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worksheet3.20100125.4b5e7a90905183.84074720

worksheet3.20100125.4b5e7a90905183.84074720 - Worksheet on...

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Worksheet on transition matrices Exploring proper orthogonal matrices A global chart Given a triplet of perpendicular lines labeled X , Y , and Z and intersecting at a point O , we saw how a global chart for the configuration of a particle located at P can be constructed using the signed distances from O to the orthogonal projections of P onto X , Y , and Z , respectively. We write Ξ for the corresponding one-to-one map and Ξ 1 = x , Ξ 2 = y , and Ξ 3 = z for the correspond- ing coordinate functions. A linear transformation Define the function f Now consider three new quantities, say Ξ 1 , Ξ 2 , and Ξ 3 and write Ξ = I Ξ 1 , Ξ 2 , Ξ 3 M T . Define the function f : R 3 fi R 3 such that f H Ξ L = I f 1 I Ξ 1 , Ξ 2 , Ξ 3 M , f 2 I Ξ 1 , Ξ 2 , Ξ 3 M , f 3 I Ξ 1 , Ξ 2 , Ξ 3 MM T , where f 1 I Ξ 1 , Ξ 2 , Ξ 3 M = h 11 Ξ 1 + h 12 Ξ 2 + h 13 Ξ 3 , f 2 I Ξ 1 , Ξ 2 , Ξ 3 M = h 21 Ξ 1 + h 22 Ξ 2 + h 23 Ξ 3 , and f 3 I Ξ 1 , Ξ 2 , Ξ 3 M = h 31 Ξ 1 + h 32 Ξ 2 + h 33 Ξ 3 . Consider the equation Ξ = f H Ξ L . For every triplet of values of Ξ 1 , Ξ 2 , and Ξ 3 there is a unique triplet of values of Ξ 1 , Ξ 2 , and Ξ 3 . Since the relationship is linear in in Ξ 1 , Ξ 2 , and Ξ 3 it can generally be inverted. Here, In[1]:= f = 8 h11 Ξ 1t + h12 Ξ 2t + h13 Ξ 3t, h21 Ξ 1t + h22 Ξ 2t + h23 Ξ 3t, h31 Ξ 1t + h32 Ξ 2t + h33 Ξ 3t < Out[1]= 8 h11 Ξ 1t + h12 Ξ 2t + h13 Ξ 3t, h21 Ξ 1t + h22 Ξ 2t + h23 Ξ 3t, h31 Ξ 1t + h32 Ξ 2t + h33 Ξ 3t < The inverse is then In[2]:= Solve @8 f @@ 1 DD x, f @@ 2 DD y, f @@ 3 DD z < , 8 Ξ 1t, Ξ 2t, Ξ 3t <D Simplify Out[2]= :: Ξ 1t fi h23h32x - h22h33x - h13h32y + h12h33y + h13h22z - h12h23z h13h22h31 - h12h23h31 - h13h21h32 + h11h23h32 + h12h21h33 - h11h22h33 , Ξ 2t fi h23h31x - h21h33x - h13h31y + h11h33y + h13h21z - h11h23z - h13h22h31 + h12h23h31 +

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