MATH 1920 Homework 2, Chapter 12.5
8. The normal direction to 3
x
+ 7
y

5
z
= 21 is
h
3
,
7
,

5
i
, then the line that paases through
(2
,
4
,
5) and points in that direction is given by
x
(
t
) = 2 + 3
t
,
y
(
t
) = 4 + 7
t
,
z
(
t
) = 5

5
t
.
12. The
z
axis is given by
x
(
t
) = 0,
y
(
t
) = 0, and
z
(
t
) =
t
.
22. The plane parallel to the plane 3
x
+
y
+
z
= 7 also has the form 3
x
+
y
+
z
=
D
for some value
D
. To get what
D
is, simply plug in the point (1
,

1
,
3) into the equation, so
D
= 3

1 + 3 = 5.
The equation is 3
x
+
y
+
z
= 5.
28. To find the point of intersection of two lines given by parametric equations, we need to find
the parameters
t
and
s
for which the corresponding coordinates all match. For the
x
coordinate,
we have
t
= 2
s
+ 2, and for
y
,

t
+ 2 =
s
+ 3. These two should suffice to give us the values for
t
and
s
. Plugging the first into the second, we have

2
s

2 + 2 =
s
+ 3, so
s
=

1, hence
t
= 0. We
have to check the
z
coordinates agree too. Indeed,
t
+ 1 = 1 and 5
s
+ 6 = 1. The point that
t
= 0
and
s
=

1 both give is (0
,
2
,
1).
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 '06
 PANTANO
 Math, Multivariable Calculus, Euclidean geometry, Parametric equation

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