Let us start with a two dimensional case. We consider a vector ( x,

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1 Roaion marices We consider now roaion marices in wo and hree dimensions. We sar wih wo dimensions since wo dimensions are easier han hree o undersand, and one dimension is a lile oo simple. However, our prime ineres is in hree dimensions, where proeins are embedded. We expec from roaion marices o have he following properies: is a square real marix such ha = I (which means ha he roaion marix does no change he lengh of he vecor. See below) and de ( ) = 1 (which means ha we do no allow mirror images). Le us sar wih a wo dimensional case. We consider a vecor ( x, y ) ha we wish o roae in he plane by an angle (in he direcion opposie o he clock). The norm of he vecor is kep fixed. The diagram below describes he operaion of ineres I is possible o hink abou he roaion of he vecor ( x, y ) as a roaion of wo vecors ( x,0) and oo!). Consider firs he vecor componen x ( x,0) and he effec of he roaion on i. Roaing by he vecor ( x,0) modifies he old ino a new vecor ( xcos, xsin ) Similarly for he y componen of he vecor ( 0, y ) we find ha couner clockwise roaion by an angle generaes he vecor ysin, ycos( ). The complee new 0, y which we will finally add up o a single operaion (i is more convenien vecor afer he roaion will be xcos ysin, xsin + ycos ( ). The above operaion which we did by using geomery and some inuiion can be pu in a more elegan form using a roaion marix cos = sin sin cos

2 y y cos sin x xcos sin sin cos = y xsin + cos Some properies of cos sin cos sin sin cos sin cos cos sin cos sin sin cos = = = = = I sin cos cos sin sin cos ( ) cos sin cos de = = cos + sin = 1 sin Roaion back by should bring us o he place in which we sared (i.e., = I =. Indeed we have cos sin cos sin = = = sin cos sin cos Le = φ1+ φ2, we inuiively expec ha he roaion by a sum can be described as sequence of roaions, i.e. ( ) = ( φ ) ( φ ) A proof: 1 2 cos( φ1) sin ( φ1) cos( φ2) sin ( φ2) ( φ1) ( φ2) = = sin ( φ1) cos( φ1) sin ( φ2) cos( φ2) cos( φ1) cos( φ2) sin ( φ1) sin ( φ2) cos( φ1) sin ( φ2) sin ( φ1) cos( φ2) sin ( φ1) cos( φ2) + cos( φ1) sin ( φ2) sin ( φ1) sin ( φ2) + cos( φ1) cos( φ2) cos( φ1+ φ2) sin ( φ1+ φ2) cos sin sin ( φ + φ ) cos( φ + φ ) sin cos = = = Consider a vecor V. A roaion marix does no change he norm of he vecor. Hence =. This is rivial o show once we remember ha V V V V V V herefore have V V = V V = V V = V V =, we

3 The disance beween any wo poins is no changing by roaion. Consider wo poins in he plane: ( x1, y 1) and ( x2, y 2). The disance beween he wo poins, d 12, is defined as x1 x2 he norm of he vecor difference d12 = = ( x1 x2 ) + ( y1 y2 ) bu we jus y1 y2 showed ha he lengh of a vecor does no change upon roaion. The disance is he lengh of he vecor difference and canno change by roaion. We define a rigid body ransformaion as a change in he coordinaes ha does no affec he disance beween any wo poins of he rigid body (we also require ha no mirror imaging akes place). Noe ha no all he marices saisfying = I are proper roaions. For example for 1 0 = 0 1 we sill have = = = I. However for he second condiion menioned above ( de ( ) = 1) we have -- de ( ) = 1 ( 1) = 1 which is no a proper roaion. Anoher way of hinking abou he condiion on he deerminan above is o consider small roaions. As we have seen a roaion by an angle can be divided o a sequence of small roaions. We can make a large number of such divisions and approach infiniesimal roaions. We wrie = lim ( ) N N i = 1,.., N or 1 N ( ) = ( N) ( N) i= 1,..., N N 1 Such decomposiion should always be possible for a real roaion (one has o be careful hough when applying such limis. The marix of an infiniesimal roaion, as wrien, does no saisfy he condiion we se for a roaion). Wihou proof we commen ha marices wih deerminan 1 canno be decomposed o a produc of infiniesimal roaion. This can be used as ye anoher definiion of roaion marices (we mus be able o decompose hem o a produc of smaller roaions). 1 0 The marix we jus considered, reflecs he Y axis hrough he origin and 0 1 generaes a mirror image. I is impossible o find a sequence of small roaions ha akes a vecor coninuously from he wo reflecion saes of he Y axis. Roaions in hree dimensions Roaions in hree dimensions are a simple exension of he roaions in wo dimensions we jus discussed. The difference is ha in hree dimensions we mus define a roaion axis. The wo dimensional roaions we considered so far can be hough as a special case

4 of roaions in hree dimensions in which he axis of roaion is he Z axis. Consider he following marix: cos sin 0 = sin cos when applied ono a vecor in hree dimensions v ( x, y, z) cos sin 0 x xcos ysin v = sin cos( ) 0 y = xsin ( ) + y cos = he resul is: 0 0 1z z which is exacly he same resul we have for a roaion in wo dimensions in he xy plane. Here Z is he roaion axis which (in general) remains unchanged when we applied he roaion on a vecor along is direcion. We can find he roaion axis of an arbirary roaion marix in hree dimension by solving he following linear problem e = 1 e where e is a vecor along he axis of roaion. Since he axis of roaion is no changing when muliplies i (oherwise i is NOT he axis of roaion), i mus be an eigenvecor of wih an eigenvalue of 1. Deermining e is a linear problem ha can be solved (for example) using he Gaussian eliminaion we discussed earlier. Noe ha if we muliply e by any scalar we sill obain a valid soluion. We herefore choose e o be real (If here is a soluion i can be made real since he marix and he eigenvalue are real) and normalized i o one: ee= 1. The axis of roaion is herefore deermined wih only wo parameers wo angles (he hird is deermined by he normalizaion). The wo parameers can be chosen as he relaive posiion of he axis of roaion wih respec o he Z axis -- cos( Θ ) and he angle of he axis of roaion wih respec o X axis. The las is obained by projecing he axis of roaion firs o he XY plane and hen measuring he relaive posiion wih respec o he X axis -- cos( Ψ ). The skech below demonsraes he process and he way we define a roaion in hree dimensions. As before is a roaion in a plane. The plane is now deermined as he se of vecors perpendicular o he axis of roaion. I.e. he se of vecors perpendicular o e Θ Z e Ψ Y X

5 A roaion in hree dimensions is herefore compleely deermined by hree parameers: ( ΘΨ,, ). Of course he choice of he 3 parameer is unique and here are oher ways of uniquely deermining a roaion in space, a common example are he Euler angles ha can be found in any exbook on classical mechanics (e.g. Goldsein). As we noed before he deerminan of a roaion marix is 1. Wha does i say on he range of eigenvalues ha a roaion marix may have? We spend considerable ime on roaion marices because we have a very concree quesion abou hese marices. Given wo proein srucures (modeled as rigid bodies) wha kind of roaion we can apply on one of hem so ha he wo srucures will overlap as bes as possible. Hence we are back o proeins and biology