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 solve the problem of finding the force \( C \) that keeps the body in equilibrium under the influence of forces \( A \) and \( B \), we can follow these steps:

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

### Step 2: Write down the forces
Given the forces:
- \( A = \hat{i} + \hat{j} + \hat{k} \)
- \( B = 2\hat{i} - \hat{j} + 3\hat{k} \)

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

### Step 4: Combine like terms
Now, we combine the like terms from \( A \) and \( B \):
\[ (\hat{i} + 2\hat{i}) + (\hat{j} - \hat{j}) + (\hat{k} + 3\hat{k}) + C = 0 \]
This simplifies to:
\[ 3\hat{i} + 0\hat{j} + 4\hat{k} + C = 0 \]

### Step 5: Isolate \( C \)
Now, we can isolate \( C \):
\[ C = - (3\hat{i} + 0\hat{j} + 4\hat{k}) \]
This simplifies to:
\[ C = -3\hat{i} - 4\hat{k} \]

### Step 6: Write the final answer
Thus, the force \( C \) that keeps the body in equilibrium is:
\[ C = -3\hat{i} + 0\hat{j} - 4\hat{k} \]