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Rotations conceptually 
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#1
Feb1611, 08:29 PM

P: 179

This isn't a homework question, but I'm having trouble understanding something about rotations conceptually. While reading about the Euler Identity online, I keep running into a few things that I can't wrap my head around and never come with an explanation.
Here are the concepts I can't understand: A halfturn is the square of a quarter turn. Rotations cannot be added, only multiplied. Why is this? Intuitively, if I'm thinking about rotating by a quarter turn, then another quarter turn, I add them together to get a half turn. Instead, it appears I have to multiply the quarter turn by itself. I don't understand this. pi equals 2(pi/2), not (pi^2)/4 Basically, I keep running across descriptions of rotations as exponential growth, whereas all I can see when looking at them is linear growth. How are rotations exponential? 


#2
Feb1611, 08:42 PM

P: 4,573

Consider the rotation matrix R that does you're quarter turn.
In a nutshell what it is saying is that your half turn is represented by R(R(X)) where X is your point assuming you are rotating the point about the origin. Consider your quarter turn to be R x X = X(0) Apply that same rotation R x X(0) applies the rotation to the result of the 1st rotation. So essentially R^2(X) is applying the same rotation "twice" hence the half turn result. If you want to prove this rigorously, create the rotation matrix R using the definition of your rotation and calculate R^2 and see what you get for each individual entry in your resultant matrix R^2. 


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