Angle between two integer-coordinate vectors $\pi \cdot r$, which $r \neq \frac{1}{2}$ is rational

Question

Are there two vectors $u=(x,y)$, $v=(z,t)$ with integer and both non-zero coordinates (for excluding trivial solutions) such that the $\widehat{uv}$, the angle between $u$ and $v$, is equal to $r \cdot \pi$, where $r \neq \frac{1}{2} \in \mathbb{Q} \cap (0,1)$?

If such, is there a rule to form all such vectors given the angle $r\cdot \pi$?

Answer 1

$(0,1), (1,1)$ would fit the bill $(r = \frac 14)$

So would $(3,2), (-2,3)$ and $(3,2), (-3,-2)$

Update

$(3,1),(1,2)$ would have non-zero coordinates and $r = \frac 14$