Question

Consider the two rotations shown. Calculate the rotation matrix for each trans- formation (ể x,y,z to ēm.y,z, then ê' r,y,z to ēr,y,z), then calculate the required rotation matrix to move from ēr,y,z to ēr,y,z. Find the eigenvalues and eigenvectors of this final matrix. Ae

Answer 1

ANSWER: are two transformations n ous Given that there They are the éx,y,z to éxi.uz From condition o @ Buruz to ē" x, y, z aez nez Ēy for Rotation 2-oxºs el e sły by 90°=8 Rotational matope becomes (about x-axls) Now, cos8 sin 8 o7 R(x, y, z) bu qoo - sin coss o Let Loo Is here 8=90° putting y = qo 1 07 RC X, Y, 2). : z by 900 - 0 0 As os qở = 0 , sn 40° = 1

condition : & y Rotation about Xaxis by 1800 ie Q = 180º en so, the rotational matrix becomes, and a R(x, y, 2)x by 180° by 180° To lo 10 cosa - sina sina cosa NOW, putting &=90, к ( у cos (180) = -1, sin (180)=0... to get rotational motore for transformation, Ecury, zto sēcm,y,z) this will become 3) * Final 0-1 0 1 ó 1- LOOIJL OO'] Therefore, we get RfPnal = eq@ x eq 0 1

then, we get Reinal as R penal = To 107 - Lo o So, equation for Eigen values for Regnale From the above the characterstic eigen values is [A - XI = 0 A - XL 0 1 0 to 10 = 0 Loo LO 0 -1 F-s 10 • 1-2 01 - 0 4 ) Loo - (1+) Therefore, by solving equation 4, then we get the values of ļ as 1=- =1 so there are the eigen values.

- 16 Now we will find eigen vector s 1/2 x,+X2 = 0; X,+ 112*2= 0; -3x3=0 From the above expressions we get besede X3=0 By solving ② and 6 equations we get WE +2X2 = 0 Ti multiplied by 2] (-) 4X, +2x2 = 0 [.! multiplied by 4] -3x, = 0 in equation © by substituting these X,=0 we get X2=0] Therefore, x1=0 , X2=0, X3 = 0 [eigen vector ③ = 5x7 r. Ooo Pn For x=1 by subs têtuting 1=1 F-1/2 10x -1 Oll. XL = 0. 0 21 X3] 0 xx+ Xo = 0, 7,-%2 = 0, 2x3 =0

tolka These fose, by the above equations we get 113 = 0] net - by solvọng flyst two equations, we get — X,+X2=0 · X -X2=0 3 one of any the above -2x,-0 => Tx;=0 S by substituting xizo in equations we get X2=0] Therefore, 17,20), x2 =0, X3=0 agen webx ® = 8:1-1871 eigen vector @ = - xx - The ou — XX -