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Mathematics 241
First Exam Solutions
Dr. Rosenberg
Friday, February 28, 2003
1.
a) Show that the points
P
1
= (

1
,

2
,
6)
, P
2
= (0
,
4
,
1)
, P
3
= (1
,
0
,
1) determine a
unique plane
P
, and determine the equation of
P
.
Solution:
To show they determine a unique plane, it’s enough to show that the line segments
P
1
P
2
and
P
1
P
3
are not parallel. But
(
P
2

P
1
)
×
(
P
3

P
1
) =
P
2
×
P
3

P
1
×
P
3
+
P
2
×
P
1
= (4
,
1
,

4) + (

26
,
2
,

4)

(2
,
7
,
2)
= (

20
,

5
,

10) =

5(4
,
1
,
2)
6
= (0
,
0
,
0)
.
So the points are not collinear and (4
,
1
,
2) is perpendicular to
P
. Then the equation of
P
is
(4
,
1
,
2)
·
(
x, y, z
) = (4
,
1
,
2)
·
(

1
,

2
,
6)
or
4
x
+
y
+ 2
z
= 6
.
b) Find the area of the triangle with vertices
P
1
,
P
2
, and
P
3
.
Solution:
1
2
±
±
(
P
2

P
1
)
×
(
P
3

P
1
)
±
±
=
5
2
±
±
(4
,
1
,
2)
±
±
=
5
2
√
21
.
2. (10 points) Let
P
1
and
P
2
be the planes with equations
3
x

4
y
+
z
= 2
,
3
x
+ 2
y

z
= 7
.
Find symmetric equations of the line
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This note was uploaded on 09/09/2009 for the course MATH 241 taught by Professor Wolfe during the Spring '08 term at Maryland.
 Spring '08
 Wolfe
 Math, Calculus

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