Question

Prove by example that the isometry preserves angles

Answer 1

Answer:

Give some information about utterly and

Answer 2

Step-by-step explanation:

ngle between the vectors is nothing but the angle between the lines represented by the vector x and y respectively. Now f preserves angle means that the angle between the lines spanned by f(0),f(x) and f(0) and f(y). Now we calculate

<f(x)−f(0),f(y)−f(0)>∥f(x)−f(0)∥∥f(y)−f(0)∥=<Ax,Ay>∥Ax∥∥Ay∥=<x,y>∥x∥∥y∥

the last step folows from the fact that A is an orthogoanl linear transformation. The definition i wrote for angle preserving maps has generalization to infinitesimal form. For example all conformal maps are angle preserving and in that case angle between lines and the angle between the images of these lines donot make much sense.