Math 153  Spring 2010
NAME
Signature
Test 5
2 June 2010
Answer the following questions. The answers must be clear, intelligible, and you must show your
work. Provide explanation for all your steps. Your grade will be determined by adherence to these
criteria. Use of books, notes and calculators is strictly forbidden. The point value of each problem
is given in the left hand margin.
1.
(10 pts.) Determine whether the lines given by the parametric equations:
x
= 1 + 2
t
x
=
±
1 + 6
s
y
= 2 + 3
t
y
= 3
±
s
z
= 3 + 4
t
z
=
±
5 + 2
s
are parallel, skew or intersecting.
The direction vectors of these lines
v
1
=
h
2
;
3
;
4
i
and
v
2
=
h
6
;
±
1
;
2
i
are not parallel. Hence
the lines are not parallel.
We check next if the lines intersect. If the two
x
coordinates are equal then we have:
1 + 2
t
=
±
1 + 6
s
which implies that
t
=
±
1 + 3
s
. Then if the
y
coordinates are equal then we must have:
2 + 3
t
= 3
±
s
2 + 3(
±
1 + 3
s
) = 3
±
s
10
s
= 4
which gives
s
=
2
5
. Finally, the equality of the
z
coordinates implies:
3 + 4
±
±
1 + 3
²
2
5
²
=
±
5 + 2
²
2
5
which is not true. Thus the lines do not intersect. They are skew line.
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 Spring '08
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 Math

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