MAE146: Astronautics – Homework
Assigned: Wednesday, January 6
th
,
2010
Due date: Friday , January 8
th
,
2010,
(at the beginning of the lecture)
Please justify all of your answers. Write your answer cleanly with sentences and diagrams to explain. Don’t forget
to write your name on your copy to receive credit.
Problem 1 (10pts): Elementary rotations
Given a reference frame
R
= (0
,
ˆ
e
1
,
ˆ
e
2
,
ˆ
e
3
), the elementary rotations are defined as the rotations transforming the
basis ( ˆ
e
1
,
ˆ
e
2
,
ˆ
e
3
) into a basis (
ˆ
f
1
,
ˆ
f
2
,
ˆ
f
3
) with one of the basis vector fixed (that is
ˆ
f
i
= ˆ
e
i
for some index
i
). The other
basis vectors are rotated by a fixed angle,
θ
, with positive angle being understood as counterclockwise rotations
when looking down the fixed vector.
For example, the rotation leaving the 3
rd
axis fixed is defined by the relations:
ˆ
f
1
=
cos(
θ
)
.
ˆ
e
1
+ sin(
θ
)
.
ˆ
e
2
(1)
ˆ
f
2
=

sin(
θ
)
.
ˆ
e
1
+ cos(
θ
)
.
ˆ
e
2
(2)
ˆ
f
3
=
ˆ
e
3
(3)
or, expressed in matrix form:
ˆ
f
1
ˆ
f
2
ˆ
f
3
=
R
3
(
θ
)
.
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 Winter '10
 Villac
 Geographic coordinate system, ECF, Coordinate system, Polar coordinate system

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