HOMEWORK 05
Solutions
(1) Suggested Problems
(2) Suggested Problems
(3)
(a) Since
v
1
is parallel to the
xz
plane, its
j
component will be zero. Also,
the ratio of the
k
component to the
i
component will be
f
x
(
P
0
) in order
to have this as its slope.
v
1
=
i
+
f
x
(
P
0
)
k
satisfies these conditions. For
v
2
, the
i
component will be zero for it to be parallel to the
yz
plane.
The ratio of the
k
component to the
j
component will be
f
y
(
P
0
). Hence,
we can use
v
2
=
j
+
f
y
(
P
0
)
k
.
(b) To find
n
we take the cross product of
v
1
and
v
2
:
n
=
v
1
×
v
2
=
i
j
k
1
0
f
x
(
P
0
)
0
1
f
y
(
P
0
)
=
−
f
x
(
P
0
)
i
−
f
y
(
P
0
)
j
+
k
(c) From our vector
n
and point
P
0
(
x
0
, y
0
, z
0
), we can write the equation of
our plane as:
−
f
x
(
P
0
)(
x
−
x
0
)
−
f
y
(
P
0
)(
y
−
y
0
)+(
z
−
z
0
) = 0. In standard
form, this is:
f
x
(
P
0
)
x
+
f
y
(
P
0
)
y
−
z
=
f
x
(
P
0
)
x
0
+
f
y
(
P
0
)
y
0
−
z
0
.
(d) First, we determine where
S
intersects the
x
,
y
, and
z
axes.
To find
the point
P
1
where
S
intersects the
z
axis, set
x
and
y
equal to zero in
our equation:
z
=
−
4
→
P
1
(0
,
0
,
−
4). Similarly, to find where
S
inter
sects the
x
axis, set
y
and
z
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 Fall '07
 ADAMNORRIS
 Euclidean geometry

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