WillJagy answered a linear relation question on Pythagorean Triples in Small linear relations between primitive Pythagorean triples $\mathsf I$.

Now let $a^2+b^2=c^2$ be a primitive Pythagorean triple and then consider the Linear Diophantine Equation $ua+vb+zc=0$ where $(u,v,z)\in\mathbb Z^3$ are variables. If $(u,v,z)\neq(0,0,0)$ then:

Is $\|(u,v,z)\|_\infty$ at least $\sqrt{\max(|a|,|b|)}$ up to constant factors or should the scale (disregarding constants) be smaller (perhaps $\sqrt[3]{\max(|a|,|b|)}$)?

What is the distribution of $\|(u,v,z)\|_\infty$?

Note if it were $ua^2+vb^2+zc^2=0$ then the answer is $O(1)$ since $(u,v,z)=(1,1,-1)$ suffices.

This is what I have $$ a = m^2 - n^2 $$ $$ b = 2mn $$ $$ c = m^2 + n^2 $$ then $$ n(m^2 - n^2 ) +(-m)(2mn) + n(m^2 + n^2) = 0 $$ or triple $$(u,v,z)=(n,-m,n)$$ works and this gives morally $\sqrt{\max(|a|,|b|)}$ ($(m,n,-m)$ also works to give morally $\sqrt{\max(|a|,|b|)}$). Could there be something smaller?