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Do the basis vectors of a rotated vector in $E^3$ transform differently than the components of the vector? I've recently come across a blog where someone rotated the i,j,k basis vector using the standard rotation matrix that is usually applied to the components of a vector in $E^3$. Is this correct?

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A link to said blog would be helpful. – user31373 Jun 12 '12 at 16:55
To be fair, he is actually transforming a quaternion basis according to the rule for transforming the components of a quaternion, but -I believe- the incorrectness of this would follow. But, even considered in isolation, it seems unlikely that the components of the quaternion would transform exactly as the basis. – bb6 Jun 12 '12 at 16:59
@Kovalev I've commented that "In Cartesian coordinates, the basis vectors transform differently than the components of the vector. Have you accounted for this? (I ask because, without checking your math in detail, it appears that you've assumed the basis transforms exactly like the components)" – bb6 Jun 12 '12 at 17:06

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