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Handin Assignment 1 Solutions (Math 20300 DD, PP, ST)
Your Name:
Foster, Antony
Your Student ID:
0000 (only last 4 digits)
Your Instructor:
Prof. A. Foster
Due Date:
Friday, February 8, 2008.
SECTION 13.1 (pages 829–834):
DO Exercises 1, 3, 4, 5, 12, 18, 19, 36, 40, and 42.
Formulas Used:
Distance Formula in Three Dimensions
:
The distance
k
P
1
P
2
k
between the points
P
1
(
x
1
, y
1
, z
1
) and
P
2
(
x
2
, y
2
, z
2
) in 3Dimensional real space or
R
3
is
k
P
1
P
2
k
=
p
(
x
2

x
1
)
2
+ (
y
2

y
1
)
2
+ (
z
2

z
1
)
2
.
(1)
Equation of a Sphere
:
An equation of a sphere with center
C
(
α, β, γ
) and radius
ρ
is
(
x

α
)
2
+ (
y

β
)
2
+ (
z

γ
)
2
=
ρ
2
.
(2)
In particular, if the center is the origin
O
(0
,
0
,
0), then an equation of the sphere is
x
2
+
y
2
+
z
2
=
ρ
2
(3)
Volume of a spherical cap
:
The volume
V
of a cap of a sphere centered at the origin
O
(0
,
0
,
0) with
radius
ρ
and height
h < ρ
of the cap (measured from the endpoint of a diameter of the sphere) is
V
=
1
3
πh
2
(3
ρ

h
)
(can be derived using calculus)
.
(4)
Distance from a Point to a Plane
:
Let
P
be a plane with equation
ax
+
by
+
cz
+
d
= 0 and
P
(
x
0
, y
0
, z
0
) is a point not in the plane
P
then the distance
d
(
p,
P
) from the point
P
to the plane
P
is given
by
d
(
P,
P
) =

ax
0
+
by
0
+
cz
0
+
d

√
a
2
+
b
2
+
c
2
.
(5)
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0.
Exercise 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?
Solution.
Without any drawings we start at the origin
O
(0
,
0
,
0) and move along the
x
axis a distance of 4 units
in the positive direction. This puts us at the point
A
(4
,
0
,
0). From
A
we move downward a distance of 3 units, this
puts our final position at the point
B
(4
,
0
,

3). Note this is a point in the
xz
plane.
tu
Exercise 3
:
Which of the points
P
(6
,
2
,
3)
, Q
(

5
,

1
,
4), and
R
(0
,
3
,
8) is closest to the
xz
plane? Which point
lies in the
yz
plane?
Solution.
Of the three points P, Q and R we can see that Q is closest to the
xz
plane.
The distance to the
xz
plane from a point
A
(
x, y, z
) in space is determined by

y

(i.e., absolute value of the
y
coordinate of the point).
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