d) Having seen how the inner product behaves under rotation we can now investigate its behaviour under the more general Euclidean group. The Euclidean group consists of all orthogonal Oij (rotations as we have seen above along with reflections) and translations aż under which a vector vi transforms as Vi → Oijvj + ai. Since the matrices Oij are orthogonal their transpose is their inverse, (4) 0-1 = OT. (5) Show whether or not the inner product is still invariant under the full Euclidean group. What about restricting it to only orthogonal transformations with no translations (ar = 0)? What about the difference between vectors ui – vi? =

d) Having seen how the inner product behaves under rotation we can now investigate its behaviour under the more general Euclidean group. The Euclidean group consists of all orthogonal Oij (rotations as we have seen above along with reflections) and translations aż under which a vector vi transforms as Vi → Oijvj + ai. Since the matrices Oij are orthogonal their transpose is their inverse, (4) 0-1 = OT. (5) Show whether or not the inner product is still invariant under the full Euclidean group. What about restricting it to only orthogonal transformations with no translations (ar = 0)? What about the difference between vectors ui – vi? =