} & {\text { (d) The } x \text { -axis }} \\ {\text { (e) The } y \text { -axis }} & {\text { (f) The } z \text { -axis }}\end{array}$

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Problem 11

Find an equation of the sphere with center $(1,-4,3)$ and
radius $5 .$ What is the intersection of this sphere with the
$x z$ -plane?

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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.

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Problem 13

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

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Problem 14

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

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Problem 15

$15-18$ Show that the equation represents a sphere, and find its
center and radius.
$$x^{2}+y^{2}+z^{2}-6 x+4 y-2 z=11$$

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Problem 16

$15-18$ Show that the equation represents a sphere, and find its
center and radius.
$$x^{2}+y^{2}+z^{2}+8 x-6 y+2 z+17=0$$

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Problem 17

$15-18$ Show that the equation represents a sphere, and find its
center and radius.
$$2 x^{2}+2 y^{2}+2 z^{2}=8 x-24 z+1$$

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Problem 18

$15-18$ Show that the equation represents a sphere, and find its
center and radius.
$$4 x^{2}+4 y^{2}+4 z^{2}-8 x+16 y=1$$

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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 $C(4,1,5)$

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Problem 20

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

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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.

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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.

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Problem 23

$23-32$ Describe in words the region of $\mathbb{R}^{3}$ represented by the equation or inequality.
$$y=-4$$

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Problem 24

$23-32$ Describe in words the region of $\mathbb{R}^{3}$ represented by the equation or inequality.
$$x=10$$

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Problem 25

$23-32$ Describe in words the region of $\mathbb{R}^{3}$ represented by the equation or inequality.
$$x>3$$

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Problem 26

$23-32$ Describe in words the region of $\mathbb{R}^{3}$ represented by the equation or inequality.
$$y \geqslant 0$$

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Problem 27

$23-32$ Describe in words the region of $\mathbb{R}^{3}$ represented by the equation or inequality.
$$0\leqslant z \leqslant 6$$

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Problem 28

$23-32$ Describe in words the region of $\mathbb{R}^{3}$ represented by the equation or inequality.
$$z^{2}=1$$

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Problem 29

$23-32$ Describe in words the region of $\mathbb{R}^{3}$ represented by the equation or inequality.
$$x^{2}+y^{2}+z^{2} \leqslant 3$$

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Problem 30

$23-32$ Describe in words the region of $\mathbb{R}^{3}$ represented by the equation or inequality.
$$\boldsymbol{X}=z$$

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Problem 31

$23-32$ Describe in words the region of $\mathbb{R}^{3}$ represented by the equation or inequality.
$$x^{2}+z^{2} \leqslant 9$$

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Problem 32

$23-32$ Describe in words the region of $\mathbb{R}^{3}$ represented by the equation or inequality.
$$x^{2}+y^{2}+z^{2}>2 z$$

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Problem 33

$33-36$ Write inequalities to describe the region.
The region between the $y z-$ plane and the vertical plane $x=5$

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Problem 34

$33-36$ 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

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Problem 35

$33-36$ 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 $r < R$

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Problem 36

$33-36$ Write inequalities to describe the region.
The solid upper hemisphere of the sphere of radius 2 centered
at the origin

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Problem 37

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.

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Problem 38

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.

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Problem 39

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

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Problem 40

Find the volume of the solid that lies inside both of the spheres
$$x^{2}+y^{2}+z^{2}+4 x-2 y+4 z+5=0 \quad and\quad x^{2}+y^{2}+z^{2}=4$$

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