AandA†have identical eigenvectors andcomplex conjugate eigenvalues...........................................................................Exercise:prove (xi,xj) = 0 ifij(see 1. in box above)...........................................................................separable

I-10Figure 1. The Cartesian coordinate system for 3-dimensional Euclidian space.x,1x,2x,3R11R21R31R12R22R32R13R23R33x1x2x2Euler Angle Rotations in 3-dimensional Euclidian Space1.In this space we candefine (r,r) =r·rand ||r|| = |r| = [r·r]½= [x² + y² + z²]½.Note that we will use|r|often and the above gives aspecificalgebraic expression.2.The linear Euler angle rotation operator,R, "maps" the components ofr=i=1,3xix^iin the fixed frame(unprimed) into the components ofr=k=1,3x'kx^k' in arotated (primed) coordinate system.TheRphysically rotates the xyz coordinate axes into the new x'y'z' axes.The proper mathematical notation for what happens to the components is:x'i=j=1,3Rijxjwhich in matrix notation means:

I-11Rz( )cossin0sincos0001Figure 2.Rx,( )1000cossin0sincosFigure 3Rz,,()cossin0sincos0001Figure 4.The Euler angle rotation of the fixed xyz (unprimed) coordinate system into the rotated x'y'z' coordinate systemconsists (by convention) ofthreeconsecutive rotations:First, a counterclockwise rotation aboutz^by(new axes are denoted byx^',y^' andz^');Second, a counterclockwise rotation aboutx^' by(new axes labelled byx^'',y^'' andz^'');Third, a counterclockwise rotation aboutz^'' by(final axes denoted byx^''',y^''' andz^''').

I-12R(, ,)cossin0sincos00011000cossin0sincoscossin0sincos0001The final form for theRR(, , )is:(see Goldstein, p 146)...........................................................................................................................................Exercise:Find the matrix components ofR(, , )by multiplying out the above matrices. (Note that it is mucheasier to remember the three product matrices than the complicated 3x3 final form forR)............................................................................................................................................note:Although the Figures show x',x'' and x''', etc., we shall now DROP the extra primes and call thefinal rotated coordinate system the PRIMED SYSTEM.

I-13x,y,z,R11R21R31xyzNOTE: We now call the final (rotated) axesx^',y^' andz^':The final form for the relationship between the components ofrin thefixed(x,y,z) frame and the components of thevectorrin therotatedframe (x'y'z') is:

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Exercise:Consider thatr= 1·x^is afixedvector. When viewed from therotated(primed) frame what are itscomponents? Use Euler angles,and .Exercise:Suppose a coordinate system is rotated by Euler angles (,,) = (/2, , /4).What are the componentsin the primed frame of the fixed vector,r= (a,0,c)?

I-14x,1x,2x,3R001R1010x1x2x3xyzR1abcUSE OF THE EULER ANGLE ROTATIONS:(1)FIND THE COMPONENTS IN A ROTATED FRAME OF A FIXED VECTOR:An example of this usage might be to find the components of a unit vector oriented along the earth's north polaraxis,rnorth pole=z^from, say, a satellite which is rotating in space.rnorth poleis "fixed" and the coordinate system on thesatellite would be rotating with respect to the earth.

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