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
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Chapter2: Second-order Linear Odes
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Question
![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($) = (qij)1<ij<3 given by
Ipp = Grr = cos(ø)
qii =1 if i # p,r
9pr =-Grp = - sin(ø)
lip = 9pi = Gir = qri = 0 i#p,r
qij =0 if i # p,r and j# p,r
%3D
%3D
will be called a plane rotation through ø in the plane span (ep, er). Let Q =
(gij)1Si,j<3 be a rotation matrix. Show that there exist angles o E [0, ), 0, Þ E
(-1, 7] called the Euler angles of Q such that
Q = Q12(4)Q23(0)Q12(4).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F154c8362-431c-4086-a9f2-ecc4f79c2de5%2Fdf01d79a-2351-4bad-b8a9-57cee7dc37ea%2Fybsqpfs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4.
We consider 3 ×3 matrices over R. An orthogonal matrix Q such
that det(Q) = 1 is called a rotation matrix. Let 1 < p <r < 3 and ø be a real
number. An orthogonal 3 x 3 matrix Qpr($) = (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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