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Euclidean Metric : Translation and Rotation

Let be the standard euclidean metric on defined by

where and are any points in .

a) a translation is a map given by

for some fixed given point . Prove, that the Euclidean metric d on is translation-invariant, ie, for any translation T it follows:

.

b) A rotation is a map given by

for some fixed real number . Prove, that the Euclidean metric d on is rotation-invariant, ie, for any rotation it follows:

.

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Solution Summary

Translation and rotation of a Euclidean metric are investigated. The solution is detailed and well presented.

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