Define scalar product of two vectors. Show that scalar product of two vectors in invariant under rotation.
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A property or rotations is that their matrices are orthogonal and their transpose is equal to their inverse so that Rt=R−1, so the scalar product is = uRR−1vt and RR−1=I (the identity matrix), so that uRRtvt=uRR−1vt=uIvt=uvt, i.e. the dot product is invariant under rotation
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