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Math 153  Spring 2010
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Test 4
19 May 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.
(6 pts.)
1.
Find the values of
x
such that the vectors
h
3
;
2
;x
i
and
h
2
x;
4
;x
i
are orthogonal.
Two orthogonal vectors have zero dot product, thus
h
3
;
2
;x
i ± h
2
x;
4
;x
i
= 0
6
x
+ 8 +
x
2
= 0
(
x
+ 2)(
x
+ 4) = 0
Answer:
x
=
²
2 and
x
=
²
4.
1
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View Full Document2.
You are given two polar curves
r
= 2 + cos 2
±
and
r
= 2 + sin
±
.
a).(5 pts.) Prove that the intersection points of these curves have coordinates
(5
=
2
;
5
²=
6)
;
(5
=
2
; ²=
6)
;
(1
;
±
²=
2)
:
Hint: the trigonometric formulas you need are provided on the last page.
The intersection points can be found by solving for
±
in the following equation:
2 + cos 2
±
= 2 + sin
±
which can be rewritten as:
1
±
2 sin
2
±
= sin
±
and also as:
2 sin
2
±
+ sin
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 Math

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