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Physics 325 Lecture 2

# Physics 325 Lecture 2 - Physics 325 Lecture 2 If we rotate...

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Physics 325 Lecture 2 If we rotate first by β and then by δ , we get the following: ' cos sin First rotate by : ' sin cos "' cos sin Now rotate by : sin cos ββ β δδ δ ⎛⎞ = ⎜⎟ ⎝⎠ = xx yy AA cos sin cos sin sin cos sin cos cos cos sin sin cos sin sin cos cos sin sin cos cos cos sin sin cos sin sin cos δβ βδ −− ⎛ ⎞ == ⎜ ⎟ ⎝ ⎠ +− +− +⎛⎞ = ++ x y x y x y A A A A A A This is equivalent to a single rotation by β+δ , as you would expect. Notice that in this case it would not matter if we did the rotation first and then the rotation, or vice versa. The rotation matrices, in the 2 dimensional case, commute. This is not so for three dimensions and higher, as is nicely shown in Figure 1.9 in the text. The rotation matrices have the property that their transpose is their inverse: 22 cos sin cos sin cos sin cos sin sin cos sin cos sin cos sin cos 10 cos sin cos sin cos sin 01 cos sin cos sin cos sin = + −+ T This property insures that dot products are invariant under rotation: () ( ) T TT T A B A B RA RB A R R B A B A B ′′ = GG ii T For instance, the length of a vector ( G G i ) does not change under rotation. Cross Products The cross product between two vectors yield a third vector, perpendicular to the first two and with magnitude given by sin CA BA B θ =×= G G G (2.1)

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where θ is the angle between and A G B G . Take for instance the three unit vectors: Cross products between different combinations of these work as follows: 1 ˆ e 3 ˆ e 2 ˆ e (2.2) 12 323 131 2 21 332 113 ˆˆ ˆˆˆ ˆˆˆ ˆ ,, ˆˆ ˆˆˆ ˆ ee eee eee e ee eee e ×= 2 We can summarize this by introducing a new, useful, item: ( ) ˆˆ ˆ i j k ijk ee e ε i where ( ) () 1 if , , (1,2,3),(3,1,2),(2,3,1) (1,3,2),(2,1,3),(3,2,1) 0 if ijk ijk i j or j k or i k += =− = === (2.3) So there are many ways to evaluate the cross product. You can use Equation 2.1 for the
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Physics 325 Lecture 2 - Physics 325 Lecture 2 If we rotate...

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