Question

3. Suppose the Earth is rotated so that its north pole goes to 20° north, 10° west in the original longitude and latitude system and that the 25° west meridian points due south (in the original system). (a) What are the Euler angles describing this rotation? (b) Find the transformation matrix S describing the rotation.

Answer 1

Sol Let us assume that, Earth is rotated so that its noith pole goes to 200 noith, 10° west in the Briginal longitude and latitude system and that the 250 west meridian points. a) from the data, Here the euler's angles are 2 = 800, B= 70° and 32-60°. b) North pole goes to 200 noith, and 100 west so that hirst first sico sina sila - cosa = 1-sina cosa o COS 800 ) sin 800 cas 800 si (x) = (-sin 800 OO - OO - es sicos S, («) 0.17186 = 1-0.9 0.985 0.174 ) -sing) Illy r cospo Sy CB) = 1 . . sinß o cosß / -sin 700 cos 700 o sin 70° o . SelB) = 0.342 1 0 10.940 0 1 o Cos 70° -0.940 0 0.342

NOW : Self). Sılar) l 0.342 0 0.940 1014 0.985 -0.905 0194 01 0.940 (0.342NO-DU)+(Ox -0910)(-0.940x0) (034270.905)40*0.04+ (. 000) (o 12800101 OXO 120+ TX-0.9.55+ OXO 0x0.985+ 1XO 14+OXO 0X0 + 110 + 0X1 0.quox0.174 + OX-0.905 70-342x0 0.940X0.98€ + OxO 120 XO 311280 0.940x0+ 0x ot O 3L12X1 0.06+0+ 0.34 +0+0 0+0+ (-0.940) 0+(-0.985440 0+0.174+ 0 0+ Oto 0.93+0+0 0+0+ Oto+0.342 s 0.06 -0.94 s' = -0.99 0:34 0.17 0.93 [ 0 0.34

Now for 250 west meridian points due south, so $363) = ( caso sind - sind cast 0 I cos (-60°) | - sin (-600) sin(600) COS (-600) 0 - r 0.5 -(-0-87) -0.87 0.5 0 cisz (3) To.5 ( 0.87 -0.87 0.5 on 0) - Therefore, final direction cosines are s(«,8,3) = 53(3). 52CB).Sica) - 53(8).s' 10.5 -0.87 ol / 0.06 0.34 -0.94 ) = 10.87 0-5 0 -0.99 0.17 0 1 O 0.93 0:34 10 :5X0.06+ (-0.87X-0-99) + 0x0 0 :5x0.347(-0-87X0:17) + 0x0-93 SC2, B8) = 1 0:5x(-0.94) + (-0-8740)H0x0:341 10.87X0:06+ 0.5x(-0-99)+ Coxo) (0.870.34) +0:5X0-12 + OX093 0.87*-0.94+0.5x0+ ox 0.34 / 0x006+ OX(-0.99)+XO 204)+180 0 X0 Q6+ 0x0-17+1x0.93 Oxl-0.94)+ oxO+ 1x0:34 10.03+0.86 +0 0.17+(-0.15) + (-0.5)+o+o SCVB 8 ) = 0.05+(-0.9)+0 0:3+ 0.09+0 -0.82+ 0+0 L0+0+0 O +0+0.93 0+0+0:34 7

0-89 0.02 =slæ,ß,d) - -0.45 0.39 0.93 -0.5 -0.82 0.34 . The transformation matrixe 's describing the rotation is s s(dißid) 0.89 -0.45 or 0 -0.5 -0.82 = 0.02 0.39 0:39 0.93 0.34 L