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- Thread starter tensor33
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Did you mean [itex]x\cdot x^T=I[/itex]??

That doesn't really explain why the determinant can't be -1, which is what the OP is asking.

The reason is that rotations preserves the orientation of a basis. If we have a right-handed basis, then rotations of this will be right-handed as well.

As an example of an orthogonal matrix that does not preserve the orientation, you can probably take a reflection.

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Fredrik

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To prove that rotations have determinant 1, you must first define the term "rotation". It's perfectly OK to just take "R is said to be a rotation if R is a member of SO(3)" as the definition, but then there's nothing to prove. Another option is "R is said to be a rotation if R is a member of the largest connected subgroup of O(3)". Then your task would be to prove that the largest connected subgroup is SO(3).

(I actually prefer the terminology that calls members of O(3) rotations, and members of SO(3) proper rotations).

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Did you mean [itex]x\cdot x^T=I[/itex]??

Ya, I forgot the transpose part.

Thanks to all those who replied, I'm pretty sure I get it now.

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