# Vector Addition in Euclidean Space

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?

Hint: $$z = x + (z-x) .$$
Then normalize $z-x$.
Let $r = u/\|u\|$. This will do the job.
• You should bring your notation in line with the question: $r$ is a scalar and $u$ a unit vector. – quid Aug 23 '16 at 1:01