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p605hw7f08wsoln

# p605hw7f08wsoln - Phys 605 Homework 7 Due 5pm Monday...

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Phys 605. Homework 7 Due 5pm, Monday, November 3, 2008 Problem 7-1: [10 pts] Make sure you understand the derivation of the “active” rotation formula (Eq. 4.62 GPS pg. 162) r 0 = r cos Φ + ˆn ( r · ˆn )(1 - cos Φ) + ( r × ˆn ) sin Φ . ( “active” rotation formula ) that expresses r 0 obtained by rotating r about an axis along ˆn = n 1 ˆ x 1 + n 2 ˆ x 2 + n 3 ˆ x 3 (where n 2 1 + n 2 2 + n 2 3 = 1) by angle Φ, where we take positive Φ to be in the negative Right Hand Rule (R.H.R.) sense around ˆn (as shown in Fig. 4.8 in GPS pg. 162) because we want to use this “active” transformation to obtain the corresponding “passive” coordinate transformation below. (a) Use the “active” rotation formula to show the general rotation matrix A , ˆn ) representing the “passive” co- ordinate transformation for a coordinate rotation by angle Φ ( positive R.H.R.) about an axis along ˆn is given by: A = cos Φ + n 2 1 (1 - cos Φ) n 1 n 2 (1 - cos Φ) + n 3 sin Φ n 1 n 3 (1 - cos Φ) - n 2 sin Φ n 2 n 1 (1 - cos Φ) - n 3 sin Φ cos Φ + n 2 2 (1 - cos Φ) n 2 n 3 (1 -

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p605hw7f08wsoln - Phys 605 Homework 7 Due 5pm Monday...

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