I'm doing this problem as part of my effort to self-study my way through the first 7 chapters of Rudin.

Suppose k

, and

. Prove (a) that if

, there are infinitely many

such that

, (b) there is exactly one such

if

and (c) there is no such z if

.

My progress:

I proved (b) and (c) using the condition for attainment in Cauchy-Schwarz. Part (a) I proved after assuming that

and

. I'm probably just being stupid, but the problem I seem to be running into is that I'm having a hard time undoing my rotation and translation without going beyond what can be done with the material learned so far. Any help would be appreciated. If anyone could give a simple justification (i.e. one that doesn't use anything but definition of the Euclidean norm and the basic algebraic properties of the dot product) for assuming WLOG that

and

, this would certainly do the trick. Any other permissible solution would be fine too.

Thanks.