Answer
$c=\pm \frac{1}{\sqrt{14}}$.
Work Step by Step
Since $\|c(1,2,3)\|=1$ and $c(1,2,3)=(c,2c,3c)$, we have
$$\sqrt{c^2+4c^2+9c^2}=1 \Longrightarrow 14 c^2=1.$$
Hence, $c=\pm \frac{1}{\sqrt{14}}$.
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