How can I show that the dot product scales up linearly with the norms of the underlying arguments?
As said in comments, it follows right from the definition of the dot product that
$$\lvert ax \cdot y \rvert = \lvert x \cdot ay \rvert = \lvert a \rvert \lvert x \cdot y\rvert$$
for all scalars $a$ and vectors $x,y$. And also for the norm -- where you seem to have forgotten the square root in the definition $\Vert x \Vert := \sqrt{x \cdot x}$ -- we have
$$\Vert a x \Vert = \lvert a \rvert \Vert x \Vert$$
(in particular, e.g. $\Vert (\sqrt2, 0) \Vert = \sqrt2 \; \Vert (1,0) \Vert$). The claim in the source follows from these equalities as in user copper.hat's comment.