In class, I justified these two equations by stating the following: the point $e$ has the potential to correspond to any point $x'$, i.e., $x'^T E e = 0$. This is because the point $e$ exists on the epipolar line associated with any point $x'$. Given that $x'^T E e = 0$ for all $x'$, it follows that $E e = \mathbf{0}$.
How did we get these equations for the epipoles?
In class, I justified these two equations by stating the following: the point $e$ has the potential to correspond to any point $x'$, i.e., $x'^T E e = 0$. This is because the point $e$ exists on the epipolar line associated with any point $x'$. Given that $x'^T E e = 0$ for all $x'$, it follows that $E e = \mathbf{0}$.