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HW1 - MAE146 Astronautics Homework Assigned Wednesday...

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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 counter-clock-wise 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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