Three ropes of the same strength pull on a ring, and the ring stays at rest…

Step 1. Square the sum. $0 = \lvert\vec a + \vec b + \vec c\rvert^2 = 1 + 1 + 1 + 2(\vec a\cdot\vec b + \vec b\cdot\vec c + \vec c\cdot\vec a)$.

Step 2. The value. So $\vec a\cdot\vec b + \vec b\cdot\vec c + \vec c\cdot\vec a = -\dfrac{3}{2} = -1.5$.

Step 3. One pair. From $\vec a + \vec b = -\vec c$ we get $\lvert\vec a + \vec b\rvert^2 = 1$, so $2 + 2\,\vec a\cdot\vec b = 1$ and $\vec a\cdot\vec b = -\dfrac{1}{2}$.

Step 4. The angle. $\cos\theta = -\dfrac{1}{2}$, so $\theta = 120^\circ$. The same argument works for each pair, so the three ropes are $120^\circ$ apart.