What the Residual Is Perpendicular To

$r = (-1,\ 2,\ -1)^\top$ and $X^\top r = (0,\ 0)^\top$

$r = (1,\ -2,\ 1)^\top$ and $X^\top r = (0,\ 0)^\top$

$r = \left(-\tfrac{1}{7},\ -\tfrac{16}{7},\ \tfrac{11}{7}\right)^\top$ and $X^\top r = \left(-\tfrac{6}{7},\ 0\right)^\top$

$r = (1,\ -2,\ 1)^\top$ and $X^\top r = (0,\ 6)^\top$