12.1 SOLUTIONS
861
CHAPTER TWELVE
Solutions for Section 12.1
Exercises
1.
The distance of a point
P
= (
x, y, z
)
from the
yz
plane is

x

, from the
xz
plane is

y

, and from the
xy
plane is

z

.
So,
B
is closest to the
yz
plane, since it has the smallest
x
coordinate in absolute value.
B
lies on the
xz
plane, since its
y
coordinate is
0
.
B
is farthest from the
xy
plane, since it has the largest
z
coordinate in absolute value.
2.
The distance of a point
P
= (
x, y, z
)
from the
yz
plane is

x

, from the
xz
plane is

y

, and from the
xy
plane is

z

.
So
A
is closest to the
yz
plane, since it has the smallest
x
coordinate in absolute value.
B
lies on the
xz
plane, since its
y
coordinate is
0
.
C
is farthest from the
xy
plane, since it has the largest
z
coordinate in absolute value.
3.
Your Fnal position is
(1
,

1
,

3)
. Therefore, you are in front of the
yz
plane, to the left of the
xz
plane, and below the
xy
plane.
4.
Your Fnal position is
(1
,

1
,
1)
. This places you in front of the
yz
plane, to the left of the
xz
plane, and above the
xy
plane.
5.
The point
P
is
√
1
2
+ 2
2
+ 1
2
=
√
6 = 2
.
45
units from the origin, and
Q
is
√
2
2
+ 0
2
+ 0
2
= 2
units from the origin.
Since
2
<
√
6
, the point
Q
is closer.
6.
The distance formula:
d
=
p
(
x
2

x
1
)
2
+ (
y
2

y
1
)
2
+ (
z
2

z
1
)
2
gives us the distance between any pair of points
(
x
1
, y
1
, z
1
)
and
(
x
2
, y
2
, z
2
)
. Thus, we Fnd
Distance from
P
1
to
P
2
= 2
√
2
Distance from
P
2
to
P
3
=
√
6
Distance from
P
1
to
P
3
=
√
10
So
P
2
and
P
3
are closest to each other.
7.
The equation is
x
2
+
y
2
+
z
2
= 25
8.
The equation is
(
x

1)
2
+ (
y

2)
2
+ (
z

3)
2
= 25
9.
The graph is a plane parallel to the
yz
plane, and passing through the point
(

3
,
0
,
0)
. See ±igure 12.1.
31
x
y
z
x
=

3

3
Figure 12.1
x
y
z
y
= 1
1
Figure 12.2
10.
The graph is a plane parallel to the
xz
plane, and passing through the point
(0
,
1
,
0)
. See ±igure 12.2.
11.
The graph is all points with
y
= 4
and
z
= 2
, i.e., a line parallel to the
x
axis and passing through the points
(0
,
4
,
2); (2
,
4
,
2); (4
,
4
,
2)
etc. See ±igure 12.3.