Vector Addition in Euclidean Space

Question

I am reading Rudin's Principles of Mathematical Analysis, and I came across something in the proof that I don't quite understand.

Let $x$ and $z$ be vectors in $\mathbb R^k$ for some $k \geq 3$ and $r >0 $ is a real number. Suppose $| z - x | = r$. Then this means that $z = x + ru$ for some unit vector $u$.

I am struggling to see why this is true. Can someone help me clarify this?

Answer 1

Hint: $$ z = x + (z-x) . $$

Then normalize $z-x$.

Answer 2

Let $r = u/\|u\|$. This will do the job.