# Forces on a ladder leaning against a box

My question is based on the following problem:

A ladder of length $l=3\mathrm{m}$ is leaning against a box with height $h=2\mathrm{m}$ so that the angle between the ladder and the ground is 60 degrees. The ladder is held in place by a wire streched from the bottom of the box to the bottom of the ladder. What is the force exerted on the ladder by the wire? Ignore all friction forces.

My initial thought relating to the forces, was that there has to be four forces acting on the ladder:

1. $m\vec{g}$ (The force of gravity)
2. $\vec{S}$ (The force from the wire)
3. $\vec{N_1}$ (The force from the ground at the base of the ladder)
4. $\vec{N_2}$ (The force from the edge of the box at which the ladder is leaning)

I then wanted to use $\sum{\vec{F}}=0$ and $\sum{\vec{\tau}} = 0$ to find $\vec{S}$, but I discovered that I needed to know the direction of $\vec{N_2}$ for this method to work. When the ladder is leaning against a wall, I know that $\vec{N_2}$ is perpendicular to the wall, but in this case it was not clear to me what the direction of $\vec{N_2}$ should be. After some trial and error, I assumed that $\vec{N_2}$ would be perpendicular to the ladder, which gave me the correct value for $\vec{S}$. But this left me wondering: Is it always the case that $\vec{N_2}$ is perpendicular to the ladder, regardless of the angle between the ladder and the ground? Furthermore, is yes: Why is that? And if no: What would be the correct, non-trial&error approach to this problem?