Answer
Any $\vec{a}$ and $\vec{b}$ that are perpendicular to each other
Work Step by Step
Notice that the second constraint is the Pythagorean theorem which suggests that we should think about right triangles.
Indeed, if $\vec{a}$ and $\vec{b}$ are perpendicular to each other, their vector sum $\vec{c}$ is the hypotenuse of the triangle.
For example, let $\vec{a} = <0, 3> $and $\vec{b} = <4,0>$,
then
$\vec{a} + \vec{b} = \vec{c} = <4,3>$
and
$|a| = 3, |b| = 4,$ and $|c| = 5$
3-4-5 is a Pythagorean triple.
