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sol5 - Math 321 Quiz 5 Name Vemg Q I“ 1(4 pt Suppose...

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Unformatted text preview: Math 321 Quiz 5 Name Vemg Q I“: 1. (4 pt) Suppose you _r>otate 7 counterclockwisely about the direction of b) by angle 04, where H) and b are independent vectors in 3d space. Write down the process to find the resulting vector. ‘ C/wcé gomv mfié ‘ 2. (a) (2 pt) In 3d space, you have an orthonormal (the vectors are unit vec— tors and orthogonal to each other) basis {?1,?2,?3}. If you get a new basis {—33, “5);, ?g} by changing only 732 to —?2 (it is a reflection), what is the matrix Qij such that v; = Qijvj, where (111,112,123) and (Din/2,129 are the components of a vector ? with respect to the two bases. «7/ «1) fix] ' el '60" &_~ (00 * new 00! (b) (2 pt) In 3d space, you have an orthonormal (the vectors are unit vectors and orthogonal to each other) basis {—31, '32, @2}. If you get a new basis {'8'1', '3'2', 73/3/} by rotating 3% and ?2 counterclockwisely about ?3 by 7r, what is the matrix Pij such that 1214’ = vaj, , where (211,212,113) and (v’l’mg’mg’) are the components of a vector 7 with respect to the two bases. ,4 0c» "Pu C 0M 00! (c) (2 pt) Calculate the determinants of Qij and Hj. Wow 0122603): I Bonus. (2 For any two sets of cartesian components a“ related by v; = Qij’Uj, what is the determinant of Qij? ~ ...
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