To determine the vector representing the third force $\vec{C}$ that will put the system into equilibrium, we need to use the concept of equilibrium in physics. For an object to be in equilibrium, the sum of all forces acting on it must be zero. This means:

$\vec{A} + \vec{B} + \vec{C} = \vec{0}$

Given: $\vec{A} = 5\hat{i} - 6\hat{j}$ $\vec{B} = 10\hat{i} + \hat{j}$

Let's denote $\vec{C}$ as: $\vec{C} = a\hat{i} + b\hat{j}$

The equilibrium condition gives us: $(5\hat{i} - 6\hat{j}) + (10\hat{i} + \hat{j}) + (a\hat{i} + b\hat{j}) = \vec{0}$

Combining the terms, we get: $(5 + 10 + a)\hat{i} + (-6 + 1 + b)\hat{j} = \vec{0}$

This simplifies to: $(15 + a)\hat{i} + (-5 + b)\hat{j} = \vec{0}$

For this equation to hold true, each component must be zero: $15 + a = 0$ $-5 + b = 0$

Solving these equations for $a$ and $b$: $a = -15$ $b = 5$

Therefore, the vector $\vec{C}$ that puts the system into equilibrium is: $\vec{C} = -15\hat{i} + 5\hat{j}$

So, the correct answer is: $\boxed{\text{c. } \vec{C} = -15\hat{i} + 5\hat{j}}$

Would you like more details or have any questions?

Here are 8 related questions for further practice:

1. If $\vec{A} = 3\hat{i} - 4\hat{j}$ and $\vec{B} = -2\hat{i} + \hat{j}$, what is $\vec{C}$ for equilibrium?
2. How do you determine the resultant vector of multiple forces?
3. What are the conditions for equilibrium in a system with three forces?
4. Given $\vec{A}$, $\vec{B}$, and $\vec{C}$, how can you verify if they are in equilibrium?
5. If $\vec{A} = \hat{i} - 2\hat{j}$ and $\vec{B} = 4\hat{i} + 3\hat{j}$, what must $\vec{C}$ be to maintain equilibrium?
6. Explain the principle of superposition in the context of force vectors.
7. How can equilibrium be applied to solve real-world problems involving forces?
8. Describe the difference between static and dynamic equilibrium.

Tip: When solving vector problems, always break down the vectors into their components and solve for each direction separately to ensure accuracy.