MAC 2313
Sept 16, 2010
Exam I and Key
Prof. S. Hudson
1) [10 pts] Find the distance from the point
P
(1
,
1)
∈
R
2
to the line
y
= 2
x
+ 1. Hint:
compute a normal vector and a scalar projection.
2) [10 pts] Show that these lines intersect and find the point of intersection.
Note: the
values of
t
in
L
1
and
L
2
do
not
have to match.
L
1
:
x
= 1 +
t, y
= 1 +
t, z
= 1 + 2
t
L
2
:
x
= 2 +
t, y
= 3
, z
= 4 +
t
3) [10 pts] Find two unit vectors in
R
2
parallel to the line
y
= 3
x
+ 1.
4) [15 pts] Find the equation of the plane that passes through the points
P
(1
,
1
,
0),
Q
(1
,
0
,
2)
and
R
(0
,

1
,
1).
5) [10 pts] Convert this equation from spherical coordinates to rectangular coordinates and
sketch the graph. Hints: We did a similar problem in class. One idea is to multiply both
sides by
ρ
first, and then use conversion formulas, such as
y
=
ρ
sin
φ
sin
θ
, though the
formula for
z
might be more useful here.
ρ
= 4 cos(
φ
)
6) [10 pts] Find an equation for the trace of 4
x
2
+
y
2
+
z
2
= 9 in the plane
y
= 2. State
what kind of conic section the trace is [is it a parabola? etc].
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 Fall '06
 GRANTCHAROV
 Scalar, pts, Parametric equation, normal vector

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