3. [10 points] For each isometry f: IS -> Ra, there is an induced map Lf: RS -> R defined by (a, b,c) > f(a, b,c) - f(0,0,0). (Hint: for all the parts below, it will probably help to draw a picture of what happens in a particular example, and to use geometric definitions of e.g. vector addition.) (a) Show that for any vector (x, y,z), we have Ly(a, b,c) = f(atx, b ty,c +z) -f(x, y,z). (b) Show that Ly is a linear map on R and is an isometry. (c) For each type of isometry of RR', say what type of isometry Ly is, with a brief justification. 4. [10 points] (a) Prove that if f: RS -> IR is an isometry such that f(0,0,0) = (0,0,0), then f preserves dot products, that is f(v) . f(w) =v. w for all v, wes. (b) Now prove the converse, namely, that if f: R -> R preserves dot products, then it is an isometry fixing the origin