Rotations in 3D A Nexus mindmap 0 1 2 This actually works 3 Angular velocity 1 There is only 1 plane 0 1 0 0 0 (or any other 3 independent planes) 2 xy xz yz 1 There are 3 planes 0 1 1 0 1 tx ty tz xy xz yz 0 There are 6 planes 0 Can't create a suitable vector space 0 1 2 This is a curiosity of 3D! 3 0 0 0 1 1 0 This argument can be extended to non-colinear rotation axes 1 Example 2 0 1 Rotations are not Abelian! 2 0 We saw that there was a really nice correspondence between rotational motion in 2D and linear motion. Can this correspondence be extended to 3D? 0 0 This is called 'Abelian' under summation. It's a defining proverty of vector spaces. 0 1 A natural extension of a location parameter to higher dimensions is by using a vector 1 Extending rotations to 3D 0 0 0 1 2 0 0 1 2 1 0 0 1 0 0 1 0 0 1 2 2 2 2 0 1 2 3 4 5 3 Cross products 1 0 0 0 0 1 Trick to derive quickly 2 1 1 Rotations in 2D 2 0 1 N.B. to stick to RH coordinate system we have to rotate this one 'backwards' 0 2 2D rotations in x-y/y-z/x-z planes 0 27 posibilities for choosing 3 angles to rotate by 0 3 of these are all around same axis 1 12 combinations involve 2 sequential rotations about same axis 2 In the end have only 27-3-12=12 legitimate combinations that could describe any 3D rotation 3 Combinations 1 Rotations in 3D 3 0 0 1 0 [Morin, Introduction to Classical Mechanics ] 1 2 4 Rotations in 3D