Three forces `A = (hat(i) + hat(j) + hat(k)), B = (2hat(i) - hat(j) + 3hat(k))` and C acting on a body to keep it in equilibrium. The C is :

Options

A. `-(3hat(i) + 4hat(k))`
B. `-(4hat(i) + 3hat(k))`
C. `3hat(i) + 4hat(k)`
D. `2hat(i) - 3hat(k)`

Answer

To find the force vector \( \mathbf{C} \) that keeps the body in equilibrium, we can follow these steps:

### Step 1: Understand the condition for equilibrium
For a body to be in equilibrium, the vector sum of all forces acting on it must equal zero. This can be expressed mathematically as:
\[ \mathbf{A} + \mathbf{B} + \mathbf{C} = \mathbf{0} \]

### Step 2: Write down the given forces
We have the following forces:
- \( \mathbf{A} = \hat{i} + \hat{j} + \hat{k} \)
- \( \mathbf{B} = 2\hat{i} - \hat{j} + 3\hat{k} \)

### Step 3: Substitute the forces into the equilibrium equation
Substituting the values of \( \mathbf{A} \) and \( \mathbf{B} \) into the equilibrium equation gives:
\[ (\hat{i} + \hat{j} + \hat{k}) + (2\hat{i} - \hat{j} + 3\hat{k}) + \mathbf{C} = \mathbf{0} \]

### Step 4: Combine the forces \( \mathbf{A} \) and \( \mathbf{B} \)
Now, we can combine the components of \( \mathbf{A} \) and \( \mathbf{B} \):
- For \( \hat{i} \): \( 1 + 2 = 3\hat{i} \)
- For \( \hat{j} \): \( 1 - 1 = 0\hat{j} \)
- For \( \hat{k} \): \( 1 + 3 = 4\hat{k} \)

Thus, we have:
\[ 3\hat{i} + 0\hat{j} + 4\hat{k} + \mathbf{C} = \mathbf{0} \]

### Step 5: Isolate \( \mathbf{C} \)
To find \( \mathbf{C} \), we can rearrange the equation:
\[ \mathbf{C} = - (3\hat{i} + 0\hat{j} + 4\hat{k}) \]
This simplifies to:
\[ \mathbf{C} = -3\hat{i} - 4\hat{k} \]

### Step 6: Write the final answer
Thus, the force vector \( \mathbf{C} \) that keeps the body in equilibrium is:
\[ \mathbf{C} = -3\hat{i} - 4\hat{k} \]