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4. We consider 3 x 3 matrices over R. An orthogonal matrix Q such that det(Q) = 1 is called a rotation matrir. Let 1

4. We consider 3 x 3 matrices over R. An orthogonal matrix Q such that det(Q) = 1 is called a rotation matrir. Let 1

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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4. We consider 3 ×3 matrices over R. An orthogonal matrix Q such that det(Q) = 1 is called a rotation matrix. Let 1 < p <r < 3 and ø be a real number. An orthogonal 3 x 3 matrix Qpr($) = (qij)1<ij<3 given by Ipp = Grr = cos(ø) qii =1 if i # p,r 9pr =-Grp = - sin(ø) lip = 9pi = Gir = qri = 0 i#p,r qij =0 if i # p,r and j# p,r %3D %3D will be called a plane rotation through ø in the plane span (ep, er). Let Q = (gij)1Si,j<3 be a rotation matrix. Show that there exist angles o E [0, ), 0, Þ E (-1, 7] called the Euler angles of Q such that Q = Q12(4)Q23(0)Q12(4).
Transcribed Image Text:4. We consider 3 ×3 matrices over R. An orthogonal matrix Q such that det(Q) = 1 is called a rotation matrix. Let 1 < p <r < 3 and ø be a real number. An orthogonal 3 x 3 matrix Qpr($) = (qij)1<ij<3 given by Ipp = Grr = cos(ø) qii =1 if i # p,r 9pr =-Grp = - sin(ø) lip = 9pi = Gir = qri = 0 i#p,r qij =0 if i # p,r and j# p,r %3D %3D will be called a plane rotation through ø in the plane span (ep, er). Let Q = (gij)1Si,j<3 be a rotation matrix. Show that there exist angles o E [0, ), 0, Þ E (-1, 7] called the Euler angles of Q such that Q = Q12(4)Q23(0)Q12(4).
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