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I will give you an example (from my recent work) where actually knowing what the components mean was useful.

I had to calculate misorientations between many pairs of orientations. If I treated quaternions as a whole, I would find their quotient and then get the angle. But since I know what the components mean, and I know I can get the misorientation angle from just the cosine (the scalar component), and I know that I can get this value from a simple dot product, I can easily save a few operations to calculate every misorientation. And since I have to calculate many of them, this change supposed a big speedup.

Black boxes are nice when learning or thinking at a high level, but if you frequently use a black box, knowing what is inside will probably become useful at some point.




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