challenge_problem_1.20100115.4b509aecddce24.09886944

challenge_problem_1.20100115.4b509aecddce24.09886944 -...

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Challenge problem Harry Dankowicz Mechanical Science and Engineering University of Illinois at Urbana-Champaign [email protected] 1. Denote by T ( r n , θ ) a transformation that rotates objects in three-dimensional space about some axis parallel to the unit vector n by an angle θ in the direction given by the right-hand rule and scales their size by a factor r . (a) Show that T ( r n , θ ) is a linear transformation. (b) Suppose that e 1 , e 2 ,and e 3 form a right-handed orthonormal basis of space (i.e., a set of mutually orthogonal vectors of unit length, such that e 1 × e 2 = e 3 ). Suppose, without loss of generality, that n = n 1 e 1 + n 2 e 2 + n 3 e 3 6 = e 1 . Show that the collection of vectors m 1 = n 3 p n 2 2 + n 2 3 e 2 n 2 p n 2 2 + n 2 3 e 3 , (1) m 2 = q n 2 2 + n 2 3 e 1 + n 1 n 2 p n 2 2 + n 2 3 e 2 + n 1 n 3 p n 2 2 + n 2 3 e 3 , (2) m 3 = n (3) forms a right-handed orthonormal basis of space. (c) Show that the matrix of coordinate projections of T ( r n , θ ) ( v ) relative to the basis m 1 , m 2 m 3
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This note was uploaded on 11/17/2011 for the course TAM 412 taught by Professor Weaver during the Spring '08 term at University of Illinois, Urbana Champaign.

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